3.2.6 \(\int (d+e x)^3 (f+g x+h x^2+i x^3) (a+b \text {ArcSin}(c x)) \, dx\) [106]

Optimal. Leaf size=684 \[ \frac {b \left (1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+360 e^3 i+588 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (30 e^2 i+49 c^2 \left (e^2 g+3 d e h+3 d^2 i\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (e h+3 d i) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+720 e^3 i+1176 c^2 e \left (e^2 g+3 d e h+3 d^2 i\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+d^3 f x (a+b \text {ArcSin}(c x))+\frac {1}{2} d^2 (3 e f+d g) x^2 (a+b \text {ArcSin}(c x))+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 (a+b \text {ArcSin}(c x))+\frac {1}{4} \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4 (a+b \text {ArcSin}(c x))+\frac {1}{5} e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5 (a+b \text {ArcSin}(c x))+\frac {1}{6} e^2 (e h+3 d i) x^6 (a+b \text {ArcSin}(c x))+\frac {1}{7} e^3 i x^7 (a+b \text {ArcSin}(c x)) \]

[Out]

-1/96*b*(24*c^4*d^2*(d*g+3*e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f))*arcsin(c*x)/c^6+d^3
*f*x*(a+b*arcsin(c*x))+1/2*d^2*(d*g+3*e*f)*x^2*(a+b*arcsin(c*x))+1/3*d*(d^2*h+3*d*e*g+3*e^2*f)*x^3*(a+b*arcsin
(c*x))+1/4*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f)*x^4*(a+b*arcsin(c*x))+1/5*e*(3*d^2*i+3*d*e*h+e^2*g)*x^5*(a+b*arcs
in(c*x))+1/6*e^2*(3*d*i+e*h)*x^6*(a+b*arcsin(c*x))+1/7*e^3*i*x^7*(a+b*arcsin(c*x))+1/11025*b*(1225*c^4*d*(d^2*
h+3*d*e*g+3*e^2*f)+360*e^3*i+588*c^2*e*(3*d^2*i+3*d*e*h+e^2*g))*x^2*(-c^2*x^2+1)^(1/2)/c^5+1/144*b*(5*e^2*(3*d
*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f))*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/1225*b*e*(30*e^2*i+49*c^2*(3*d^2
*i+3*d*e*h+e^2*g))*x^4*(-c^2*x^2+1)^(1/2)/c^3+1/36*b*e^2*(3*d*i+e*h)*x^5*(-c^2*x^2+1)^(1/2)/c+1/49*b*e^3*i*x^6
*(-c^2*x^2+1)^(1/2)/c+1/352800*b*(352800*c^6*d^3*f+78400*c^4*d*(d^2*h+3*d*e*g+3*e^2*f)+23040*e^3*i+37632*c^2*e
*(3*d^2*i+3*d*e*h+e^2*g)+3675*c^2*(24*c^4*d^2*(d*g+3*e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e
^3*f))*x)*(-c^2*x^2+1)^(1/2)/c^7

________________________________________________________________________________________

Rubi [A]
time = 3.17, antiderivative size = 684, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4833, 12, 1823, 794, 222} \begin {gather*} d^3 f x (a+b \text {ArcSin}(c x))+\frac {1}{3} d x^3 (a+b \text {ArcSin}(c x)) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 (a+b \text {ArcSin}(c x)) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) (a+b \text {ArcSin}(c x))+\frac {1}{4} x^4 (a+b \text {ArcSin}(c x)) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) (a+b \text {ArcSin}(c x))+\frac {1}{7} e^3 i x^7 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{96 c^6}+\frac {b e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{1225 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{144 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )}{11025 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (3675 c^2 x \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )+32 \left (11025 c^6 d^3 f+2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+720 e^3 i\right )\right )}{352800 c^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

(b*(1225*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 360*e^3*i + 588*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i))*x^2*Sqrt[1 - c
^2*x^2])/(11025*c^5) + (b*(5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x^3*Sqrt[1 - c
^2*x^2])/(144*c^3) + (b*e*(30*e^2*i + 49*c^2*(e^2*g + 3*d*e*h + 3*d^2*i))*x^4*Sqrt[1 - c^2*x^2])/(1225*c^3) +
(b*e^2*(e*h + 3*d*i)*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*e^3*i*x^6*Sqrt[1 - c^2*x^2])/(49*c) + (b*(32*(11025*c^
6*d^3*f + 2450*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 720*e^3*i + 1176*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i)) + 3675*
c^2*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x)*Sqrt[1
- c^2*x^2])/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*
d^2*e*h + d^3*i))*ArcSin[c*x])/(96*c^6) + d^3*f*x*(a + b*ArcSin[c*x]) + (d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c
*x]))/2 + (d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + ((e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)
*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*(e*h + 3*d*i)*x
^6*(a + b*ArcSin[c*x]))/6 + (e^3*i*x^7*(a + b*ArcSin[c*x]))/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 4833

