Optimal. Leaf size=200 \[ \frac {4 B (b c-a d) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(b f-a g) (d f-c g)}+\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(f+g x) (b f-a g)}+\frac {8 B^2 (b c-a d) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g) (d f-c g)} \]
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Rubi [B] time = 1.30, antiderivative size = 620, normalized size of antiderivative = 3.10, number of steps used = 32, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {8 b B^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g (b f-a g)}+\frac {8 B^2 d \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g (d f-c g)}-\frac {8 B^2 (b c-a d) \text {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{(b f-a g) (d f-c g)}+\frac {4 b B \log (a+b x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g (b f-a g)}-\frac {4 B d \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g (d f-c g)}+\frac {4 B (b c-a d) \log (f+g x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(b f-a g) (d f-c g)}-\frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g (f+g x)}+\frac {8 B^2 d \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g (d f-c g)}+\frac {8 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {8 B^2 (b c-a d) \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{(b f-a g) (d f-c g)}-\frac {4 b B^2 \log ^2(a+b x)}{g (b f-a g)}-\frac {4 B^2 d \log ^2(c+d x)}{g (d f-c g)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^2} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {(2 B) \int \frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {(4 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) (c+d x) (f+g x)} \, dx}{g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {(4 B (b c-a d)) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (b f-a g) (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (-d f+c g) (c+d x)}+\frac {g^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b f-a g) (d f-c g) (f+g x)}\right ) \, dx}{g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {\left (4 b^2 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{g (b f-a g)}-\frac {\left (4 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{g (d f-c g)}+\frac {(4 B (b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (4 b B^2\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{g (b f-a g)}+\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{g (d f-c g)}-\frac {\left (4 B^2 (b c-a d)\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (f+g x)}{e (a+b x)^2} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (4 b B^2\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{e g (b f-a g)}+\frac {\left (4 B^2 d\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{e g (d f-c g)}-\frac {\left (4 B^2 (b c-a d)\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (f+g x)}{(a+b x)^2} \, dx}{e (b f-a g) (d f-c g)}\\ &=\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (4 b B^2\right ) \int \left (\frac {2 b e \log (a+b x)}{a+b x}-\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{e g (b f-a g)}+\frac {\left (4 B^2 d\right ) \int \left (\frac {2 b e \log (c+d x)}{a+b x}-\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{e g (d f-c g)}-\frac {\left (4 B^2 (b c-a d)\right ) \int \left (\frac {2 b e \log (f+g x)}{a+b x}-\frac {2 d e \log (f+g x)}{c+d x}\right ) \, dx}{e (b f-a g) (d f-c g)}\\ &=\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (8 b^2 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{g (b f-a g)}+\frac {\left (8 b B^2 d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{g (b f-a g)}+\frac {\left (8 b B^2 d\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{g (d f-c g)}-\frac {\left (8 B^2 d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{g (d f-c g)}-\frac {\left (8 b B^2 (b c-a d)\right ) \int \frac {\log (f+g x)}{a+b x} \, dx}{(b f-a g) (d f-c g)}+\frac {\left (8 B^2 d (b c-a d)\right ) \int \frac {\log (f+g x)}{c+d x} \, dx}{(b f-a g) (d f-c g)}\\ &=\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}+\frac {8 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {8 B^2 (b c-a d) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (8 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{g (b f-a g)}-\frac {\left (8 b^2 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{g (b f-a g)}-\frac {\left (8 B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{g (d f-c g)}-\frac {\left (8 B^2 d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{g (d f-c g)}+\frac {\left (8 B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}-\frac {\left (8 B^2 (b c-a d) g\right ) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx}{(b f-a g) (d f-c g)}\\ &=-\frac {4 b B^2 \log ^2(a+b x)}{g (b f-a g)}+\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}-\frac {4 B^2 d \log ^2(c+d x)}{g (d f-c g)}+\frac {8 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {8 B^2 (b c-a d) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}-\frac {\left (8 b B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{g (b f-a g)}-\frac {\left (8 B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{g (d f-c g)}+\frac {\left (8 B^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g) (d f-c g)}-\frac {\left (8 B^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{(b f-a g) (d f-c g)}\\ &=-\frac {4 b B^2 \log ^2(a+b x)}{g (b f-a g)}+\frac {4 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (b f-a g)}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{g (f+g x)}+\frac {8 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)}-\frac {4 B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{g (d f-c g)}-\frac {4 B^2 d \log ^2(c+d x)}{g (d f-c g)}+\frac {8 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g (b f-a g)}-\frac {8 B^2 (b c-a d) \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {4 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g) (d f-c g)}+\frac {8 b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{g (b f-a g)}+\frac {8 B^2 d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{g (d f-c g)}-\frac {8 B^2 (b c-a d) \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{(b f-a g) (d f-c g)}+\frac {8 B^2 (b c-a d) \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{(b f-a g) (d f-c g)}\\ \end {align*}
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Mathematica [B] time = 0.59, size = 409, normalized size = 2.04 \[ \frac {\frac {4 B \left (b \log (a+b x) (d f-c g) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-d (b f-a g) \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+g (b c-a d) \log (f+g x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-b B (d f-c g) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+B d (b f-a g) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B g (b c-a d) \left (\log (f+g x) \left (\log \left (\frac {g (a+b x)}{a g-b f}\right )-\log \left (\frac {g (c+d x)}{c g-d f}\right )\right )+\text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )-\text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g) (d f-c g)}-\frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{f+g x}}{g} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, A B \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + A^{2}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (\frac {\left (b x +a \right )^{2} e}{\left (d x +c \right )^{2}}\right )+A \right )^{2}}{\left (g x +f \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, A B {\left (\frac {2 \, b \log \left (b x + a\right )}{b f g - a g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{d f g - c g^{2}} + \frac {2 \, {\left (b c - a d\right )} \log \left (g x + f\right )}{b d f^{2} + a c g^{2} - {\left (b c + a d\right )} f g} - \frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{g^{2} x + f g}\right )} - B^{2} {\left (\frac {4 \, \log \left (d x + c\right )^{2}}{g^{2} x + f g} + \int -\frac {d g x \log \relax (e)^{2} + c g \log \relax (e)^{2} + 4 \, {\left (d g x + c g\right )} \log \left (b x + a\right )^{2} + 4 \, {\left (d g x \log \relax (e) + c g \log \relax (e)\right )} \log \left (b x + a\right ) - 4 \, {\left ({\left (g \log \relax (e) - 2 \, g\right )} d x + c g \log \relax (e) - 2 \, d f + 2 \, {\left (d g x + c g\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{d g^{3} x^{3} + c f^{2} g + {\left (2 \, d f g^{2} + c g^{3}\right )} x^{2} + {\left (d f^{2} g + 2 \, c f g^{2}\right )} x}\,{d x}\right )} - \frac {A^{2}}{g^{2} x + f g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2}{{\left (f+g\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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