3.475 \(\int \frac {\sqrt {\sinh ^{-1}(a x)}}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac {x \sqrt {\sinh ^{-1}(a x)}}{c \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 x^2+1} \text {Int}\left (\frac {x}{\left (a^2 x^2+1\right ) \sqrt {\sinh ^{-1}(a x)}},x\right )}{2 c \sqrt {a^2 c x^2+c}} \]

[Out]

x*arcsinh(a*x)^(1/2)/c/(a^2*c*x^2+c)^(1/2)-1/2*a*(a^2*x^2+1)^(1/2)*Unintegrable(x/(a^2*x^2+1)/arcsinh(a*x)^(1/
2),x)/c/(a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2),x]

[Out]

(x*Sqrt[ArcSinh[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[1 + a^2*x^2]*Defer[Int][x/((1 + a^2*x^2)*Sqrt[ArcSinh
[a*x]]), x])/(2*c*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 )^{3/2}} \, dx &=\frac {x \sqrt {\sinh ^{-1}(a x)}}{c \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}{2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2),x]

[Out]

Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2), x]

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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(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsinh(a*x))/(a^2*c*x^2 + c)^(3/2), x)

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maple [A]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arcsinh \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arcsinh(a*x))/(a^2*c*x^2 + c)^(3/2), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {asinh}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(3/2),x)

[Out]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(3/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {asinh}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(1/2)/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(sqrt(asinh(a*x))/(c*(a**2*x**2 + 1))**(3/2), x)

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