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

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

[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)

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Rubi [A]  time = 0.508353, antiderivative size = 380, normalized size of antiderivative = 1., 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 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 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 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 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 )^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*}

Mathematica [A]  time = 0.24171, size = 276, normalized size = 0.73 \[ \frac{c x \left (1920 a c^9 x^3 \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )+b \sqrt{1-c^2 x^2} \left (16 c^8 \left (400 d^2 e x^4+300 d^3 x^2+225 d e^2 x^6+48 e^3 x^8\right )+8 c^6 \left (1000 d^2 e x^2+900 d^3+525 d e^2 x^4+108 e^3 x^6\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+315 c^2 e^2 \left (25 d+4 e x^2\right )+1890 e^3\right )\right )+15 b \sin ^{-1}(c x) \left (128 c^{10} x^4 \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )-800 c^4 d^2 e-480 c^6 d^3-525 c^2 d e^2-126 e^3\right )}{76800 c^{10}} \]

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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Maple [A]  time = 0.006, size = 449, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{10}{x}^{10}}{10}}+{\frac{3\,{c}^{10}d{e}^{2}{x}^{8}}{8}}+{\frac{{c}^{10}{d}^{2}e{x}^{6}}{2}}+{\frac{{x}^{4}{c}^{10}{d}^{3}}{4}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}{c}^{10}{x}^{10}}{10}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{10}d{e}^{2}{x}^{8}}{8}}+{\frac{\arcsin \left ( cx \right ){c}^{10}{d}^{2}e{x}^{6}}{2}}+{\frac{\arcsin \left ( cx \right ){c}^{10}{x}^{4}{d}^{3}}{4}}-{\frac{{e}^{3}}{10} \left ( -{\frac{{c}^{9}{x}^{9}}{10}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{9\,{c}^{7}{x}^{7}}{80}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{21\,{c}^{5}{x}^{5}}{160}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{21\,{c}^{3}{x}^{3}}{128}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{63\,cx}{256}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{63\,\arcsin \left ( cx \right ) }{256}} \right ) }-{\frac{3\,{c}^{2}d{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}^{4}{d}^{2}e}{2} \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}^{3}{c}^{6}}{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)^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 = 1.48915, size = 639, normalized size = 1.68 \begin{align*} \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b e^{3} \end{align*}

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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^3 + 1/96*(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/sqrt(c^2))/(sqrt(c^2)*c^6))*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*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*d*e^2 + 1/12800*(1280*x^10*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^10))*c)*b*e^3

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Fricas [A]  time = 2.21083, size = 776, normalized size = 2.04 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 46.39, size = 597, normalized size = 1.57 \begin{align*} \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} \end{align*}

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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Giac [B]  time = 1.32868, size = 1386, normalized size = 3.65 \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)^3*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

-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 + 1/4*(c^2*x^2 - 1)^2*a*d^3/c^4 + 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 + 1/2*(c^2*x^2 - 1)*a*d^3/c^4 + 5/32*b*d^3*arcsin(c*x)/c^4 + 1/2*(c^2*x^2 - 1)^3*a*d^2*e/c^6 + 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)^2*a*d^2*e/c^6 + 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 + 3/8*(c^2*x
^2 - 1)^4*a*d*e^2/c^8 + 3/2*(c^2*x^2 - 1)^3*b*d*arcsin(c*x)*e^2/c^8 + 3/2*(c^2*x^2 - 1)*a*d^2*e/c^6 + 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 + 1/10*(c^2*x^2 - 1)^5*b*arcsin(c*x)*e^3/c^10 + 3/2*(c^2*x^2 - 1)^3*a*d*e^2/c^8 + 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 + 1/10*(c^2*x^2 - 1)^5*a*e^3/c^10 + 1/2*(c^2*x^2 - 1)^4*b*arcsin(c*x)*e^3/c^10 + 9/4*(c^2*x^
2 - 1)^2*a*d*e^2/c^8 + 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 + 1/2*(c^2*x^2 - 1)^4*a*e^3/c^10 + (c^2*x^2 - 1)^3*b*arcsin(c*x)*e^3/c^10 + 3/2*(c^2*x^2 - 1)*a*d
*e^2/c^8 + 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)^3*a*e^
3/c^10 + (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 + (c^2*x^2 - 1)^2*a*
e^3/c^10 + 1/2*(c^2*x^2 - 1)*b*arcsin(c*x)*e^3/c^10 + 1/2*(c^2*x^2 - 1)*a*e^3/c^10 + 193/2560*b*arcsin(c*x)*e^
3/c^10