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHide[ExpandExpression[Px, x], x]}, Dis
t[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b,
c}, x] && PolynomialQ[Px, x]

Rubi steps

\begin {align*} \int (d+e x)^3 \left (f+g x+h x^2+106 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{420 \sqrt {1-c^2 x^2}} \, dx\\ &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{420} (b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-2940 c^2 d^3 f-1470 c^2 d^2 (3 e f+d g) x-980 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-735 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^3-12 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4-490 c^2 e^2 (318 d+e h) x^5\right )}{\sqrt {1-c^2 x^2}} \, dx}{2940 c}\\ &=\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (17640 c^4 d^3 f+8820 c^4 d^2 (3 e f+d g) x+5880 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2+490 c^2 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3+72 c^2 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{17640 c^3}\\ &=\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-88200 c^6 d^3 f-44100 c^6 d^2 (3 e f+d g) x-24 c^2 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2-2450 c^4 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{88200 c^5}\\ &=\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (352800 c^8 d^3 f+7350 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x+96 c^4 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{352800 c^7}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-96 c^4 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )-22050 c^6 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{1058400 c^9}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) \sin ^{-1}(c x)}{96 c^6}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 619, normalized size = 0.90 \begin {gather*} a d^3 f x+\frac {1}{2} a d^2 (3 e f+d g) x^2+\frac {1}{3} a d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3+\frac {1}{4} a \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right ) x^4+\frac {1}{5} a e \left (e^2 g+3 d e h+3 d^2 i\right ) x^5+\frac {1}{6} a e^2 (e h+3 d i) x^6+\frac {1}{7} a e^3 i x^7+\frac {b \sqrt {1-c^2 x^2} \left (23040 e^3 i+3 c^2 e \left (37632 d^2 i+147 d e (256 h+125 i x)+e^2 (12544 g+5 x (1225 h+768 i x))\right )+c^4 \left (1225 d^3 (64 h+27 i x)+147 d^2 e \left (1600 g+675 h x+384 i x^2\right )+147 d e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )+e^3 x \left (33075 f+2 x \left (9408 g+6125 h x+4320 i x^2\right )\right )\right )+2 c^6 \left (1225 d^3 (144 f+x (36 g+x (16 h+9 i x)))+147 d^2 e x (900 f+x (400 g+9 x (25 h+16 i x)))+147 d e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))+e^3 x^3 (11025 f+4 x (1764 g+25 x (49 h+36 i x)))\right )\right )}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (e h+3 d i)+9 c^2 \left (e^3 f+3 d e^2 g+3 d^2 e h+d^3 i\right )\right ) \text {ArcSin}(c x)}{96 c^6}+\frac {1}{420} b x \left (35 d^3 (12 f+x (6 g+x (4 h+3 i x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+4 i x)))+21 d e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right ) \text {ArcSin}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

a*d^3*f*x + (a*d^2*(3*e*f + d*g)*x^2)/2 + (a*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3)/3 + (a*(e^3*f + 3*d*e^2*g + 3*
d^2*e*h + d^3*i)*x^4)/4 + (a*e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5)/5 + (a*e^2*(e*h + 3*d*i)*x^6)/6 + (a*e^3*i*x^7
)/7 + (b*Sqrt[1 - c^2*x^2]*(23040*e^3*i + 3*c^2*e*(37632*d^2*i + 147*d*e*(256*h + 125*i*x) + e^2*(12544*g + 5*
x*(1225*h + 768*i*x))) + c^4*(1225*d^3*(64*h + 27*i*x) + 147*d^2*e*(1600*g + 675*h*x + 384*i*x^2) + 147*d*e^2*
(1600*f + x*(675*g + 384*h*x + 250*i*x^2)) + e^3*x*(33075*f + 2*x*(9408*g + 6125*h*x + 4320*i*x^2))) + 2*c^6*(
1225*d^3*(144*f + x*(36*g + x*(16*h + 9*i*x))) + 147*d^2*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))) + 147*d
*e^2*x^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))) + e^3*x^3*(11025*f + 4*x*(1764*g + 25*x*(49*h + 36*i*x)))))
)/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d
^3*i))*ArcSin[c*x])/(96*c^6) + (b*x*(35*d^3*(12*f + x*(6*g + x*(4*h + 3*i*x))) + 21*d^2*e*x*(30*f + x*(20*g +
3*x*(5*h + 4*i*x))) + 21*d*e^2*x^2*(20*f + x*(15*g + 2*x*(6*h + 5*i*x))) + e^3*x^3*(105*f + 2*x*(42*g + 5*x*(7
*h + 6*i*x))))*ArcSin[c*x])/420

