3.7.6 \(\int x^3 (d+e x^2)^2 (a+b \text {ArcSin}(c x)) \, dx\) [606]

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

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 241, normalized size of antiderivative = 1.00, 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 = {272, 45, 4815, 12, 1281, 470, 327, 222} \begin {gather*} \frac {1}{4} d^2 x^4 (a+b \text {ArcSin}(c x))+\frac {1}{3} d e x^6 (a+b \text {ArcSin}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^8}+\frac {b e^2 x^7 \sqrt {1-c^2 x^2}}{64 c}+\frac {b e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{1152 c^3}+\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 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} \end {gather*}

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 12

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

Rule 45

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 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 272

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 327

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 470

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 1281

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 4815

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]))

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*}

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Mathematica [A]
time = 0.11, size = 190, normalized size = 0.79 \begin {gather*} \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 (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+3 b \left (-288 c^4 d^2-320 c^2 d e-105 e^2+128 c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )\right ) \text {ArcSin}(c x)}{9216 c^8} \end {gather*}

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 = 0.13, size = 303, normalized size = 1.26

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}-\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}\right )}{c^{4}}}{c^{4}}\)	\(303\)
default	\(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arcsin \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}-\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}\right )}{c^{4}}}{c^{4}}\)	\(303\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 284, normalized size = 1.18 \begin {gather*} \frac {1}{8} \, a x^{8} e^{2} + \frac {1}{3} \, a d x^{6} e + \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 (c x\right )}{c^{5}}\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 (c x\right )}{c^{7}}\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 (c x\right )}{c^{9}}\right )} c\right )} b e^{2} \end {gather*}

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*x^8*e^2 + 1/3*a*d*x^6*e + 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*x)/c^5)*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*x)/c^7)*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*x)/c^9)*c)*b*e^2

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Fricas [A]
time = 1.21, size = 214, normalized size = 0.89 \begin {gather*} \frac {1152 \, a c^{8} x^{8} e^{2} + 3072 \, a c^{8} d x^{6} e + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} + 3 \, {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} e^{2} + 64 \, {\left (16 \, b c^{8} d x^{6} - 5 \, b c^{2} d\right )} e\right )} \arcsin \left (c x\right ) + {\left (576 \, b c^{7} d^{2} x^{3} + 864 \, b c^{5} d^{2} x + 3 \, {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} e^{2} + 64 \, {\left (8 \, b c^{7} d x^{5} + 10 \, b c^{5} d x^{3} + 15 \, b c^{3} d x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{8}} \end {gather*}

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*x^8*e^2 + 3072*a*c^8*d*x^6*e + 2304*a*c^8*d^2*x^4 + 3*(768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 +
3*(128*b*c^8*x^8 - 35*b)*e^2 + 64*(16*b*c^8*d*x^6 - 5*b*c^2*d)*e)*arcsin(c*x) + (576*b*c^7*d^2*x^3 + 864*b*c^5
*d^2*x + 3*(48*b*c^7*x^7 + 56*b*c^5*x^5 + 70*b*c^3*x^3 + 105*b*c*x)*e^2 + 64*(8*b*c^7*d*x^5 + 10*b*c^5*d*x^3 +
15*b*c^3*d*x)*e)*sqrt(-c^2*x^2 + 1))/c^8

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Sympy [A]
time = 1.13, size = 382, normalized size = 1.59 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (217) = 434\).
time = 0.44, size = 498, normalized size = 2.07 \begin {gather*} \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 {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e x}{18 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d e \arcsin \left (c x\right )}{3 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e x}{72 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{64 \, c^{7}} + \frac {5 \, b d^{2} \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d e \arcsin \left (c x\right )}{c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{2} \arcsin \left (c x\right )}{8 \, c^{8}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b d e x}{48 \, c^{5}} + \frac {25 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{384 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e \arcsin \left (c x\right )}{c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} \arcsin \left (c x\right )}{2 \, c^{8}} - \frac {163 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2} x}{1536 \, c^{7}} + \frac {11 \, b d e \arcsin \left (c x\right )}{48 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} \arcsin \left (c x\right )}{4 \, c^{8}} + \frac {93 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{1024 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} \arcsin \left (c x\right )}{2 \, c^{8}} + \frac {93 \, b e^{2} \arcsin \left (c x\right )}{1024 \, c^{8}} \end {gather*}

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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