3.1.74 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx\) [74]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}},x\right ) \]

[Out]

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

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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 {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 43.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/6*((2*(15*(x*e + d)^2*e^2 - 10*(x*e + d)*d*e^2 - 2*d^2*e^2)/((x*e + d)^(5/2)*d^3 - (x*e + d)^(3/2)*d^4) + 15
*e^2*log((sqrt(x*e + d) - sqrt(d))/(sqrt(x*e + d) + sqrt(d)))/d^(7/2))*e^(-1)*log(c) - 6*integrate(log(x)/(sqr
t(x*e + d)*x^4*e^2 + 2*sqrt(x*e + d)*d*x^3*e + sqrt(x*e + d)*d^2*x^2), x) + 6*integrate(log(sqrt(c^2*x^2 + 1)
+ 1)/(sqrt(x*e + d)*x^4*e^2 + 2*sqrt(x*e + d)*d*x^3*e + sqrt(x*e + d)*d^2*x^2), x))*b - 1/6*a*(2*(15*(x*e + d)
^2*e - 10*(x*e + d)*d*e - 2*d^2*e)/((x*e + d)^(5/2)*d^3 - (x*e + d)^(3/2)*d^4) + 15*e*log((sqrt(x*e + d) - sqr
t(d))/(sqrt(x*e + d) + sqrt(d)))/d^(7/2))

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

[Out]

integral((b*arccsch(c*x) + a)*sqrt(x*e + d)/(x^5*e^3 + 3*d*x^4*e^2 + 3*d^2*x^3*e + d^3*x^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**2/(e*x+d)**(5/2),x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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