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Maple [A]
time = 0.15, size = 932, normalized size = 1.36 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(a/c^6*(1/7*e^3*i*c^7*x^7+1/6*(3*c*d*e^2*i+c*e^3*h)*c^6*x^6+1/5*(3*c^2*d^2*e*i+3*c^2*d*e^2*h+c^2*e^3*g)*c^
5*x^5+1/4*(c^3*d^3*i+3*c^3*d^2*e*h+3*c^3*d*e^2*g+c^3*e^3*f)*c^4*x^4+1/3*(c^4*d^3*h+3*c^4*d^2*e*g+3*c^4*d*e^2*f
)*c^3*x^3+1/2*(c^5*d^3*g+3*c^5*d^2*e*f)*c^2*x^2+d^3*c^7*f*x)+b/c^6*(1/7*arcsin(c*x)*e^3*i*c^7*x^7+1/2*arcsin(c
*x)*c^7*d*e^2*i*x^6+1/6*arcsin(c*x)*c^7*e^3*h*x^6+3/5*arcsin(c*x)*c^7*d^2*e*i*x^5+3/5*arcsin(c*x)*c^7*d*e^2*h*
x^5+1/5*arcsin(c*x)*c^7*e^3*g*x^5+1/4*arcsin(c*x)*c^7*d^3*i*x^4+3/4*arcsin(c*x)*c^7*d^2*e*h*x^4+3/4*arcsin(c*x
)*c^7*d*e^2*g*x^4+1/4*arcsin(c*x)*c^7*e^3*f*x^4+1/3*arcsin(c*x)*c^7*d^3*h*x^3+arcsin(c*x)*c^7*d^2*e*g*x^3+arcs
in(c*x)*c^7*d*e^2*f*x^3+1/2*arcsin(c*x)*c^7*d^3*g*x^2+3/2*arcsin(c*x)*c^7*d^2*e*f*x^2+arcsin(c*x)*d^3*c^7*f*x-
1/7*e^3*i*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/
35*(-c^2*x^2+1)^(1/2))-1/420*(210*c*d*e^2*i+70*c*e^3*h)*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^
2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))-1/420*(252*c^2*d^2*e*i+252*c^2*d*e^2*h+84*c^2*e^3*g)*
(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/420*(105*c^3*d^3*i
+315*c^3*d^2*e*h+315*c^3*d*e^2*g+105*c^3*e^3*f)*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/
8*arcsin(c*x))-1/420*(140*c^4*d^3*h+420*c^4*d^2*e*g+420*c^4*d*e^2*f)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^
2*x^2+1)^(1/2))-1/420*(210*c^5*d^3*g+630*c^5*d^2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+d^3*c^6*f*
(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.49, size = 1215, normalized size = 1.78 \begin {gather*} \frac {1}{6} \, a h x^{6} e^{3} + \frac {1}{7} i \, a x^{7} e^{3} + \frac {3}{5} \, a d h x^{5} e^{2} + \frac {1}{2} i \, a d x^{6} e^{2} + \frac {3}{4} \, a d^{2} h x^{4} e + \frac {3}{5} i \, a d^{2} x^{5} e + \frac {1}{3} \, a d^{3} h x^{3} + \frac {1}{4} i \, a d^{3} x^{4} + \frac {1}{5} \, a g x^{5} e^{3} + \frac {3}{4} \, a d g x^{4} e^{2} + a d^{2} g x^{3} e + \frac {1}{2} \, a d^{3} g x^{2} + \frac {1}{4} \, a f x^{4} e^{3} + a d f x^{3} e^{2} + \frac {3}{2} \, a d^{2} f x^{2} e + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} h + a d^{3} f x + \frac {3}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} f e + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} g e + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} h e + \frac {1}{32} i \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d f e^{2} + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d g e^{2} + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d h e^{2} + \frac {1}{25} i \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b f e^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b g e^{3} + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b h e^{3} + \frac {1}{96} i \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{245} i \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/6*a*h*x^6*e^3 + 1/7*I*a*x^7*e^3 + 3/5*a*d*h*x^5*e^2 + 1/2*I*a*d*x^6*e^2 + 3/4*a*d^2*h*x^4*e + 3/5*I*a*d^2*x^
5*e + 1/3*a*d^3*h*x^3 + 1/4*I*a*d^3*x^4 + 1/5*a*g*x^5*e^3 + 3/4*a*d*g*x^4*e^2 + a*d^2*g*x^3*e + 1/2*a*d^3*g*x^
2 + 1/4*a*f*x^4*e^3 + a*d*f*x^3*e^2 + 3/2*a*d^2*f*x^2*e + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2
- arcsin(c*x)/c^3))*b*d^3*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c
^4))*b*d^3*h + a*d^3*f*x + 3/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d^2*f*e
+ 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^2*g*e + 3/32*(8*x^4*
arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d^2*h*e + 1
/32*I*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*
b*d^3 + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3*f/c + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/
c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d*f*e^2 + 3/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(
-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d*g*e^2 + 1/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c
^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d*h*e^2 + 1/25*I*(15*x^5*arcsin(c*x) + (3*s
qrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^2*e + 1/32*(8*x^4*
arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*f*e^3 + 1/7
5*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^
6)*c)*b*g*e^3 + 1/288*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15
*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*h*e^3 + 1/96*I*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1
)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*d*e^2 + 1/2
45*I*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)
*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*e^3

