3.606 \(\int x^3 (d+e x^2)^2 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=241 \[ \frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^3 \sqrt{1-c^2 x^2} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5}+\frac{b x \sqrt{1-c^2 x^2} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7}-\frac{b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \sin ^{-1}(c x)}{3072 c^8}+\frac{b e x^5 \sqrt{1-c^2 x^2} \left (64 c^2 d+21 e\right )}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c} \]

[Out]

(b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(288*c^4*d^2 + 320*c^2*d*e + 105
*e^2)*x^3*Sqrt[1 - c^2*x^2])/(4608*c^5) + (b*e*(64*c^2*d + 21*e)*x^5*Sqrt[1 - c^2*x^2])/(1152*c^3) + (b*e^2*x^
7*Sqrt[1 - c^2*x^2])/(64*c) - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*ArcSin[c*x])/(3072*c^8) + (d^2*x^4*(a +
b*ArcSin[c*x]))/4 + (d*e*x^6*(a + b*ArcSin[c*x]))/3 + (e^2*x^8*(a + b*ArcSin[c*x]))/8

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Rubi [A]  time = 0.250571, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {266, 43, 4731, 12, 1267, 459, 321, 216} \[ \frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^3 \sqrt{1-c^2 x^2} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5}+\frac{b x \sqrt{1-c^2 x^2} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7}-\frac{b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \sin ^{-1}(c x)}{3072 c^8}+\frac{b e x^5 \sqrt{1-c^2 x^2} \left (64 c^2 d+21 e\right )}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(288*c^4*d^2 + 320*c^2*d*e + 105
*e^2)*x^3*Sqrt[1 - c^2*x^2])/(4608*c^5) + (b*e*(64*c^2*d + 21*e)*x^5*Sqrt[1 - c^2*x^2])/(1152*c^3) + (b*e^2*x^
7*Sqrt[1 - c^2*x^2])/(64*c) - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*ArcSin[c*x])/(3072*c^8) + (d^2*x^4*(a +
b*ArcSin[c*x]))/4 + (d*e*x^6*(a + b*ArcSin[c*x]))/3 + (e^2*x^8*(a + b*ArcSin[c*x]))/8

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[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, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

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

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

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]

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{24} (b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x^4 \left (-48 c^2 d^2-e \left (64 c^2 d+21 e\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{192 c}\\ &=\frac{b e \left (64 c^2 d+21 e\right ) x^5 \sqrt{1-c^2 x^2}}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right )\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{1152 c^3}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \sqrt{1-c^2 x^2}}{4608 c^5}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \sqrt{1-c^2 x^2}}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right )\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{1536 c^5}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \sqrt{1-c^2 x^2}}{3072 c^7}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \sqrt{1-c^2 x^2}}{4608 c^5}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \sqrt{1-c^2 x^2}}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3072 c^7}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \sqrt{1-c^2 x^2}}{3072 c^7}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \sqrt{1-c^2 x^2}}{4608 c^5}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \sqrt{1-c^2 x^2}}{1152 c^3}+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}-\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \sin ^{-1}(c x)}{3072 c^8}+\frac{1}{4} d^2 x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.162572, size = 190, normalized size = 0.79 \[ \frac{384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+b c x \sqrt{1-c^2 x^2} \left (16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+30 c^2 e \left (32 d+7 e x^2\right )+315 e^2\right )+3 b \sin ^{-1}(c x) \left (128 c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-288 c^4 d^2-320 c^2 d e-105 e^2\right )}{9216 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b*c*x*Sqrt[1 - c^2*x^2]*(315*e^2 + 30*c^2*e*(32*d + 7*e*x^2)
+ 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 3*b*(-288*c^4*d^
2 - 320*c^2*d*e - 105*e^2 + 128*c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcSin[c*x])/(9216*c^8)

