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

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 = 0.11, 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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \]

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 = 28.47, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx \]

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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fricas [A]  time = 2.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}}, x\right ) \]

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

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

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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maple [F(-2)]  time = 180.00, 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 \[ \frac {1}{6} \, b {\left (\frac {{\left (\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} e^{2} - 10 \, {\left (e x + d\right )} d e^{2} - 2 \, d^{2} e^{2}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} d^{3} - {\left (e x + d\right )}^{\frac {3}{2}} d^{4}} + \frac {15 \, e^{2} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )} \log \relax (c)}{e} - 6 \, \int \frac {\log \relax (x)}{\sqrt {e x + d} e^{2} x^{4} + 2 \, \sqrt {e x + d} d e x^{3} + \sqrt {e x + d} d^{2} x^{2}}\,{d x} + 6 \, \int \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{2} x^{4} + 2 \, \sqrt {e x + d} d e x^{3} + \sqrt {e x + d} d^{2} x^{2}}\,{d x}\right )} - \frac {1}{6} \, a {\left (\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} e - 10 \, {\left (e x + d\right )} d e - 2 \, d^{2} e\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} d^{3} - {\left (e x + d\right )}^{\frac {3}{2}} d^{4}} + \frac {15 \, e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )} \]

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

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

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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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**2/(e*x+d)**(5/2),x)

[Out]

Timed out

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