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Fricas [A]
time = 2.54, size = 885, normalized size = 1.29 \begin {gather*} \frac {235200 \, a c^{7} d^{3} h x^{3} + 176400 i \, a c^{7} d^{3} x^{4} + 352800 \, a c^{7} d^{3} g x^{2} + 705600 \, a c^{7} d^{3} f x + 1680 \, {\left (70 \, a c^{7} h x^{6} + 60 i \, a c^{7} x^{7} + 84 \, a c^{7} g x^{5} + 105 \, a c^{7} f x^{4}\right )} e^{3} + 35280 \, {\left (12 \, a c^{7} d h x^{5} + 10 i \, a c^{7} d x^{6} + 15 \, a c^{7} d g x^{4} + 20 \, a c^{7} d f x^{3}\right )} e^{2} + 35280 \, {\left (15 \, a c^{7} d^{2} h x^{4} + 12 i \, a c^{7} d^{2} x^{5} + 20 \, a c^{7} d^{2} g x^{3} + 30 \, a c^{7} d^{2} f x^{2}\right )} e - 105 \, {\left (1120 i \, b c^{7} d^{3} h x^{3} - 840 \, b c^{7} d^{3} x^{4} + 1680 i \, b c^{7} d^{3} g x^{2} + 3360 i \, b c^{7} d^{3} f x - 840 i \, b c^{5} d^{3} g + 315 \, b c^{3} d^{3} + {\left (560 i \, b c^{7} h x^{6} - 480 \, b c^{7} x^{7} + 672 i \, b c^{7} g x^{5} + 840 i \, b c^{7} f x^{4} - 315 i \, b c^{3} f - 175 i \, b c h\right )} e^{3} + 21 \, {\left (96 i \, b c^{7} d h x^{5} - 80 \, b c^{7} d x^{6} + 120 i \, b c^{7} d g x^{4} + 160 i \, b c^{7} d f x^{3} - 45 i \, b c^{3} d g + 25 \, b c d\right )} e^{2} + 21 \, {\left (120 i \, b c^{7} d^{2} h x^{4} - 96 \, b c^{7} d^{2} x^{5} + 160 i \, b c^{7} d^{2} g x^{3} + 240 i \, b c^{7} d^{2} f x^{2} - 120 i \, b c^{5} d^{2} f - 45 i \, b c^{3} d^{2} h\right )} e\right )} \log \left (-2 \, c^{2} x^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} c x + 1\right ) - 2 \, {\left (-39200 i \, b c^{6} d^{3} h x^{2} + 22050 \, b c^{6} d^{3} x^{3} - 352800 i \, b c^{6} d^{3} f - 78400 i \, b c^{4} d^{3} h + 11025 \, {\left (-8 i \, b c^{6} d^{3} g + 3 \, b c^{4} d^{3}\right )} x + {\left (-9800 i \, b c^{6} h x^{5} + 7200 \, b c^{6} x^{6} + 288 \, {\left (-49 i \, b c^{6} g + 30 \, b c^{4}\right )} x^{4} - 37632 i \, b c^{2} g + 2450 \, {\left (-9 i \, b c^{6} f - 5 i \, b c^{4} h\right )} x^{3} + 384 \, {\left (-49 i \, b c^{4} g + 30 \, b c^{2}\right )} x^{2} + 3675 \, {\left (-9 i \, b c^{4} f - 5 i \, b c^{2} h\right )} x + 23040 \, b\right )} e^{3} + 147 \, {\left (-288 i \, b c^{6} d h x^{4} + 200 \, b c^{6} d x^{5} - 1600 i \, b c^{4} d f - 768 i \, b c^{2} d h + 50 \, {\left (-9 i \, b c^{6} d g + 5 \, b