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

Optimal. Leaf size=216 \[ -\frac {15 \sqrt {\pi } b^{5/2} \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}-\frac {15 \sqrt {\pi } b^{5/2} \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^2 \sqrt {a+b \sin ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]

[Out]

-1/4*(a+b*arcsin(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arcsin(c*x))^(5/2)-15/128*b^(5/2)*cos(2*a/b)*FresnelC(2*(a+b*arc
sin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/c^2-15/128*b^(5/2)*FresnelS(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1
/2))*sin(2*a/b)*Pi^(1/2)/c^2+5/8*b*x*(a+b*arcsin(c*x))^(3/2)*(-c^2*x^2+1)^(1/2)/c+15/64*b^2*(a+b*arcsin(c*x))^
(1/2)/c^2-15/32*b^2*x^2*(a+b*arcsin(c*x))^(1/2)

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Rubi [A]  time = 0.74, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4629, 4707, 4641, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac {15 \sqrt {\pi } b^{5/2} \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{128 c^2}-\frac {15 \sqrt {\pi } b^{5/2} \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 c^2}+\frac {15 b^2 \sqrt {a+b \sin ^{-1}(c x)}}{64 c^2}-\frac {15}{32} b^2 x^2 \sqrt {a+b \sin ^{-1}(c x)}+\frac {5 b x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac {\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(15*b^2*Sqrt[a + b*ArcSin[c*x]])/(64*c^2) - (15*b^2*x^2*Sqrt[a + b*ArcSin[c*x]])/32 + (5*b*x*Sqrt[1 - c^2*x^2]
*(a + b*ArcSin[c*x])^(3/2))/(8*c) - (a + b*ArcSin[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcSin[c*x])^(5/2))/2 - (1
5*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*c^2) - (15*b^(5
/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*c^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 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[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 3351

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

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

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Mathematica [C]  time = 0.10, size = 141, normalized size = 0.65 \[ \frac {e^{-\frac {2 i a}{b}} \left (a+b \sin ^{-1}(c x)\right )^{5/2} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {7}{2},\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{32 \sqrt {2} c^2 \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*ArcSin[c*x])^(5/2)*(Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] + E^(((
4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[7/2, ((2*I)*(a + b*ArcSin[c*x]))/b]))/(32*Sqrt[2]*c^2*E^((
(2*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(3/2))

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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*(a+b*arcsin(c*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 = 2.73, size = 1307, normalized size = 6.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*sqrt(pi)*a^3*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^
(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 3/8*sqrt(pi)*a^2*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(
b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 1/4*I*sqrt(pi)*a^3*b^(3/2)*erf(-sq
rt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c
^2) - 3/8*sqrt(pi)*a^2*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b)
)*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2) - 1/8*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(2*I*arcsin(c*x))/
c^2 - 1/8*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(-2*I*arcsin(c*x))/c^2 + 3/8*sqrt(pi)*a^2*b^2*erf(-sqrt(
b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b
))*c^2) - 9/64*I*sqrt(pi)*a*b^3*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b
))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b))*c^2) + 1/4*I*sqrt(pi)*a^3*b*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b)
+ I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*c^2) + 3/8*sqrt(pi)*a^2
*b^2*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) -
I*b^(5/2)/abs(b))*c^2) + 9/64*I*sqrt(pi)*a*b^3*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) +
a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*c^2) - 1/4*I*sqrt(pi)*a^3*sqrt(b)*erf(-sqrt(b*ar
csin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 9/64
*I*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*
a/b)/((b + I*b^2/abs(b))*c^2) + 15/256*sqrt(pi)*b^(7/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin
(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) - 9/64*I*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin
(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) + 15/256*
sqrt(pi)*b^(7/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)
/((b - I*b^2/abs(b))*c^2) - 1/4*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(2*I*arcsin(c*x))/c^2 - 5/32*I*sqrt(
b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(2*I*arcsin(c*x))/c^2 - 1/4*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(-2
*I*arcsin(c*x))/c^2 + 5/32*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)*e^(-2*I*arcsin(c*x))/c^2 - 1/8*sqrt(b*arc
sin(c*x) + a)*a^2*e^(2*I*arcsin(c*x))/c^2 - 5/32*I*sqrt(b*arcsin(c*x) + a)*a*b*e^(2*I*arcsin(c*x))/c^2 + 15/12
8*sqrt(b*arcsin(c*x) + a)*b^2*e^(2*I*arcsin(c*x))/c^2 - 1/8*sqrt(b*arcsin(c*x) + a)*a^2*e^(-2*I*arcsin(c*x))/c
^2 + 5/32*I*sqrt(b*arcsin(c*x) + a)*a*b*e^(-2*I*arcsin(c*x))/c^2 + 15/128*sqrt(b*arcsin(c*x) + a)*b^2*e^(-2*I*
arcsin(c*x))/c^2

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maple [B]  time = 0.11, size = 394, normalized size = 1.82 \[ -\frac {15 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{3}+15 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{3}+32 \arcsin \left (c x \right )^{3} \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (c x \right )^{2} \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}-40 \arcsin \left (c x \right )^{2} \sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (c x \right ) \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a^{2} b -30 \arcsin \left (c x \right ) \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b^{3}-80 \arcsin \left (c x \right ) \sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a^{3}-30 \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a \,b^{2}-40 \sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a^{2} b}{128 c^{2} \sqrt {a +b \arcsin \left (c x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128/c^2*(15*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*ar
csin(c*x))^(1/2)/b)*b^3+15*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(
1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^3+32*arcsin(c*x)^3*cos(2*(a+b*arcsin(c*x))/b-2*a/b)*b^3+96*arcsin(c*x)^2*cos
(2*(a+b*arcsin(c*x))/b-2*a/b)*a*b^2-40*arcsin(c*x)^2*sin(2*(a+b*arcsin(c*x))/b-2*a/b)*b^3+96*arcsin(c*x)*cos(2
*(a+b*arcsin(c*x))/b-2*a/b)*a^2*b-30*arcsin(c*x)*cos(2*(a+b*arcsin(c*x))/b-2*a/b)*b^3-80*arcsin(c*x)*sin(2*(a+
b*arcsin(c*x))/b-2*a/b)*a*b^2+32*cos(2*(a+b*arcsin(c*x))/b-2*a/b)*a^3-30*cos(2*(a+b*arcsin(c*x))/b-2*a/b)*a*b^
2-40*sin(2*(a+b*arcsin(c*x))/b-2*a/b)*a^2*b)/(a+b*arcsin(c*x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c*x))^(5/2),x)

[Out]

int(x*(a + b*asin(c*x))^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*asin(c*x))**(5/2), x)

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