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

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

[Out]

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

________________________________________________________________________________________

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 28.84, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 3.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x \left (e x +d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, b {\left (\frac {{\left (\frac {3 \, e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )} e + d e\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} d^{2}}\right )} \log \relax (c)}{e} + 3 \, \int \frac {\log \relax (x)}{\sqrt {e x + d} e^{2} x^{3} + 2 \, \sqrt {e x + d} d e x^{2} + \sqrt {e x + d} d^{2} x}\,{d x} - 3 \, \int \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{2} x^{3} + 2 \, \sqrt {e x + d} d e x^{2} + \sqrt {e x + d} d^{2} x}\,{d x}\right )} + \frac {1}{3} \, a {\left (\frac {3 \, \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, e x + 4 \, d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________