\(\int \frac {\operatorname {PolyLog}(2,1+\frac {b c-a d}{d (a+b x)})}{(a+b x) (c+d x)} \, dx\) [102]

Optimal result

Integrand size = 38, antiderivative size = 35 \[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=-\frac {\operatorname {PolyLog}\left (3,1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6745} \[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=-\frac {\operatorname {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]

[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 6745

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\frac {\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{-b c+a d} \]

[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)

Maple [A] (verified)

Time = 8.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {\operatorname {Li}_{3}\left (1-\frac {a d -c b}{d \left (b x +a \right )}\right )}{a d -c b}\)	\(36\)
default	\(\frac {\operatorname {Li}_{3}\left (1-\frac {a d -c b}{d \left (b x +a \right )}\right )}{a d -c b}\)	\(36\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {{\rm Li}_2\left (\frac {b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\mathrm {polylog}\left (2,1-\frac {a\,d-b\,c}{d\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[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)