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

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 15.68, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**2/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*acsch(c*x))/(x**2*(d + e*x)**(3/2)), x)

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