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

Optimal. Leaf size=380 \[ \frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}+\frac {b x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sin ^{-1}(c x)}{5120 c^{10} e^2}-\frac {b x \sqrt {1-c^2 x^2} \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{38400 c^7 e}-\frac {b x \sqrt {1-c^2 x^2} \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{76800 c^9 e} \]

[Out]

1/5120*b*(128*c^10*d^5-480*c^6*d^3*e^2-800*c^4*d^2*e^3-525*c^2*d*e^4-126*e^5)*arcsin(c*x)/c^10/e^2-1/8*d*(e*x^
2+d)^4*(a+b*arcsin(c*x))/e^2+1/10*(e*x^2+d)^5*(a+b*arcsin(c*x))/e^2-1/76800*b*(1232*c^8*d^4-2536*c^6*d^3*e-775
8*c^4*d^2*e^2-6615*c^2*d*e^3-1890*e^4)*x*(-c^2*x^2+1)^(1/2)/c^9/e-1/38400*b*(136*c^6*d^3-1096*c^4*d^2*e-1617*c
^2*d*e^2-630*e^3)*x*(e*x^2+d)*(-c^2*x^2+1)^(1/2)/c^7/e+1/9600*b*(26*c^4*d^2+201*c^2*d*e+126*e^2)*x*(e*x^2+d)^2
*(-c^2*x^2+1)^(1/2)/c^5/e+1/1600*b*(11*c^2*d+18*e)*x*(e*x^2+d)^3*(-c^2*x^2+1)^(1/2)/c^3/e+1/100*b*x*(e*x^2+d)^
4*(-c^2*x^2+1)^(1/2)/c/e

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Rubi [A]  time = 0.51, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 43, 4731, 12, 528, 388, 216} \[ \frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {b x \sqrt {1-c^2 x^2} \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e}-\frac {b x \sqrt {1-c^2 x^2} \left (-1096 c^4 d^2 e+136 c^6 d^3-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{38400 c^7 e}-\frac {b x \sqrt {1-c^2 x^2} \left (-7758 c^4 d^2 e^2-2536 c^6 d^3 e+1232 c^8 d^4-6615 c^2 d e^3-1890 e^4\right )}{76800 c^9 e}+\frac {b \left (-480 c^6 d^3 e^2-800 c^4 d^2 e^3+128 c^{10} d^5-525 c^2 d e^4-126 e^5\right ) \sin ^{-1}(c x)}{5120 c^{10} e^2}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}+\frac {b x \sqrt {1-c^2 x^2} \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{1600 c^3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*Sqrt[1 - c^2*x^2])/(76800
*c^9*e) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(38400
*c^7*e) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/(9600*c^5*e) + (b*(11*c^2
*d + 18*e)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(1600*c^3*e) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^4)/(100*c*e) +
(b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*ArcSin[c*x])/(5120*c^10*e^2)
- (d*(d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcSin[c*x]))/(10*e^2)

Rule 12

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

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 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]

