3.96 \(\int \text{PolyLog}(\frac{3}{2},a x) \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\text{PolyLog}\left (-\frac{1}{2},a x\right ),x\right )-x \text{PolyLog}\left (\frac{1}{2},a x\right )+x \text{PolyLog}\left (\frac{3}{2},a x\right ) \]

[Out]

-(x*PolyLog[1/2, a*x]) + x*PolyLog[3/2, a*x] + Unintegrable[PolyLog[-1/2, a*x], x]

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Rubi [A]  time = 0.0076448, 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 = {} \[ \int \text{PolyLog}\left (\frac{3}{2},a x\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[PolyLog[3/2, a*x],x]

[Out]

-(x*PolyLog[1/2, a*x]) + x*PolyLog[3/2, a*x] + Defer[Int][PolyLog[-1/2, a*x], x]

Rubi steps

\begin{align*} \int \text{Li}_{\frac{3}{2}}(a x) \, dx &=x \text{Li}_{\frac{3}{2}}(a x)-\int \text{Li}_{\frac{1}{2}}(a x) \, dx\\ &=-x \text{Li}_{\frac{1}{2}}(a x)+x \text{Li}_{\frac{3}{2}}(a x)+\int \text{Li}_{-\frac{1}{2}}(a x) \, dx\\ \end{align*}

Mathematica [A]  time = 0.0066395, size = 0, normalized size = 0. \[ \int \text{PolyLog}\left (\frac{3}{2},a x\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[PolyLog[3/2, a*x],x]

[Out]

Integrate[PolyLog[3/2, a*x], x]

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Maple [A]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\it polylog} \left ({\frac{3}{2}},ax \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3/2,a*x),x)

[Out]

int(polylog(3/2,a*x),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{\frac{3}{2}}(a x)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3/2,a*x),x, algorithm="maxima")

[Out]

integrate(polylog(3/2, a*x), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm polylog}\left (\frac{3}{2}, a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3/2,a*x),x, algorithm="fricas")

[Out]

integral(polylog(3/2, a*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{\frac{3}{2}}\left (a x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3/2,a*x),x)

[Out]

Integral(polylog(3/2, a*x), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{\frac{3}{2}}(a x)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3/2,a*x),x, algorithm="giac")

[Out]

integrate(polylog(3/2, a*x), x)