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

Optimal. Leaf size=30 \[ \text {Int}\left (\text {Li}_{-\frac {1}{2}}(a x),x\right )-x \text {Li}_{\frac {1}{2}}(a x)+x \text {Li}_{\frac {3}{2}}(a x) \]

[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 = 0.01, 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 = {} \[ \int \text {PolyLog}\left (\frac {3}{2},a x\right ) \, dx \]

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*}

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Mathematica [A]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \text {Li}_{\frac {3}{2}}(a x) \, dx \]

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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fricas [A]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm polylog}\left (\frac {3}{2}, a x\right ), x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Li}_{\frac {3}{2}}(a x)\,{d x} \]

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)

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maple [A]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \polylog \left (\frac {3}{2}, a x \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Li}_{\frac {3}{2}}(a x)\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {polylog}\left (\frac {3}{2},a\,x\right ) \,d x \]

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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Li}_{\frac {3}{2}}\left (a x\right )\, dx \]

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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