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

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

[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 = 0.09, 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 = {} \[ \int \frac {\sinh ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

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 = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

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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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)^(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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(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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maple [A]  time = 0.38, 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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^(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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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