c^{4} d\right )} x^{3} + 32 \, {\left (-25 i \, b c^{6} d f - 12 i \, b c^{4} d h\right )} x^{2} + 75 \, {\left (-9 i \, b c^{4} d g + 5 \, b c^{2} d\right )} x\right )} e^{2} + 147 \, {\left (-450 i \, b c^{6} d^{2} h x^{3} + 288 \, b c^{6} d^{2} x^{4} - 1600 i \, b c^{4} d^{2} g + 768 \, b c^{2} d^{2} + 32 \, {\left (-25 i \, b c^{6} d^{2} g + 12 \, b c^{4} d^{2}\right )} x^{2} + 225 \, {\left (-8 i \, b c^{6} d^{2} f - 3 i \, b c^{4} d^{2} h\right )} x\right )} e\right )} \sqrt {c^{2} x^{2} - 1}}{705600 \, c^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/705600*(235200*a*c^7*d^3*h*x^3 + 176400*I*a*c^7*d^3*x^4 + 352800*a*c^7*d^3*g*x^2 + 705600*a*c^7*d^3*f*x + 16
80*(70*a*c^7*h*x^6 + 60*I*a*c^7*x^7 + 84*a*c^7*g*x^5 + 105*a*c^7*f*x^4)*e^3 + 35280*(12*a*c^7*d*h*x^5 + 10*I*a
*c^7*d*x^6 + 15*a*c^7*d*g*x^4 + 20*a*c^7*d*f*x^3)*e^2 + 35280*(15*a*c^7*d^2*h*x^4 + 12*I*a*c^7*d^2*x^5 + 20*a*
c^7*d^2*g*x^3 + 30*a*c^7*d^2*f*x^2)*e - 105*(1120*I*b*c^7*d^3*h*x^3 - 840*b*c^7*d^3*x^4 + 1680*I*b*c^7*d^3*g*x
^2 + 3360*I*b*c^7*d^3*f*x - 840*I*b*c^5*d^3*g + 315*b*c^3*d^3 + (560*I*b*c^7*h*x^6 - 480*b*c^7*x^7 + 672*I*b*c
^7*g*x^5 + 840*I*b*c^7*f*x^4 - 315*I*b*c^3*f - 175*I*b*c*h)*e^3 + 21*(96*I*b*c^7*d*h*x^5 - 80*b*c^7*d*x^6 + 12
0*I*b*c^7*d*g*x^4 + 160*I*b*c^7*d*f*x^3 - 45*I*b*c^3*d*g + 25*b*c*d)*e^2 + 21*(120*I*b*c^7*d^2*h*x^4 - 96*b*c^
7*d^2*x^5 + 160*I*b*c^7*d^2*g*x^3 + 240*I*b*c^7*d^2*f*x^2 - 120*I*b*c^5*d^2*f - 45*I*b*c^3*d^2*h)*e)*log(-2*c^
2*x^2 - 2*sqrt(c^2*x^2 - 1)*c*x + 1) - 2*(-39200*I*b*c^6*d^3*h*x^2 + 22050*b*c^6*d^3*x^3 - 352800*I*b*c^6*d^3*
f - 78400*I*b*c^4*d^3*h + 11025*(-8*I*b*c^6*d^3*g + 3*b*c^4*d^3)*x + (-9800*I*b*c^6*h*x^5 + 7200*b*c^6*x^6 + 2
88*(-49*I*b*c^6*g + 30*b*c^4)*x^4 - 37632*I*b*c^2*g + 2450*(-9*I*b*c^6*f - 5*I*b*c^4*h)*x^3 + 384*(-49*I*b*c^4
*g + 30*b*c^2)*x^2 + 3675*(-9*I*b*c^4*f - 5*I*b*c^2*h)*x + 23040*b)*e^3 + 147*(-288*I*b*c^6*d*h*x^4 + 200*b*c^
6*d*x^5 - 1600*I*b*c^4*d*f - 768*I*b*c^2*d*h + 50*(-9*I*b*c^6*d*g + 5*b*c^4*d)*x^3 + 32*(-25*I*b*c^6*d*f - 12*
I*b*c^4*d*h)*x^2 + 75*(-9*I*b*c^4*d*g + 5*b*c^2*d)*x)*e^2 + 147*(-450*I*b*c^6*d^2*h*x^3 + 288*b*c^6*d^2*x^4 -
1600*I*b*c^4*d^2*g + 768*b*c^2*d^2 + 32*(-25*I*b*c^6*d^2*g + 12*b*c^4*d^2)*x^2 + 225*(-8*I*b*c^6*d^2*f - 3*I*b
*c^4*d^2*h)*x)*e)*sqrt(c^2*x^2 - 1))/c^7