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 388

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

Rule 528

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

Rubi steps

\begin {align*} \int x^3 \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{40 e^2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{40 e^2}\\ &=\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}+\frac {b \int \frac {\left (d+e x^2\right )^3 \left (2 d \left (5 c^2 d-2 e\right )-2 e \left (11 c^2 d+18 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{400 c e^2}\\ &=\frac {b \left (11 c^2 d+18 e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^2 \left (-2 d \left (40 c^4 d^2-27 c^2 d e-18 e^2\right )+2 e \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{3200 c^3 e^2}\\ &=\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{9600 c^5 e}+\frac {b \left (11 c^2 d+18 e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}+\frac {b \int \frac {\left (d+e x^2\right ) \left (2 d \left (240 c^6 d^3-188 c^4 d^2 e-309 c^2 d e^2-126 e^3\right )+2 e \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{19200 c^5 e^2}\\ &=-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{38400 c^7 e}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{9600 c^5 e}+\frac {b \left (11 c^2 d+18 e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}-\frac {b \int \frac {-2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )-2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{76800 c^7 e^2}\\ &=-\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \sqrt {1-c^2 x^2}}{76800 c^9 e}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{38400 c^7 e}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{9600 c^5 e}+\frac {b \left (11 c^2 d+18 e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}+\frac {\left (b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{5120 c^9 e^2}\\ &=-\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \sqrt {1-c^2 x^2}}{76800 c^9 e}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{38400 c^7 e}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{9600 c^5 e}+\frac {b \left (11 c^2 d+18 e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{1600 c^3 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^4}{100 c e}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sin ^{-1}(c x)}{5120 c^{10} e^2}-\frac {d \left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e^2}+\frac {\left (d+e x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 e^2}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 276, normalized size = 0.73 \[ \frac {c x \left (1920 a c^9 x^3 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (16 c^8 \left (300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right )+8 c^6 \left (900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+315 c^2 e^2 \left (25 d+4 e x^2\right )+1890 e^3\right )\right )+15 b \sin ^{-1}(c x) \left (128 c^{10} x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )-480 c^6 d^3-800 c^4 d^2 e-525 c^2 d e^2-126 e^3\right )}{76800 c^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(1920*a*c^9*x^3*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) + b*Sqrt[1 - c^2*x^2]*(1890*e^3 + 315*
c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x^2 +
525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e^2*x^6 + 48*e^3*x^8))) + 15*b*(-48
0*c^6*d^3 - 800*c^4*d^2*e - 525*c^2*d*e^2 - 126*e^3 + 128*c^10*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e
^3*x^6))*ArcSin[c*x])/(76800*c^10)

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fricas [A]  time = 0.57, size = 318, normalized size = 0.84 \[ \frac {7680 \, a c^{10} e^{3} x^{10} + 28800 \, a c^{10} d e^{2} x^{8} + 38400 \, a c^{10} d^{2} e x^{6} + 19200 \, a c^{10} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} e^{3} x^{10} + 1920 \, b c^{10} d e^{2} x^{8} + 2560 \, b c^{10} d^{2} e x^{6} + 1280 \, b c^{10} d^{3} x^{4} - 480 \, b c^{6} d^{3} - 800 \, b c^{4} d^{2} e - 525 \, b c^{2} d e^{2} - 126 \, b e^{3}\right )} \arcsin \left (c x\right ) + {\left (768 \, b c^{9} e^{3} x^{9} + 144 \, {\left (25 \, b c^{9} d e^{2} + 6 \, b c^{7} e^{3}\right )} x^{7} + 8 \, {\left (800 \, b c^{9} d^{2} e + 525 \, b c^{7} d e^{2} + 126 \, b c^{5} e^{3}\right )} x^{5} + 10 \, {\left (480 \, b c^{9} d^{3} + 800 \, b c^{7} d^{2} e + 525 \, b c^{5} d e^{2} + 126 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (480 \, b c^{7} d^{3} + 800 \, b c^{5} d^{2} e + 525 \, b c^{3} d e^{2} + 126 \, b c e^{3}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{76800 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76800*(7680*a*c^10*e^3*x^10 + 28800*a*c^10*d*e^2*x^8 + 38400*a*c^10*d^2*e*x^6 + 19200*a*c^10*d^3*x^4 + 15*(5
12*b*c^10*e^3*x^10 + 1920*b*c^10*d*e^2*x^8 + 2560*b*c^10*d^2*e*x^6 + 1280*b*c^10*d^3*x^4 - 480*b*c^6*d^3 - 800
*b*c^4*d^2*e - 525*b*c^2*d*e^2 - 126*b*e^3)*arcsin(c*x) + (768*b*c^9*e^3*x^9 + 144*(25*b*c^9*d*e^2 + 6*b*c^7*e
^3)*x^7 + 8*(800*b*c^9*d^2*e + 525*b*c^7*d*e^2 + 126*b*c^5*e^3)*x^5 + 10*(480*b*c^9*d^3 + 800*b*c^7*d^2*e + 52
5*b*c^5*d*e^2 + 126*b*c^3*e^3)*x^3 + 15*(480*b*c^7*d^3 + 800*b*c^5*d^2*e + 525*b*c^3*d*e^2 + 126*b*c*e^3)*x)*s
qrt(-c^2*x^2 + 1))/c^10