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Maple [A]  time = 0.007, size = 303, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}ed{x}^{6}}{3}}+{\frac{{x}^{4}{c}^{8}{d}^{2}}{4}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{\arcsin \left ( cx \right ){e}^{2}{c}^{8}{x}^{8}}{8}}+{\frac{\arcsin \left ( cx \right ){c}^{8}ed{x}^{6}}{3}}+{\frac{\arcsin \left ( cx \right ){d}^{2}{c}^{8}{x}^{4}}{4}}-{\frac{{e}^{2}}{8} \left ( -{\frac{{c}^{7}{x}^{7}}{8}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{7\,{c}^{5}{x}^{5}}{48}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{35\,{c}^{3}{x}^{3}}{192}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{35\,cx}{128}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{35\,\arcsin \left ( cx \right ) }{128}} \right ) }-{\frac{{c}^{2}ed}{3} \left ( -{\frac{{c}^{5}{x}^{5}}{6}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,{c}^{3}{x}^{3}}{24}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,cx}{16}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,\arcsin \left ( cx \right ) }{16}} \right ) }-{\frac{{d}^{2}{c}^{4}}{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^2*(a+b*arcsin(c*x)),x)

[Out]

1/c^4*(a/c^4*(1/8*e^2*c^8*x^8+1/3*c^8*e*d*x^6+1/4*x^4*c^8*d^2)+b/c^4*(1/8*arcsin(c*x)*e^2*c^8*x^8+1/3*arcsin(c
*x)*c^8*e*d*x^6+1/4*arcsin(c*x)*d^2*c^8*x^4-1/8*e^2*(-1/8*c^7*x^7*(-c^2*x^2+1)^(1/2)-7/48*c^5*x^5*(-c^2*x^2+1)
^(1/2)-35/192*c^3*x^3*(-c^2*x^2+1)^(1/2)-35/128*c*x*(-c^2*x^2+1)^(1/2)+35/128*arcsin(c*x))-1/3*c^2*e*d*(-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))-1/4*d^
2*c^4*(-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))))

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Maxima [A]  time = 1.47477, size = 432, normalized size = 1.79 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{2} + \frac{1}{144} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b d e + \frac{1}{3072} \,{\left (384 \, x^{8} \arcsin \left (c x\right ) +{\left (\frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{8}} - \frac{105 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sq
rt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^2 + 1/144*(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^2*x/sqr
t(c^2))/(sqrt(c^2)*c^6))*c)*b*d*e + 1/3072*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^
2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c^2*x/sqrt(c^2)
)/(sqrt(c^2)*c^8))*c)*b*e^2

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Fricas [A]  time = 2.14481, size = 517, normalized size = 2.15 \begin{align*} \frac{1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \,{\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \arcsin \left (c x\right ) +{\left (144 \, b c^{7} e^{2} x^{7} + 8 \,{\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \,{\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \,{\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{9216 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*(384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x
^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 - 320*b*c^2*d*e - 105*b*e^2)*arcsin(c*x) + (144*b*c^7*e^2*x^7 + 8*(64*b
*c^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5*d*e + 105*b*c^3*e^2)*x^3 + 3*(288*b*c^5*d^2 + 320*
b*c^3*d*e + 105*b*c*e^2)*x)*sqrt(-c^2*x^2 + 1))/c^8

