Optimal. Leaf size=243 \[ \frac {b^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {b^2 B \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g i^3 (b c-a d)^3}-\frac {B \left (4 b-\frac {d (a+b x)}{c+d x}\right )^2}{4 g i^3 (b c-a d)^3} \]
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Rubi [C] time = 0.90, antiderivative size = 535, normalized size of antiderivative = 2.20, number of steps used = 28, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44} \[ \frac {b^2 B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {b^2 B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {b^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}-\frac {b^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (c+d x) (b c-a d)^2}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 g i^3 (c+d x)^2 (b c-a d)}-\frac {b^2 B \log ^2(a+b x)}{2 g i^3 (b c-a d)^3}-\frac {b^2 B \log ^2(c+d x)}{2 g i^3 (b c-a d)^3}-\frac {3 b^2 B \log (a+b x)}{2 g i^3 (b c-a d)^3}+\frac {b^2 B \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {3 b^2 B \log (c+d x)}{2 g i^3 (b c-a d)^3}+\frac {b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}-\frac {3 b B}{2 g i^3 (c+d x) (b c-a d)^2}-\frac {B}{4 g i^3 (c+d x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(51 c+51 d x)^3 (a g+b g x)} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d) g (c+d x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\frac {b^3 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{132651 (b c-a d)^3 g}-\frac {\left (b^2 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{132651 (b c-a d)^3 g}-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{132651 (b c-a d)^2 g}-\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{132651 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{132651 (b c-a d)^3 g}+\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{132651 (b c-a d)^3 g}-\frac {(b B) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{132651 (b c-a d)^2 g}-\frac {B \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{265302 (b c-a d) g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {B \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{265302 g}-\frac {(b B) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{132651 (b c-a d) g}-\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{132651 (b c-a d)^3 e g}+\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{132651 (b c-a d)^3 e g}\\ &=\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {B \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{265302 g}-\frac {(b B) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{132651 (b c-a d) g}-\frac {\left (b^2 B\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{132651 (b c-a d)^3 e g}+\frac {\left (b^2 B\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{132651 (b c-a d)^3 e g}\\ &=-\frac {B}{530604 (b c-a d) g (c+d x)^2}-\frac {b B}{88434 (b c-a d)^2 g (c+d x)}-\frac {b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {\left (b^3 B\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{132651 (b c-a d)^3 g}+\frac {\left (b^3 B\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{132651 (b c-a d)^3 g}+\frac {\left (b^2 B d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{132651 (b c-a d)^3 g}\\ &=-\frac {B}{530604 (b c-a d) g (c+d x)^2}-\frac {b B}{88434 (b c-a d)^2 g (c+d x)}-\frac {b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac {b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}+\frac {b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{132651 (b c-a d)^3 g}-\frac {\left (b^3 B\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{132651 (b c-a d)^3 g}\\ &=-\frac {B}{530604 (b c-a d) g (c+d x)^2}-\frac {b B}{88434 (b c-a d)^2 g (c+d x)}-\frac {b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}-\frac {b^2 B \log ^2(a+b x)}{265302 (b c-a d)^3 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac {b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {b^2 B \log ^2(c+d x)}{265302 (b c-a d)^3 g}+\frac {b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{132651 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{132651 (b c-a d)^3 g}\\ &=-\frac {B}{530604 (b c-a d) g (c+d x)^2}-\frac {b B}{88434 (b c-a d)^2 g (c+d x)}-\frac {b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}-\frac {b^2 B \log ^2(a+b x)}{265302 (b c-a d)^3 g}+\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac {b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac {b^2 B \log ^2(c+d x)}{265302 (b c-a d)^3 g}+\frac {b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}+\frac {b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 418, normalized size = 1.72 \[ \frac {4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 b^2 B (c+d x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{4 g i^3 (c+d x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 355, normalized size = 1.46 \[ \frac {{\left (6 \, A - 7 \, B\right )} b^{2} c^{2} - 8 \, {\left (A - B\right )} a b c d + {\left (2 \, A - B\right )} a^{2} d^{2} + 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + B