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

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

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 29.96, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x (d+e x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [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*arccsc(c*x))/x/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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