3.89 \(\int x \sin ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=119 \[ -\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^2}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {15 \sqrt {\sin ^{-1}(a x)}}{64 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)} \]

[Out]

-1/4*arcsin(a*x)^(5/2)/a^2+1/2*x^2*arcsin(a*x)^(5/2)-15/128*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^
2+5/8*x*arcsin(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a+15/64*arcsin(a*x)^(1/2)/a^2-15/32*x^2*arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.31, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4629, 4707, 4641, 4723, 3312, 3304, 3352} \[ -\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^2}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {15 \sqrt {\sin ^{-1}(a x)}}{64 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSin[a*x]^(5/2),x]

[Out]

(15*Sqrt[ArcSin[a*x]])/(64*a^2) - (15*x^2*Sqrt[ArcSin[a*x]])/32 + (5*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(8
*a) - ArcSin[a*x]^(5/2)/(4*a^2) + (x^2*ArcSin[a*x]^(5/2))/2 - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt
[Pi]])/(128*a^2)

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 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 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rubi steps

\begin {align*} \int x \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {1}{4} (5 a) \int \frac {x^2 \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15}{16} \int x \sqrt {\sin ^{-1}(a x)} \, dx-\frac {5 \int \frac {\sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}+\frac {1}{64} (15 a) \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx\\ &=-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}+\frac {15 \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{64 a^2}\\ &=\frac {15 \sqrt {\sin ^{-1}(a x)}}{64 a^2}-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a^2}\\ &=\frac {15 \sqrt {\sin ^{-1}(a x)}}{64 a^2}-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{64 a^2}\\ &=\frac {15 \sqrt {\sin ^{-1}(a x)}}{64 a^2}-\frac {15}{32} x^2 \sqrt {\sin ^{-1}(a x)}+\frac {5 x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{8 a}-\frac {\sin ^{-1}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \sin ^{-1}(a x)^{5/2}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a^2}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 81, normalized size = 0.68 \[ \frac {\sqrt {\sin ^{-1}(a x)} \left (\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},2 i \sin ^{-1}(a x)\right )\right )}{32 \sqrt {2} a^2 \sqrt {\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*ArcSin[a*x]^(5/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (2*
I)*ArcSin[a*x]]))/(32*Sqrt[2]*a^2*Sqrt[ArcSin[a*x]^2])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [C]  time = 0.28, size = 143, normalized size = 1.20 \[ -\frac {\arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\arcsin \left (a x\right )^{\frac {5}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {5 i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac {5 i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac {\left (15 i + 15\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{512 \, a^{2}} - \frac {\left (15 i - 15\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{512 \, a^{2}} + \frac {15 \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{128 \, a^{2}} + \frac {15 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{128 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

-1/8*arcsin(a*x)^(5/2)*e^(2*I*arcsin(a*x))/a^2 - 1/8*arcsin(a*x)^(5/2)*e^(-2*I*arcsin(a*x))/a^2 - 5/32*I*arcsi
n(a*x)^(3/2)*e^(2*I*arcsin(a*x))/a^2 + 5/32*I*arcsin(a*x)^(3/2)*e^(-2*I*arcsin(a*x))/a^2 + (15/512*I + 15/512)
*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^2 - (15/512*I - 15/512)*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a^
2 + 15/128*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^2 + 15/128*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^2

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maple [A]  time = 0.07, size = 79, normalized size = 0.66 \[ -\frac {32 \arcsin \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arcsin \left (a x \right )\right ) \sqrt {\pi }-40 \arcsin \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arcsin \left (a x \right )\right ) \sqrt {\pi }-30 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arcsin \left (a x \right )\right )+15 \pi \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )}{128 a^{2} \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(a*x)^(5/2),x)

[Out]

-1/128/a^2/Pi^(1/2)*(32*arcsin(a*x)^(5/2)*cos(2*arcsin(a*x))*Pi^(1/2)-40*arcsin(a*x)^(3/2)*sin(2*arcsin(a*x))*
Pi^(1/2)-30*arcsin(a*x)^(1/2)*Pi^(1/2)*cos(2*arcsin(a*x))+15*Pi*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {asin}\left (a\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*asin(a*x)^(5/2),x)

[Out]

int(x*asin(a*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(a*x)**(5/2),x)

[Out]

Integral(x*asin(a*x)**(5/2), x)

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