b^{2} c^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (2 \, A - 3 \, B\right )} b^{2} c d - {\left (2 \, A - 3 \, B\right )} a b d^{2}\right )} x + 2 \, {\left ({\left (2 \, A - 3 \, B\right )} b^{2} d^{2} x^{2} + 2 \, A b^{2} c^{2} - 4 \, B a b c d + B a^{2} d^{2} + 2 \, {\left (2 \, {\left (A - B\right )} b^{2} c d - B a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 340, normalized size = 1.40 \[ \frac {{\left (2 \, B b^{2} i e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 4 \, A b^{2} i e^{2} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {8 \, {\left (b x e + a e\right )} B b d i e \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {8 \, {\left (b x e + a e\right )} A b d i e}{d x + c} + \frac {8 \, {\left (b x e + a e\right )} B b d i e}{d x + c} + \frac {2 \, {\left (b x e + a e\right )}^{2} B d^{2} i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {2 \, {\left (b x e + a e\right )}^{2} A d^{2} i}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x e + a e\right )}^{2} B d^{2} i}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (b^{2} c^{2} g e - 2 \, a b c d g e + a^{2} d^{2} g e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1287, normalized size = 5.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.69, size = 885, normalized size = 3.64 \[ \frac {1}{2} \, B {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g i^{3}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {1}{2} \, A {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g i^{3}}\right )} - \frac {{\left (7 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )^{2} + 6 \, {\left (b^{2} c d - a b d^{2}\right )} x + 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x + 3 \, b^{2} c^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{4 \, {\left (b^{3} c^{5} g i^{3} - 3 \, a b^{2} c^{4} d g i^{3} + 3 \, a^{2} b c^{3} d^{2} g i^{3} - a^{3} c^{2} d^{3} g i^{3} + {\left (b^{3} c^{3} d^{2} g i^{3} - 3 \, a b^{2} c^{2} d^{3} g i^{3} + 3 \, a^{2} b c d^{4} g i^{3} - a^{3} d^{5} g i^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d g i^{3} - 3 \, a b^{2} c^{3} d^{2} g i^{3} + 3 \, a^{2} b c^{2} d^{3} g i^{3} - a^{3} c d^{4} g i^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.08, size = 545, normalized size = 2.24 \[ \frac {3\,A\,b\,c}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {A\,a\,d}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,b^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3}+\frac {B\,a\,d}{4\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {7\,B\,b\,c}{4\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,a^2\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,c^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {A\,b\,d\,x}{g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b\,d\,x}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}-\frac {B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {2\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3}-\frac {B\,b^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.99, size = 889, normalized size = 3.66 \[ - \frac {B b^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d^{3} g i^{3} - 6 a^{2} b c d^{2} g i^{3} + 6 a b^{2} c^{2} d g i^{3} - 2 b^{3} c^{3} g i^{3}} + \frac {b^{2} \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A a b^{2} d + 2 A b^{3} c - 3 B a b^{2} d - 3 B b^{3} c - \frac {a^{4} b^{2} d^{4} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c d^{3} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{4} c^{2} d^{2} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a b^{5} c^{3} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{4} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b^{3} d - 6 B b^{3} d} \right )}}{2 g i^{3} \left (a d - b c\right )^{3}} - \frac {b^{2} \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A a b^{2} d + 2 A b^{3} c - 3 B a b^{2} d - 3 B b^{3} c + \frac {a^{4} b^{2} d^{4} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{3} c d^{3} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{4} c^{2} d^{2} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a b^{5} c^{3} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{4} \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b^{3} d - 6 B b^{3} d} \right )}}{2 g i^{3} \left (a d - b c\right )^{3}} + \frac {\left (- B a d + 3 B b c + 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} c^{2} d^{2} g i^{3} + 4 a^{2} c d^{3} g i^{3} x + 2 a^{2} d^{4} g i^{3} x^{2} - 4 a b c^{3} d g i^{3} - 8 a b c^{2} d^{2} g i^{3} x - 4 a b c d^{3} g i^{3} x^{2} + 2 b^{2} c^{4} g i^{3} + 4 b^{2} c^{3} d g i^{3} x + 2 b^{2} c^{2} d^{2} g i^{3} x^{2}} + \frac {- 2 A a d + 6 A b c + B a d - 7 B b c + x \left (4 A b d - 6 B b d\right )}{4 a^{2} c^{2} d^{2} g i^{3} - 8 a b c^{3} d g i^{3} + 4 b^{2} c^{4} g i^{3} + x^{2} \left (4 a^{2} d^{4} g i^{3} - 8 a b c d^{3} g i^{3} + 4 b^{2} c^{2} d^{2} g i^{3}\right ) + x \left (8 a^{2} c d^{3} g i^{3} - 16 a b c^{2} d^{2} g i^{3} + 8 b^{2} c^{3} d g i^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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