Optimal. Leaf size=188 \[ -\frac {2 B g (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A+2 B\right )}{b d^2}-\frac {2 B g (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b d}+\frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 b}-\frac {4 B^2 g (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2} \]
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Rubi [A] time = 0.49, antiderivative size = 291, normalized size of antiderivative = 1.55, number of steps used = 16, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac {4 B^2 g (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d^2}+\frac {2 B g (b c-a d)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b d^2}+\frac {g (a+b x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 b}-\frac {2 A B g x (b c-a d)}{d}+\frac {2 B^2 g (b c-a d)^2 \log ^2(c+d x)}{b d^2}+\frac {4 B^2 g (b c-a d)^2 \log (c+d x)}{b d^2}-\frac {4 B^2 g (b c-a d)^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b d^2}-\frac {2 B^2 g (a+b x) (b c-a d) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}-\frac {B \int \frac {2 (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{c+d x} \, dx}{b g}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}-\frac {(2 B (b c-a d) g) \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{c+d x} \, dx}{b}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}-\frac {(2 B (b c-a d) g) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d}+\frac {(-b c+a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{d (c+d x)}\right ) \, dx}{b}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}-\frac {(2 B (b c-a d) g) \int \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{d}+\frac {\left (2 B (b c-a d)^2 g\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{b d}\\ &=-\frac {2 A B (b c-a d) g x}{d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (2 B^2 (b c-a d) g\right ) \int \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \, dx}{d}-\frac {\left (2 B^2 (b c-a d)^2 g\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}{b d^2}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}+\frac {\left (4 B^2 (b c-a d)^2 g\right ) \int \frac {1}{c+d x} \, dx}{b d}-\frac {\left (2 B^2 (b c-a d)^2 g\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}{b d^2 e}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {4 B^2 (b c-a d)^2 g \log (c+d x)}{b d^2}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (2 B^2 (b c-a d)^2 g\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}{b d^2 e}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {4 B^2 (b c-a d)^2 g \log (c+d x)}{b d^2}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (4 B^2 (b c-a d)^2 g\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d^2}+\frac {\left (4 B^2 (b c-a d)^2 g\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b d}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {4 B^2 (b c-a d)^2 g \log (c+d x)}{b d^2}-\frac {4 B^2 (b c-a d)^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}+\frac {\left (4 B^2 (b c-a d)^2 g\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b d^2}+\frac {\left (4 B^2 (b c-a d)^2 g\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {4 B^2 (b c-a d)^2 g \log (c+d x)}{b d^2}-\frac {4 B^2 (b c-a d)^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}+\frac {2 B^2 (b c-a d)^2 g \log ^2(c+d x)}{b d^2}+\frac {\left (4 B^2 (b c-a d)^2 g\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b d^2}\\ &=-\frac {2 A B (b c-a d) g x}{d}-\frac {2 B^2 (b c-a d) g (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 b}+\frac {4 B^2 (b c-a d)^2 g \log (c+d x)}{b d^2}-\frac {4 B^2 (b c-a d)^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b d^2}+\frac {2 B^2 (b c-a d)^2 g \log ^2(c+d x)}{b d^2}-\frac {4 B^2 (b c-a d)^2 g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b d^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 207, normalized size = 1.10 \[ \frac {g \left ((a+b x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2-\frac {4 B (b c-a d) \left (-(b c-a d) \log (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+B d (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+B (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B (b c-a d) \log (c+d x)+A b d x\right )}{d^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ), x\right ) \]
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 \[ \int {\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right ) \left (B \ln \left (\frac {\left (b x +a \right )^{2} e}{\left (d x +c \right )^{2}}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.42, size = 727, normalized size = 3.87 \[ \frac {1}{2} \, A^{2} b g x^{2} + 2 \, {\left (x \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 ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} A B a g + {\left (x^{2} \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 ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} A B b g + A^{2} a g x + \frac {2 \, {\left ({\left (g \log \relax (e) + 2 \, g\right )} b c^{2} - 2 \, {\left (g \log \relax (e) + g\right )} a c d\right )} B^{2} \log \left (d x + c\right )}{d^{2}} + \frac {4 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{b d^{2}} + \frac {B^{2} b^{2} d^{2} g x^{2} \log \relax (e)^{2} - 2 \, {\left (2 \, b^{2} c d g \log \relax (e) - {\left (g \log \relax (e)^{2} + 2 \, g \log \relax (e)\right )} a b d^{2}\right )} B^{2} x + 4 \, {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x + B^{2} a^{2} d^{2} g\right )} \log \left (b x + a\right )^{2} + 4 \, {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x - {\left (b^{2} c^{2} g - 2 \, a b c d g\right )} B^{2}\right )} \log \left (d x + c\right )^{2} + 4 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \relax (e) + 2 \, {\left ({\left (g \log \relax (e) + g\right )} a b d^{2} - b^{2} c d g\right )} B^{2} x + {\left ({\left (g \log \relax (e) + 2 \, g\right )} a^{2} d^{2} - 2 \, a b c d g\right )} B^{2}\right )} \log \left (b x + a\right ) - 4 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \relax (e) + 2 \, {\left ({\left (g \log \relax (e) + g\right )} a b d^{2} - b^{2} c d g\right )} B^{2} x + 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x + B^{2} a^{2} d^{2} g\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{2 \, b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a\,g+b\,g\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2 \,d x \]
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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