3.5.84 \(\int \frac {\sinh ^{-1}(a x)^{5/2}}{(c+a^2 c x^2)^{3/2}} \, dx\) [484]

Optimal. Leaf size=87 \[ \frac {x \sinh ^{-1}(a x)^{5/2}}{c \sqrt {c+a^2 c x^2}}-\frac {5 a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x \sinh ^{-1}(a x)^{3/2}}{1+a^2 x^2},x\right )}{2 c \sqrt {c+a^2 c x^2}} \]

[Out]

x*arcsinh(a*x)^(5/2)/c/(a^2*c*x^2+c)^(1/2)-5/2*a*(a^2*x^2+1)^(1/2)*Unintegrable(x*arcsinh(a*x)^(3/2)/(a^2*x^2+
1),x)/c/(a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.07, 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 = {} \begin {gather*} \int \frac {\sinh ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

(x*ArcSinh[a*x]^(5/2))/(c*Sqrt[c + a^2*c*x^2]) - (5*a*Sqrt[1 + a^2*x^2]*Defer[Int][(x*ArcSinh[a*x]^(3/2))/(1 +
a^2*x^2), x])/(2*c*Sqrt[c + a^2*c*x^2])

Rubi steps

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

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Mathematica [A]
time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

Integrate[ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2), x]

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Maple [A]
time = 4.97, size = 0, normalized size = 0.00 \[\int \frac {\arcsinh \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(3/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2 + c)^(3/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(5/2)/(a**2*c*x**2+c)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2 + c)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^(5/2)/(c + a^2*c*x^2)^(3/2),x)

[Out]

int(asinh(a*x)^(5/2)/(c + a^2*c*x^2)^(3/2), x)

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