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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0938657, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.9046, size = 0, normalized size = 0. \[ \int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.789, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsc} \left (cx\right )}{{x}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{3} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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