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (688) = 1376\).
time = 1.19, size = 1809, normalized size = 2.64 \begin {gather*} \begin {cases} a d^{3} f x + \frac {a d^{3} g x^{2}}{2} + \frac {a d^{3} h x^{3}}{3} + \frac {a d^{3} i x^{4}}{4} + \frac {3 a d^{2} e f x^{2}}{2} + a d^{2} e g x^{3} + \frac {3 a d^{2} e h x^{4}}{4} + \frac {3 a d^{2} e i x^{5}}{5} + a d e^{2} f x^{3} + \frac {3 a d e^{2} g x^{4}}{4} + \frac {3 a d e^{2} h x^{5}}{5} + \frac {a d e^{2} i x^{6}}{2} + \frac {a e^{3} f x^{4}}{4} + \frac {a e^{3} g x^{5}}{5} + \frac {a e^{3} h x^{6}}{6} + \frac {a e^{3} i x^{7}}{7} + b d^{3} f x \operatorname {asin}{\left (c x \right )} + \frac {b d^{3} g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b d^{3} h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d^{3} i x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d^{2} e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + b d^{2} e g x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d^{2} e h x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d^{2} e i x^{5} \operatorname {asin}{\left (c x \right )}}{5} + b d e^{2} f x^{3} \operatorname {asin}{\left (c x \right )} + \frac {3 b d e^{2} g x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b d e^{2} h x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b d e^{2} i x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{3} f x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e^{3} g x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {b e^{3} h x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b e^{3} i x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d^{3} f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d^{3} g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{3} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {b d^{3} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b d^{2} e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d^{2} e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e i x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b d e^{2} f x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {3 b d e^{2} g x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {3 b d e^{2} h x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b d e^{2} i x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {b e^{3} f x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e^{3} g x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {b e^{3} h x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} + \frac {b e^{3} i x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {b d^{3} g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {3 b d^{2} e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d^{3} h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b d^{3} i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {2 b d^{2} e g \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {9 b d^{2} e h x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b d^{2} e i x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {2 b d e^{2} f \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {9 b d e^{2} g x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b d e^{2} h x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} + \frac {5 b d e^{2} i x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {3 b e^{3} f x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {4 b e^{3} g x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {5 b e^{3} h x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} + \frac {6 b e^{3} i x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {3 b d^{3} i \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {9 b d^{2} e h \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {9 b d e^{2} g \operatorname {asin}{\left (c x \right )}}{32 c^{4}} - \frac {3 b e^{3} f \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {8 b d^{2} e i \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {8 b d e^{2} h \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} + \frac {5 b d e^{2} i x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {8 b e^{3} g \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {5 b e^{3} h x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} + \frac {8 b e^{3} i