3.102 \(\int \frac {\text {Li}_2(1+\frac {b c-a d}{d (a+b x)})}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=35 \[ -\frac {\text {Li}_3\left (\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]

[Out]

-polylog(3,1+(-a*d+b*c)/d/(b*x+a))/(-a*d+b*c)

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Rubi [A]  time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6610} \[ -\frac {\text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)),x]

[Out]

-(PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))]/(b*c - a*d))

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx &=-\frac {\text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.86 \[ \frac {\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{a d-b c} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)),x]

[Out]

PolyLog[3, (b*(c + d*x))/(d*(a + b*x))]/(-(b*c) + a*d)

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (\frac {b c - a d}{b d x + a d} + 1\right )}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

integral(dilog((b*c - a*d)/(b*d*x + a*d) + 1)/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(dilog((b*c - a*d)/((b*x + a)*d) + 1)/((b*x + a)*(d*x + c)), x)

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maple [A]  time = 0.05, size = 36, normalized size = 1.03 \[ \frac {\polylog \left (3, -\frac {a d -b c}{\left (b x +a \right ) d}+1\right )}{a d -b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x)

[Out]

1/(a*d-b*c)*polylog(3,1-(a*d-b*c)/d/(b*x+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

integrate(dilog((b*c - a*d)/((b*x + a)*d) + 1)/((b*x + a)*(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, 1 - (a*d - b*c)/(d*(a + b*x)))/((a + b*x)*(c + d*x)),x)

[Out]

int(polylog(2, 1 - (a*d - b*c)/(d*(a + b*x)))/((a + b*x)*(c + d*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{2}\left (- \frac {a d}{a d + b d x} + \frac {b c}{a d + b d x} + 1\right )}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x)

[Out]

Integral(polylog(2, -a*d/(a*d + b*d*x) + b*c/(a*d + b*d*x) + 1)/((a + b*x)*(c + d*x)), x)

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