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Sympy [A]  time = 15.6783, size = 382, normalized size = 1.59 \begin{align*} \begin{cases} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \frac{b d^{2} x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d e x^{6} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e^{2} x^{8} \operatorname{asin}{\left (c x \right )}}{8} + \frac{b d^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} + \frac{b d e x^{5} \sqrt{- c^{2} x^{2} + 1}}{18 c} + \frac{b e^{2} x^{7} \sqrt{- c^{2} x^{2} + 1}}{64 c} + \frac{3 b d^{2} x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} + \frac{5 b d e x^{3} \sqrt{- c^{2} x^{2} + 1}}{72 c^{3}} + \frac{7 b e^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{384 c^{3}} - \frac{3 b d^{2} \operatorname{asin}{\left (c x \right )}}{32 c^{4}} + \frac{5 b d e x \sqrt{- c^{2} x^{2} + 1}}{48 c^{5}} + \frac{35 b e^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac{5 b d e \operatorname{asin}{\left (c x \right )}}{48 c^{6}} + \frac{35 b e^{2} x \sqrt{- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac{35 b e^{2} \operatorname{asin}{\left (c x \right )}}{1024 c^{8}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{2} x^{4}}{4} + \frac{d e x^{6}}{3} + \frac{e^{2} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*asin(c*x)/4 + b*d*e*x**6*asin(c*x)/3 + b
*e**2*x**8*asin(c*x)/8 + b*d**2*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*d*e*x**5*sqrt(-c**2*x**2 + 1)/(18*c) + b*
e**2*x**7*sqrt(-c**2*x**2 + 1)/(64*c) + 3*b*d**2*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*d*e*x**3*sqrt(-c**2*x*
*2 + 1)/(72*c**3) + 7*b*e**2*x**5*sqrt(-c**2*x**2 + 1)/(384*c**3) - 3*b*d**2*asin(c*x)/(32*c**4) + 5*b*d*e*x*s
qrt(-c**2*x**2 + 1)/(48*c**5) + 35*b*e**2*x**3*sqrt(-c**2*x**2 + 1)/(1536*c**5) - 5*b*d*e*asin(c*x)/(48*c**6)
+ 35*b*e**2*x*sqrt(-c**2*x**2 + 1)/(1024*c**7) - 35*b*e**2*asin(c*x)/(1024*c**8), Ne(c, 0)), (a*(d**2*x**4/4 +
d*e*x**6/3 + e**2*x**8/8), True))

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Giac [B]  time = 1.25017, size = 861, normalized size = 3.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(-c^2*x^2 + 1)^(3/2)*b*d^2*x/c^3 + 1/4*(c^2*x^2 - 1)^2*b*d^2*arcsin(c*x)/c^4 + 5/32*sqrt(-c^2*x^2 + 1)*b
*d^2*x/c^3 + 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*x*e/c^5 + 1/4*(c^2*x^2 - 1)^2*a*d^2/c^4 + 1/2*(c^2*x^
2 - 1)*b*d^2*arcsin(c*x)/c^4 + 1/3*(c^2*x^2 - 1)^3*b*d*arcsin(c*x)*e/c^6 - 13/72*(-c^2*x^2 + 1)^(3/2)*b*d*x*e/
c^5 + 1/2*(c^2*x^2 - 1)*a*d^2/c^4 + 5/32*b*d^2*arcsin(c*x)/c^4 + 1/3*(c^2*x^2 - 1)^3*a*d*e/c^6 + (c^2*x^2 - 1)
^2*b*d*arcsin(c*x)*e/c^6 + 1/64*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*x*e^2/c^7 + 11/48*sqrt(-c^2*x^2 + 1)*b*d*
x*e/c^5 + 1/8*(c^2*x^2 - 1)^4*b*arcsin(c*x)*e^2/c^8 + (c^2*x^2 - 1)^2*a*d*e/c^6 + (c^2*x^2 - 1)*b*d*arcsin(c*x
)*e/c^6 + 25/384*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*x*e^2/c^7 + 1/8*(c^2*x^2 - 1)^4*a*e^2/c^8 + 1/2*(c^2*x^2
- 1)^3*b*arcsin(c*x)*e^2/c^8 + (c^2*x^2 - 1)*a*d*e/c^6 + 11/48*b*d*arcsin(c*x)*e/c^6 - 163/1536*(-c^2*x^2 + 1
)^(3/2)*b*x*e^2/c^7 + 1/2*(c^2*x^2 - 1)^3*a*e^2/c^8 + 3/4*(c^2*x^2 - 1)^2*b*arcsin(c*x)*e^2/c^8 + 93/1024*sqrt
(-c^2*x^2 + 1)*b*x*e^2/c^7 + 3/4*(c^2*x^2 - 1)^2*a*e^2/c^8 + 1/2*(c^2*x^2 - 1)*b*arcsin(c*x)*e^2/c^8 + 1/2*(c^
2*x^2 - 1)*a*e^2/c^8 + 93/1024*b*arcsin(c*x)*e^2/c^8