x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {5 b d e^{2} i \operatorname {asin}{\left (c x \right )}}{32 c^{6}} - \frac {5 b e^{3} h \operatorname {asin}{\left (c x \right )}}{96 c^{6}} + \frac {16 b e^{3} i \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\a \left (d^{3} f x + \frac {d^{3} g x^{2}}{2} + \frac {d^{3} h x^{3}}{3} + \frac {d^{3} i x^{4}}{4} + \frac {3 d^{2} e f x^{2}}{2} + d^{2} e g x^{3} + \frac {3 d^{2} e h x^{4}}{4} + \frac {3 d^{2} e i x^{5}}{5} + d e^{2} f x^{3} + \frac {3 d e^{2} g x^{4}}{4} + \frac {3 d e^{2} h x^{5}}{5} + \frac {d e^{2} i x^{6}}{2} + \frac {e^{3} f x^{4}}{4} + \frac {e^{3} g x^{5}}{5} + \frac {e^{3} h x^{6}}{6} + \frac {e^{3} i x^{7}}{7}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**3*f*x + a*d**3*g*x**2/2 + a*d**3*h*x**3/3 + a*d**3*i*x**4/4 + 3*a*d**2*e*f*x**2/2 + a*d**2*e*g
*x**3 + 3*a*d**2*e*h*x**4/4 + 3*a*d**2*e*i*x**5/5 + a*d*e**2*f*x**3 + 3*a*d*e**2*g*x**4/4 + 3*a*d*e**2*h*x**5/
5 + a*d*e**2*i*x**6/2 + a*e**3*f*x**4/4 + a*e**3*g*x**5/5 + a*e**3*h*x**6/6 + a*e**3*i*x**7/7 + b*d**3*f*x*asi
n(c*x) + b*d**3*g*x**2*asin(c*x)/2 + b*d**3*h*x**3*asin(c*x)/3 + b*d**3*i*x**4*asin(c*x)/4 + 3*b*d**2*e*f*x**2
*asin(c*x)/2 + b*d**2*e*g*x**3*asin(c*x) + 3*b*d**2*e*h*x**4*asin(c*x)/4 + 3*b*d**2*e*i*x**5*asin(c*x)/5 + b*d
*e**2*f*x**3*asin(c*x) + 3*b*d*e**2*g*x**4*asin(c*x)/4 + 3*b*d*e**2*h*x**5*asin(c*x)/5 + b*d*e**2*i*x**6*asin(
c*x)/2 + b*e**3*f*x**4*asin(c*x)/4 + b*e**3*g*x**5*asin(c*x)/5 + b*e**3*h*x**6*asin(c*x)/6 + b*e**3*i*x**7*asi
n(c*x)/7 + b*d**3*f*sqrt(-c**2*x**2 + 1)/c + b*d**3*g*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d**3*h*x**2*sqrt(-c**2*
x**2 + 1)/(9*c) + b*d**3*i*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*f*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d*
*2*e*g*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d**2*e*h*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*i*x**4*sqr
t(-c**2*x**2 + 1)/(25*c) + b*d*e**2*f*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d*e**2*g*x**3*sqrt(-c**2*x**2 + 1)
/(16*c) + 3*b*d*e**2*h*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + b*d*e**2*i*x**5*sqrt(-c**2*x**2 + 1)/(12*c) + b*e**3
*f*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*e**3*g*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + b*e**3*h*x**5*sqrt(-c**2*x**
2 + 1)/(36*c) + b*e**3*i*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - b*d**3*g*asin(c*x)/(4*c**2) - 3*b*d**2*e*f*asin(c*
x)/(4*c**2) + 2*b*d**3*h*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*b*d**3*i*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 2*b*d**
2*e*g*sqrt(-c**2*x**2 + 1)/(3*c**3) + 9*b*d**2*e*h*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4*b*d**2*e*i*x**2*sqrt(-
c**2*x**2 + 1)/(25*c**3) + 2*b*d*e**2*f*sqrt(-c**2*x**2 + 1)/(3*c**3) + 9*b*d*e**2*g*x*sqrt(-c**2*x**2 + 1)/(3
2*c**3) + 4*b*d*e**2*h*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) + 5*b*d*e**2*i*x**3*sqrt(-c**2*x**2 + 1)/(48*c**3)
+ 3*b*e**3*f*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4*b*e**3*g*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 5*b*e**3*h*x*
*3*sqrt(-c**2*x**2 + 1)/(144*c**3) + 6*b*e**3*i*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) - 3*b*d**3*i*asin(c*x)/(3
2*c**4) - 9*b*d**2*e*h*asin(c*x)/(32*c**4) - 9*b*d*e**2*g*asin(c*x)/(32*c**4) - 3*b*e**3*f*asin(c*x)/(32*c**4)
+ 8*b*d**2*e*i*sqrt(-c**2*x**2 + 1)/(25*c**5) + 8*b*d*e**2*h*sqrt(-c**2*x**2 + 1)/(25*c**5) + 5*b*d*e**2*i*x*
sqrt(-c**2*x**2 + 1)/(32*c**5) + 8*b*e**3*g*sqrt(-c**2*x**2 + 1)/(75*c**5) + 5*b*e**3*h*x*sqrt(-c**2*x**2 + 1)
/(96*c**5) + 8*b*e**3*i*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 5*b*d*e**2*i*asin(c*x)/(32*c**6) - 5*b*e**3*h*a
sin(c*x)/(96*c**6) + 16*b*e**3*i*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d**3*f*x + d**3*g*x**2/2 + d*
*3*h*x**3/3 + d**3*i*x**4/4 + 3*d**2*e*f*x**2/2 + d**2*e*g*x**3 + 3*d**2*e*h*x**4/4 + 3*d**2*e*i*x**5/5 + d*e*
*2*f*x**3 + 3*d*e**2*g*x**4/4 + 3*d*e**2*h*x**5/5 + d*e**2*i*x**6/2 + e**3*f*x**4/4 + e**3*g*x**5/5 + e**3*h*x
**6/6 + e**3*i*x**7/7), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2010 vs. \(2 (643) = 1286\).
time = 0.44, size = 2010, normalized size = 2.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

