3.160 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^3 (d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x)

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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)),x]

[Out]

Defer[Int][(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)), x]

Rubi steps

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

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Mathematica [A]  time = 59.84, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^3 \left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)),x]

[Out]

Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)), x]

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fricas [A]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{3} x^{9} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{5} + d^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arccsch(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/((e*x^2 + d)^(5/2)*x^3), x)

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maple [A]  time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, a {\left (\frac {15 \, e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {7}{2}}} - \frac {15 \, e}{\sqrt {e x^{2} + d} d^{3}} - \frac {5 \, e}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2}} - \frac {3}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d x^{2}}\right )} + b \int \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/6*a*(15*e*arcsinh(d/(sqrt(d*e)*abs(x)))/d^(7/2) - 15*e/(sqrt(e*x^2 + d)*d^3) - 5*e/((e*x^2 + d)^(3/2)*d^2) -
3/((e*x^2 + d)^(3/2)*d*x^2)) + b*integrate(log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/((e*x^2 + d)^(5/2)*x^3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^3\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c*x)))/(x^3*(d + e*x^2)^(5/2)),x)

[Out]

int((a + b*asinh(1/(c*x)))/(x^3*(d + e*x^2)^(5/2)), 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((a+b*acsch(c*x))/x**3/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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