3.85 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=351 \[ \frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}} \]

[Out]

(5*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1
- c^2*x^2]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*
x^2])/(64*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (5*d^2*x^3*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/64 + (5*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/48 + (x^3*(d - c^2
*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^3*Sqrt[1 - c
^2*x^2])

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Rubi [A]  time = 0.472728, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {4699, 4697, 4707, 4641, 30, 14, 266, 43} \[ \frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(5*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1
- c^2*x^2]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*
x^2])/(64*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (5*d^2*x^3*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/64 + (5*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/48 + (x^3*(d - c^2
*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/8 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^3*Sqrt[1 - c
^2*x^2])

Rule 4699

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n -
1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4697

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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])

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} (5 d) \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{16} \left (5 d^2\right ) \int x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \, dx}{48 \sqrt{1-c^2 x^2}}\\ &=\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (x-2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (x^3-c^2 x^5\right ) \, dx}{48 \sqrt{1-c^2 x^2}}\\ &=-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b d^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt{1-c^2 x^2}}\\ &=\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{256 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.197845, size = 196, normalized size = 0.56 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (45 a^2+6 a b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )+6 b \sin ^{-1}(c x) \left (15 a+b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )+b^2 c^2 x^2 \left (-36 c^6 x^6+136 c^4 x^4-177 c^2 x^2+45\right )+45 b^2 \sin ^{-1}(c x)^2\right )}{2304 b c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(45*a^2 + b^2*c^2*x^2*(45 - 177*c^2*x^2 + 136*c^4*x^4 - 36*c^6*x^6) + 6*a*b*c*x*Sqrt[
1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) + 6*b*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c
^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(2304*b*c^3*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.313, size = 620, normalized size = 1.8 \begin{align*} -{\frac{ax}{8\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{7}{2}}}}+{\frac{ax}{48\,{c}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{5\,adx}{192\,{c}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,a{d}^{2}x}{128\,{c}^{2}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{5\,a{d}^{3}}{128\,{c}^{2}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+{\frac{b{d}^{2}{c}^{6}\arcsin \left ( cx \right ){x}^{9}}{8\,{c}^{2}{x}^{2}-8}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{23\,b{d}^{2}{c}^{4}\arcsin \left ( cx \right ){x}^{7}}{48\,{c}^{2}{x}^{2}-48}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{127\,b{c}^{2}{d}^{2}\arcsin \left ( cx \right ){x}^{5}}{192\,{c}^{2}{x}^{2}-192}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{133\,b{d}^{2}\arcsin \left ( cx \right ){x}^{3}}{384\,{c}^{2}{x}^{2}-384}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,b{d}^{2}\arcsin \left ( cx \right ) x}{128\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{359\,b{d}^{2}}{73728\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b{d}^{2}{c}^{5}{x}^{8}}{64\,{c}^{2}{x}^{2}-64}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{17\,b{d}^{2}{c}^{3}{x}^{6}}{288\,{c}^{2}{x}^{2}-288}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{59\,b{d}^{2}c{x}^{4}}{768\,{c}^{2}{x}^{2}-768}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,b{d}^{2}{x}^{2}}{256\,c \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,b \left ( \arcsin \left ( cx \right ) \right ) ^{2}{d}^{2}}{256\,{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x)

[Out]

-1/8*a*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a/c^2*x*(-c^2*d*x^2+d)^(5/2)+5/192*a/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/1
28*a/c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/128*a/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))
+1/8*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)*x^9-23/48*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*
x^2-1)*arcsin(c*x)*x^7+127/192*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5-133/384*b*(-d*(c^2
*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(c*x)*x^3+5/128*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)*
x-359/73728*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+1/64*b*(-d*(c^2*x^2-1))^(1/2)*d^2*
c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8-17/288*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*
x^6+59/768*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-5/256*b*(-d*(c^2*x^2-1))^(1/2)*d^
2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-5/256*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsi
n(c*x)^2*d^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a c^{4} d^{2} x^{6} - 2 \, a c^{2} d^{2} x^{4} + a d^{2} x^{2} +{\left (b c^{4} d^{2} x^{6} - 2 \, b c^{2} d^{2} x^{4} + b d^{2} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^6 - 2*a*c^2*d^2*x^4 + a*d^2*x^2 + (b*c^4*d^2*x^6 - 2*b*c^2*d^2*x^4 + b*d^2*x^2)*arcsin(c
*x))*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)*x^2, x)