3.129 \(\int \frac{x^6 (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=293 \[ -\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac{5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b x^2 \sqrt{1-c^2 x^2}}{4 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b}{6 c^7 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{7 b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

-b/(6*c^7*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[1 - c^2*x^2])/(4*c^5*d^2*Sqrt[d - c^2*d*x^2
]) + (x^5*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d
- c^2*d*x^2]) - (5*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^6*d^3) + (5*Sqrt[1 - c^2*x^2]*(a + b*ArcSi
n[c*x])^2)/(4*b*c^7*d^2*Sqrt[d - c^2*d*x^2]) - (7*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(6*c^7*d^2*Sqrt[d - c^
2*d*x^2])

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Rubi [A]  time = 0.442504, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {4703, 4707, 4643, 4641, 30, 266, 43} \[ -\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac{5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b x^2 \sqrt{1-c^2 x^2}}{4 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b}{6 c^7 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{7 b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

-b/(6*c^7*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[1 - c^2*x^2])/(4*c^5*d^2*Sqrt[d - c^2*d*x^2
]) + (x^5*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d
- c^2*d*x^2]) - (5*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^6*d^3) + (5*Sqrt[1 - c^2*x^2]*(a + b*ArcSi
n[c*x])^2)/(4*b*c^7*d^2*Sqrt[d - c^2*d*x^2]) - (7*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(6*c^7*d^2*Sqrt[d - c^
2*d*x^2])

Rule 4703

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p +
1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*f*n*d^IntPart[p]*(d + e*x^2
)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 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 4643

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

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 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 \frac{x^6 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{x^5}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx}{c^4 d^2}+\frac{\left (5 b \sqrt{1-c^2 x^2}\right ) \int \frac{x^3}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 c d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac{5 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d-c^2 d x^2}} \, dx}{2 c^6 d^2}+\frac{\left (5 b \sqrt{1-c^2 x^2}\right ) \int x \, dx}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4}+\frac{1}{c^4 \left (-1+c^2 x\right )^2}+\frac{2}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b}{6 c^7 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{13 b x^2 \sqrt{1-c^2 x^2}}{12 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}-\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{3 c^7 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b}{6 c^7 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{1-c^2 x^2}}{4 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^5 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{5 x^3 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c^6 d^3}+\frac{5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^7 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.634476, size = 253, normalized size = 0.86 \[ \frac{\sqrt{d} \left (4 a c x \left (3 c^4 x^4-20 c^2 x^2+15\right )+b \left (6 c^4 x^4-9 c^2 x^2+7\right ) \sqrt{1-c^2 x^2}+28 b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )\right )-60 a \left (c^2 x^2-1\right ) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-30 b \sqrt{d} \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2+4 b c \sqrt{d} x \left (3 c^4 x^4-20 c^2 x^2+15\right ) \sin ^{-1}(c x)}{24 c^7 d^{5/2} \left (c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

(4*b*c*Sqrt[d]*x*(15 - 20*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 30*b*Sqrt[d]*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 -
60*a*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*(
4*a*c*x*(15 - 20*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(7 - 9*c^2*x^2 + 6*c^4*x^4) + 28*b*(1 - c^2*x^2)^(
3/2)*Log[1 - c^2*x^2]))/(24*c^7*d^(5/2)*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])

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Maple [C]  time = 0.403, size = 1716, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-5/2*a/c^6/d^2*x/(-c^2*d*x^2+d)^(1/2)+5/2*a/c^6/d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))
-1/2*a*x^5/c^2/d/(-c^2*d*x^2+d)^(3/2)+5/6*a/c^4*x^3/d/(-c^2*d*x^2+d)^(3/2)-406*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/
(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4+1120/3*I*b*(-d*(c^2
*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^5*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2+
147*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c*arcsin(c*x)*(-c^2*x^2
+1)^(1/2)*x^6+91/6*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^4*(-c^
2*x^2+1)*x^3-7*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^6*(-c^2*x^
2+1)*x-343/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^7*arcsin(c*x
)*(-c^2*x^2+1)^(1/2)-14/3*I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^7/d^3/(c^2*x^2-1)*arcsin(c*x)-49/6*I
*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^2*(-c^2*x^2+1)*x^5-49/6*b*
(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^7*(-c^2*x^2+1)^(1/2)+1/8*b*(-
d*(c^2*x^2-1))^(1/2)/c^7/d^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+147*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c
^6*x^6+334*c^4*x^4-209*c^2*x^2+49)*arcsin(c*x)*x^7-49/6*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6
+334*c^4*x^4-209*c^2*x^2+49)*x^7+37/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2
*x^2+49)/c^5*x^2*(-c^2*x^2+1)^(1/2)+7/3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^7/d^3/(c^2*x^2-1)*ln(1+(
I*c*x+(-c^2*x^2+1)^(1/2))^2)+1009/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x
^2+49)/c^4*arcsin(c*x)*x^3-98*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)
/c^6*arcsin(c*x)*x-1/2*b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arcsin(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/
2)/c^6/d^3/(c^2*x^2-1)*arcsin(c*x)*x-385*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*
c^2*x^2+49)/c^2*arcsin(c*x)*x^5-21/2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*
x^2+49)/c^3*(-c^2*x^2+1)^(1/2)*x^4-5/4*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^7/d^3/(c^2*x^2-1)*arcsin(
c*x)^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-133/6*I*b*(-d*(c^2*x^2-1))^(1/2
)/d^3/(63*c^8*x^8-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^4*x^3+7*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8
-237*c^6*x^6+334*c^4*x^4-209*c^2*x^2+49)/c^6*x+70/3*I*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(63*c^8*x^8-237*c^6*x^6+334
*c^4*x^4-209*c^2*x^2+49)/c^2*x^5

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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^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),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 (-\frac{{\left (b x^{6} \arcsin \left (c x\right ) + a x^{6}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(b*x^6*arcsin(c*x) + a*x^6)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)
, 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**6*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{6}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)*x^6/(-c^2*d*x^2 + d)^(5/2), x)