1/7*a*e^3*i*x^7 + 1/6*a*e^3*h*x^6 + 1/2*a*d*e^2*i*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 3/5*a*d^2*e*i*x^
5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + 3/4*a*d^2*e*h*x^4 + 1/4*a*d^3*i*x^4 + a*d*e^2*f*x^3 + a*d^2*e*g*x^3
+ 1/3*a*d^3*h*x^3 + b*d^3*f*x*arcsin(c*x) + a*d^3*f*x + (c^2*x^2 - 1)*b*d*e^2*f*x*arcsin(c*x)/c^2 + (c^2*x^2 -
1)*b*d^2*e*g*x*arcsin(c*x)/c^2 + 1/3*(c^2*x^2 - 1)*b*d^3*h*x*arcsin(c*x)/c^2 + 3/4*sqrt(-c^2*x^2 + 1)*b*d^2*e
*f*x/c + 1/4*sqrt(-c^2*x^2 + 1)*b*d^3*g*x/c + 3/2*(c^2*x^2 - 1)*b*d^2*e*f*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1)*
b*d^3*g*arcsin(c*x)/c^2 + b*d*e^2*f*x*arcsin(c*x)/c^2 + b*d^2*e*g*x*arcsin(c*x)/c^2 + 1/5*(c^2*x^2 - 1)^2*b*e^
3*g*x*arcsin(c*x)/c^4 + 1/3*b*d^3*h*x*arcsin(c*x)/c^2 + 3/5*(c^2*x^2 - 1)^2*b*d*e^2*h*x*arcsin(c*x)/c^4 + 3/5*
(c^2*x^2 - 1)^2*b*d^2*e*i*x*arcsin(c*x)/c^4 + sqrt(-c^2*x^2 + 1)*b*d^3*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*e^3*f
*x/c^3 - 3/16*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*g*x/c^3 - 3/16*(-c^2*x^2 + 1)^(3/2)*b*d^2*e*h*x/c^3 - 1/16*(-c^2*x^
2 + 1)^(3/2)*b*d^3*i*x/c^3 + 3/2*(c^2*x^2 - 1)*a*d^2*e*f/c^2 + 1/2*(c^2*x^2 - 1)*a*d^3*g/c^2 + 3/4*b*d^2*e*f*a
rcsin(c*x)/c^2 + 1/4*(c^2*x^2 - 1)^2*b*e^3*f*arcsin(c*x)/c^4 + 1/4*b*d^3*g*arcsin(c*x)/c^2 + 3/4*(c^2*x^2 - 1)
^2*b*d*e^2*g*arcsin(c*x)/c^4 + 3/4*(c^2*x^2 - 1)^2*b*d^2*e*h*arcsin(c*x)/c^4 + 1/4*(c^2*x^2 - 1)^2*b*d^3*i*arc
sin(c*x)/c^4 + 2/5*(c^2*x^2 - 1)*b*e^3*g*x*arcsin(c*x)/c^4 + 6/5*(c^2*x^2 - 1)*b*d*e^2*h*x*arcsin(c*x)/c^4 + 6
/5*(c^2*x^2 - 1)*b*d^2*e*i*x*arcsin(c*x)/c^4 + 1/7*(c^2*x^2 - 1)^3*b*e^3*i*x*arcsin(c*x)/c^6 - 1/3*(-c^2*x^2 +
1)^(3/2)*b*d*e^2*f/c^3 - 1/3*(-c^2*x^2 + 1)^(3/2)*b*d^2*e*g/c^3 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^3*h/c^3 + 5/32
*sqrt(-c^2*x^2 + 1)*b*e^3*f*x/c^3 + 15/32*sqrt(-c^2*x^2 + 1)*b*d*e^2*g*x/c^3 + 15/32*sqrt(-c^2*x^2 + 1)*b*d^2*
e*h*x/c^3 + 1/36*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3*h*x/c^5 + 5/32*sqrt(-c^2*x^2 + 1)*b*d^3*i*x/c^3 + 1/
12*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2*i*x/c^5 + 1/2*(c^2*x^2 - 1)*b*e^3*f*arcsin(c*x)/c^4 + 3/2*(c^2*x
^2 - 1)*b*d*e^2*g*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)*b*d^2*e*h*arcsin(c*x)/c^4 + 1/6*(c^2*x^2 - 1)^3*b*e^3*h*
arcsin(c*x)/c^6 + 1/2*(c^2*x^2 - 1)*b*d^3*i*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^3*b*d*e^2*i*arcsin(c*x)/c^6 +
1/5*b*e^3*g*x*arcsin(c*x)/c^4 + 3/5*b*d*e^2*h*x*arcsin(c*x)/c^4 + 3/5*b*d^2*e*i*x*arcsin(c*x)/c^4 + 3/7*(c^2*x
^2 - 1)^2*b*e^3*i*x*arcsin(c*x)/c^6 + sqrt(-c^2*x^2 + 1)*b*d*e^2*f/c^3 + sqrt(-c^2*x^2 + 1)*b*d^2*e*g/c^3 + 1/
25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3*g/c^5 + 1/3*sqrt(-c^2*x^2 + 1)*b*d^3*h/c^3 + 3/25*(c^2*x^2 - 1)^2*
sqrt(-c^2*x^2 + 1)*b*d*e^2*h/c^5 + 3/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2*e*i/c^5 - 13/144*(-c^2*x^2 +
1)^(3/2)*b*e^3*h*x/c^5 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*i*x/c^5 + 5/32*b*e^3*f*arcsin(c*x)/c^4 + 15/32*b*d
*e^2*g*arcsin(c*x)/c^4 + 15/32*b*d^2*e*h*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^2*b*e^3*h*arcsin(c*x)/c^6 + 5/32*
b*d^3*i*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)^2*b*d*e^2*i*arcsin(c*x)/c^6 + 3/7*(c^2*x^2 - 1)*b*e^3*i*x*arcsin(c
*x)/c^6 - 2/15*(-c^2*x^2 + 1)^(3/2)*b*e^3*g/c^5 - 2/5*(-c^2*x^2 + 1)^(3/2)*b*d*e^2*h/c^5 - 2/5*(-c^2*x^2 + 1)^
(3/2)*b*d^2*e*i/c^5 + 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3*i/c^7 + 11/96*sqrt(-c^2*x^2 + 1)*b*e^3*h*x
/c^5 + 11/32*sqrt(-c^2*x^2 + 1)*b*d*e^2*i*x/c^5 + 1/2*(c^2*x^2 - 1)*b*e^3*h*arcsin(c*x)/c^6 + 3/2*(c^2*x^2 - 1
)*b*d*e^2*i*arcsin(c*x)/c^6 + 1/7*b*e^3*i*x*arcsin(c*x)/c^6 + 1/5*sqrt(-c^2*x^2 + 1)*b*e^3*g/c^5 + 3/5*sqrt(-c
^2*x^2 + 1)*b*d*e^2*h/c^5 + 3/5*sqrt(-c^2*x^2 + 1)*b*d^2*e*i/c^5 + 3/35*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e
^3*i/c^7 + 11/96*b*e^3*h*arcsin(c*x)/c^6 + 11/32*b*d*e^2*i*arcsin(c*x)/c^6 - 1/7*(-c^2*x^2 + 1)^(3/2)*b*e^3*i/
c^7 + 1/7*sqrt(-c^2*x^2 + 1)*b*e^3*i/c^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3),x)

[Out]

int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3), x)

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