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giac [B]  time = 0.40, size = 793, normalized size = 2.09 \[ \frac {1}{10} \, a x^{10} e^{3} + \frac {3}{8} \, a d x^{8} e^{2} + \frac {1}{2} \, a d^{2} x^{6} e + \frac {1}{4} \, a d^{3} x^{4} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3} x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2} x e}{12 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{3} \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} \arcsin \left (c x\right ) e}{2 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} x e}{48 \, c^{5}} + \frac {5 \, b d^{3} \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} \arcsin \left (c x\right ) e}{2 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d x e^{2}}{64 \, c^{7}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x e}{32 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{4} b d \arcsin \left (c x\right ) e^{2}}{8 \, c^{8}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} \arcsin \left (c x\right ) e}{2 \, c^{6}} + \frac {25 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d x e^{2}}{128 \, c^{7}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d \arcsin \left (c x\right ) e^{2}}{2 \, c^{8}} + \frac {11 \, b d^{2} \arcsin \left (c x\right ) e}{32 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b x e^{3}}{100 \, c^{9}} - \frac {163 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d x e^{2}}{512 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b \arcsin \left (c x\right ) e^{3}}{10 \, c^{10}} + \frac {9 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right ) e^{2}}{4 \, c^{8}} + \frac {41 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b x e^{3}}{800 \, c^{9}} + \frac {279 \, \sqrt {-c^{2} x^{2} + 1} b d x e^{2}}{1024 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b \arcsin \left (c x\right ) e^{3}}{2 \, c^{10}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right ) e^{2}}{2 \, c^{8}} + \frac {171 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b x e^{3}}{1600 \, c^{9}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b \arcsin \left (c x\right ) e^{3}}{c^{10}} + \frac {279 \, b d \arcsin \left (c x\right ) e^{2}}{1024 \, c^{8}} - \frac {149 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b x e^{3}}{1280 \, c^{9}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b \arcsin \left (c x\right ) e^{3}}{c^{10}} + \frac {193 \, \sqrt {-c^{2} x^{2} + 1} b x e^{3}}{2560 \, c^{9}} + \frac {{\left (c^{2} x^{2} - 1\right )} b \arcsin \left (c x\right ) e^{3}}{2 \, c^{10}} + \frac {193 \, b \arcsin \left (c x\right ) e^{3}}{2560 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a*x^10*e^3 + 3/8*a*d*x^8*e^2 + 1/2*a*d^2*x^6*e + 1/4*a*d^3*x^4 - 1/16*(-c^2*x^2 + 1)^(3/2)*b*d^3*x/c^3 +
1/4*(c^2*x^2 - 1)^2*b*d^3*arcsin(c*x)/c^4 + 5/32*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 + 1/12*(c^2*x^2 - 1)^2*sqrt(-c
^2*x^2 + 1)*b*d^2*x*e/c^5 + 1/2*(c^2*x^2 - 1)*b*d^3*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^3*b*d^2*arcsin(c*x)*e/
c^6 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d^2*x*e/c^5 + 5/32*b*d^3*arcsin(c*x)/c^4 + 3/2*(c^2*x^2 - 1)^2*b*d^2*arcsin
(c*x)*e/c^6 + 3/64*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d*x*e^2/c^7 + 11/32*sqrt(-c^2*x^2 + 1)*b*d^2*x*e/c^5 +
3/8*(c^2*x^2 - 1)^4*b*d*arcsin(c*x)*e^2/c^8 + 3/2*(c^2*x^2 - 1)*b*d^2*arcsin(c*x)*e/c^6 + 25/128*(c^2*x^2 - 1
)^2*sqrt(-c^2*x^2 + 1)*b*d*x*e^2/c^7 + 3/2*(c^2*x^2 - 1)^3*b*d*arcsin(c*x)*e^2/c^8 + 11/32*b*d^2*arcsin(c*x)*e
/c^6 + 1/100*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*x*e^3/c^9 - 163/512*(-c^2*x^2 + 1)^(3/2)*b*d*x*e^2/c^7 + 1/1
0*(c^2*x^2 - 1)^5*b*arcsin(c*x)*e^3/c^10 + 9/4*(c^2*x^2 - 1)^2*b*d*arcsin(c*x)*e^2/c^8 + 41/800*(c^2*x^2 - 1)^
3*sqrt(-c^2*x^2 + 1)*b*x*e^3/c^9 + 279/1024*sqrt(-c^2*x^2 + 1)*b*d*x*e^2/c^7 + 1/2*(c^2*x^2 - 1)^4*b*arcsin(c*
x)*e^3/c^10 + 3/2*(c^2*x^2 - 1)*b*d*arcsin(c*x)*e^2/c^8 + 171/1600*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*x*e^3/
c^9 + (c^2*x^2 - 1)^3*b*arcsin(c*x)*e^3/c^10 + 279/1024*b*d*arcsin(c*x)*e^2/c^8 - 149/1280*(-c^2*x^2 + 1)^(3/2
)*b*x*e^3/c^9 + (c^2*x^2 - 1)^2*b*arcsin(c*x)*e^3/c^10 + 193/2560*sqrt(-c^2*x^2 + 1)*b*x*e^3/c^9 + 1/2*(c^2*x^
2 - 1)*b*arcsin(c*x)*e^3/c^10 + 193/2560*b*arcsin(c*x)*e^3/c^10

