3.476 \(\int \frac {\sqrt {\sinh ^{-1}(a x)}}{(c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=179 \[ -\frac {a \sqrt {a^2 x^2+1} \text {Int}\left (\frac {x}{\left (a^2 x^2+1\right )^2 \sqrt {\sinh ^{-1}(a x)}},x\right )}{6 c^2 \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 x^2+1} \text {Int}\left (\frac {x}{\left (a^2 x^2+1\right ) \sqrt {\sinh ^{-1}(a x)}},x\right )}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \sqrt {\sinh ^{-1}(a x)}}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x \sqrt {\sinh ^{-1}(a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

1/3*x*arcsinh(a*x)^(1/2)/c/(a^2*c*x^2+c)^(3/2)+2/3*x*arcsinh(a*x)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-1/6*a*(a^2*x^2
+1)^(1/2)*Unintegrable(x/(a^2*x^2+1)^2/arcsinh(a*x)^(1/2),x)/c^2/(a^2*c*x^2+c)^(1/2)-1/3*a*(a^2*x^2+1)^(1/2)*U
nintegrable(x/(a^2*x^2+1)/arcsinh(a*x)^(1/2),x)/c^2/(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2),x]

[Out]

(x*Sqrt[ArcSinh[a*x]])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[a*x]])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a
*Sqrt[1 + a^2*x^2]*Defer[Int][x/((1 + a^2*x^2)^2*Sqrt[ArcSinh[a*x]]), x])/(6*c^2*Sqrt[c + a^2*c*x^2]) - (a*Sqr
t[1 + a^2*x^2]*Defer[Int][x/((1 + a^2*x^2)*Sqrt[ArcSinh[a*x]]), x])/(3*c^2*Sqrt[c + a^2*c*x^2])

Rubi steps

\begin {align*} \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sqrt {\sinh ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\sinh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x \sqrt {\sinh ^{-1}(a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\sinh ^{-1}(a x)}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\sinh ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right ) \sqrt {\sinh ^{-1}(a x)}} \, dx}{3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.42, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2), x]

________________________________________________________________________________________

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(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsinh(a*x))/(a^2*c*x^2 + c)^(5/2), x)

________________________________________________________________________________________

maple [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arcsinh \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)

[Out]

int(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arcsinh(a*x))/(a^2*c*x^2 + c)^(5/2), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {asinh}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(5/2),x)

[Out]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(5/2), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {asinh}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(1/2)/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(sqrt(asinh(a*x))/(c*(a**2*x**2 + 1))**(5/2), x)

________________________________________________________________________________________