\(\int (a+i b \arcsin (1-i d x^2))^{5/2} \, dx\) [328]

Optimal result

Integrand size = 22, antiderivative size = 348 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=15 b^2 x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}-\frac {5 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2}+\frac {15 b^2 \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {15 \sqrt {-\frac {i}{b}} b^3 \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )} \]

[Out]

x*(a-I*b*arcsin(-1+I*d*x^2))^(5/2)+15*b^2*x*FresnelS((-I/b)^(1/2)*(a-I*b*arcsin(-1+I*d*x^2))^(1/2)/Pi^(1/2))*(
cosh(1/2*a/b)+I*sinh(1/2*a/b))*Pi^(1/2)/(cos(1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-1+I*d*x^2)))/(-I/b)^(1/2)
-15*b^3*x*FresnelC((-I/b)^(1/2)*(a-I*b*arcsin(-1+I*d*x^2))^(1/2)/Pi^(1/2))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*(-I
/b)^(1/2)*Pi^(1/2)/(cos(1/2*arcsin(-1+I*d*x^2))+sin(1/2*arcsin(-1+I*d*x^2)))-5*b*(a-I*b*arcsin(-1+I*d*x^2))^(3
/2)*(2*I*d*x^2+d^2*x^4)^(1/2)/d/x+15*b^2*x*(a-I*b*arcsin(-1+I*d*x^2))^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4898, 4895} \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=-\frac {15 \sqrt {\pi } \sqrt {-\frac {i}{b}} b^3 x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}+\frac {15 \sqrt {\pi } b^2 x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}+15 b^2 x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}-\frac {5 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \]

[In]

Int[(a + I*b*ArcSin[1 - I*d*x^2])^(5/2),x]

[Out]

15*b^2*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]] - (5*b*Sqrt[(2*I)*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2])^(3
/2))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2) + (15*b^2*Sqrt[Pi]*x*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*Ar
cSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[(-I)/b]*(Cos[ArcSin[1 - I*d*x^2]/2] - S
in[ArcSin[1 - I*d*x^2]/2])) - (15*Sqrt[(-I)/b]*b^3*Sqrt[Pi]*x*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I
*d*x^2]])/Sqrt[Pi]]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2
])

Rule 4895

Int[Sqrt[(a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[x*Sqrt[a + b*ArcSin[c + d*x^2]], x] + (
-Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(FresnelC[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt
[c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))), x] + Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] - c*Sin[a/
(2*b)])*(FresnelS[Sqrt[c/(Pi*b)]*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[c/b]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[A
rcSin[c + d*x^2]/2]))), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]

Rule 4898

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*(a + b*ArcSin[c + d*x^2])^n, x] + (-
Dist[4*b^2*n*(n - 1), Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x], x] + Simp[2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*((
a + b*ArcSin[c + d*x^2])^(n - 1)/(d*x)), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {5 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2}+\left (15 b^2\right ) \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx \\ & = 15 b^2 x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}-\frac {5 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2}+\frac {15 b^2 \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {15 \sqrt {-\frac {i}{b}} b^3 \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.97 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=-\frac {5 b \sqrt {d x^2 \left (2 i+d x^2\right )} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{3/2}}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2}+\frac {15 b^2 x \left (\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )-\sqrt {\pi } \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]

[In]

Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(5/2),x]

[Out]

(-5*b*Sqrt[d*x^2*(2*I + d*x^2)]*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(
5/2) + (15*b^2*x*(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 -
I*d*x^2]/2]) - Sqrt[Pi]*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I
*Sinh[a/(2*b)]) + Sqrt[Pi]*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)]
+ I*Sinh[a/(2*b)])))/(Sqrt[(-I)/b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))

Maple [F]

\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}+i\right )\right )}^{\frac {5}{2}}d x\]

[In]

int((a+b*arcsinh(I+d*x^2))^(5/2),x)

[Out]

int((a+b*arcsinh(I+d*x^2))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(I+d*x^2))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*asinh(I+d*x**2))**(5/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real I

Maxima [F]

\[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*arcsinh(I+d*x^2))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arcsinh(d*x^2 + I) + a)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(I+d*x^2))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^{5/2} \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^{5/2} \,d x \]

[In]

int((a + b*asinh(d*x^2 + 1i))^(5/2),x)

[Out]

int((a + b*asinh(d*x^2 + 1i))^(5/2), x)