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maple [A]  time = 0.01, size = 449, normalized size = 1.18 \[ \frac {\frac {a \left (\frac {1}{10} e^{3} c^{10} x^{10}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {1}{4} x^{4} c^{10} d^{3}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} c^{10} x^{10}}{10}+\frac {3 \arcsin \left (c x \right ) c^{10} d \,e^{2} x^{8}}{8}+\frac {\arcsin \left (c x \right ) c^{10} d^{2} e \,x^{6}}{2}+\frac {\arcsin \left (c x \right ) c^{10} x^{4} d^{3}}{4}-\frac {e^{3} \left (-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{10}-\frac {9 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{80}-\frac {21 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{160}-\frac {21 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{128}-\frac {63 c x \sqrt {-c^{2} x^{2}+1}}{256}+\frac {63 \arcsin \left (c x \right )}{256}\right )}{10}-\frac {3 c^{2} d \,e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}-\frac {c^{4} d^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{2}-\frac {d^{3} c^{6} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}\right )}{c^{6}}}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(a/c^6*(1/10*e^3*c^10*x^10+3/8*c^10*d*e^2*x^8+1/2*c^10*d^2*e*x^6+1/4*x^4*c^10*d^3)+b/c^6*(1/10*arcsin(c*
x)*e^3*c^10*x^10+3/8*arcsin(c*x)*c^10*d*e^2*x^8+1/2*arcsin(c*x)*c^10*d^2*e*x^6+1/4*arcsin(c*x)*c^10*x^4*d^3-1/
10*e^3*(-1/10*c^9*x^9*(-c^2*x^2+1)^(1/2)-9/80*c^7*x^7*(-c^2*x^2+1)^(1/2)-21/160*c^5*x^5*(-c^2*x^2+1)^(1/2)-21/
128*c^3*x^3*(-c^2*x^2+1)^(1/2)-63/256*c*x*(-c^2*x^2+1)^(1/2)+63/256*arcsin(c*x))-3/8*c^2*d*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))-1/2*c^4*d^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))-1/4*d^3*c^6*(-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 = 1.14, size = 425, normalized size = 1.12 \[ \frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {1}{96} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d^{2} e + \frac {1}{1024} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b d e^{2} + \frac {1}{12800} \, {\left (1280 \, x^{10} \arcsin \left (c x\right ) + {\left (\frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \arcsin \left (c x\right )}{c^{11}}\right )} c\right )} b e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 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*d^3 + 1/96*(48*x^6*arcsin(c*x) + (8*sqr
t(-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^2*e + 1/1024*(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*s
qrt(-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*d*e^2 + 1/12800*(1280*x^1
0*arcsin(c*x) + (128*sqrt(-c^2*x^2 + 1)*x^9/c^2 + 144*sqrt(-c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/
c^6 + 210*sqrt(-c^2*x^2 + 1)*x^3/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsin(c*x)/c^11)*c)*b*e^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 27.78, size = 597, normalized size = 1.57 \[ \begin {cases} \frac {a d^{3} x^{4}}{4} + \frac {a d^{2} e x^{6}}{2} + \frac {3 a d e^{2} x^{8}}{8} + \frac {a e^{3} x^{10}}{10} + \frac {b d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d^{2} e x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 b d e^{2} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b e^{3} x^{10} \operatorname {asin}{\left (c x \right )}}{10} + \frac {b d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d^{2} e x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {3 b d e^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} + \frac {b e^{3} x^{9} \sqrt {- c^{2} x^{2} + 1}}{100 c} + \frac {3 b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b d^{2} e x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {7 b d e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{128 c^{3}} + \frac {9 b e^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{800 c^{3}} - \frac {3 b d^{3} \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b d^{2} e x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {35 b d e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{512 c^{5}} + \frac {21 b e^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{1600 c^{5}} - \frac {5 b d^{2} e \operatorname {asin}{\left (c x \right )}}{32 c^{6}} + \frac {105 b d e^{2} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} + \frac {21 b e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1280 c^{7}} - \frac {105 b d e^{2} \operatorname {asin}{\left (c x \right )}}{1024 c^{8}} + \frac {63 b e^{3} x \sqrt {- c^{2} x^{2} + 1}}{2560 c^{9}} - \frac {63 b e^{3} \operatorname {asin}{\left (c x \right )}}{2560 c^{10}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{4}}{4} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**4/4 + a*d**2*e*x**6/2 + 3*a*d*e**2*x**8/8 + a*e**3*x**10/10 + b*d**3*x**4*asin(c*x)/4 + b
*d**2*e*x**6*asin(c*x)/2 + 3*b*d*e**2*x**8*asin(c*x)/8 + b*e**3*x**10*asin(c*x)/10 + b*d**3*x**3*sqrt(-c**2*x*
*2 + 1)/(16*c) + b*d**2*e*x**5*sqrt(-c**2*x**2 + 1)/(12*c) + 3*b*d*e**2*x**7*sqrt(-c**2*x**2 + 1)/(64*c) + b*e
**3*x**9*sqrt(-c**2*x**2 + 1)/(100*c) + 3*b*d**3*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*d**2*e*x**3*sqrt(-c**2
*x**2 + 1)/(48*c**3) + 7*b*d*e**2*x**5*sqrt(-c**2*x**2 + 1)/(128*c**3) + 9*b*e**3*x**7*sqrt(-c**2*x**2 + 1)/(8
00*c**3) - 3*b*d**3*asin(c*x)/(32*c**4) + 5*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(32*c**5) + 35*b*d*e**2*x**3*sqrt(
-c**2*x**2 + 1)/(512*c**5) + 21*b*e**3*x**5*sqrt(-c**2*x**2 + 1)/(1600*c**5) - 5*b*d**2*e*asin(c*x)/(32*c**6)
+ 105*b*d*e**2*x*sqrt(-c**2*x**2 + 1)/(1024*c**7) + 21*b*e**3*x**3*sqrt(-c**2*x**2 + 1)/(1280*c**7) - 105*b*d*
e**2*asin(c*x)/(1024*c**8) + 63*b*e**3*x*sqrt(-c**2*x**2 + 1)/(2560*c**9) - 63*b*e**3*asin(c*x)/(2560*c**10),
Ne(c, 0)), (a*(d**3*x**4/4 + d**2*e*x**6/2 + 3*d*e**2*x**8/8 + e**3*x**10/10), True))

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