Optimal. Leaf size=275 \[ \frac {b g^2 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{d^3 i^2}+\frac {g^2 (a+b x) (2 A+B n) (b c-a d)}{d^2 i^2 (c+d x)}+\frac {g^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (c+d x)}+\frac {2 b B g^2 n (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {2 B g^2 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^2 (c+d x)}-\frac {2 B g^2 n (a+b x) (b c-a d)}{d^2 i^2 (c+d x)} \]
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Rubi [A] time = 0.53, antiderivative size = 351, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 13, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {2 b B g^2 n (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^3 i^2}-\frac {g^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^2 (c+d x)}-\frac {2 b g^2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^2}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^2}+\frac {B g^2 n (b c-a d)^2}{d^3 i^2 (c+d x)}-\frac {b B g^2 n (b c-a d) \log ^2(c+d x)}{d^3 i^2}+\frac {b B g^2 n (b c-a d) \log (a+b x)}{d^3 i^2}-\frac {2 b B g^2 n (b c-a d) \log (c+d x)}{d^3 i^2}+\frac {2 b B g^2 n (b c-a d) \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^3 i^2}+\frac {A b^2 g^2 x}{d^2 i^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 44
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 \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(144 c+144 d x)^2} \, dx &=\int \left (\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^2}+\frac {(-b c+a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^2 (c+d x)^2}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10368 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{20736 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{10368 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{20736 d^2}\\ &=\frac {A b^2 g^2 x}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}+\frac {\left (b^2 B g^2\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{20736 d^2}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{10368 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {1}{c+d x} \, dx}{20736 d^2}+\frac {\left (B (b c-a d)^3 g^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{20736 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}+\frac {\left (b^2 B (b c-a d) g^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{10368 d^2}+\frac {\left (B (b c-a d)^3 g^2 n\right ) \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}{20736 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{10368 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{10368 d^2}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}-\frac {b B (b c-a d) g^2 n \log ^2(c+d x)}{20736 d^3}-\frac {\left (b B (b c-a d) g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{10368 d^3}\\ &=\frac {A b^2 g^2 x}{20736 d^2}+\frac {B (b c-a d)^2 g^2 n}{20736 d^3 (c+d x)}+\frac {b B (b c-a d) g^2 n \log (a+b x)}{20736 d^3}+\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{20736 d^2}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20736 d^3 (c+d x)}-\frac {b B (b c-a d) g^2 n \log (c+d x)}{10368 d^3}+\frac {b B (b c-a d) g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{10368 d^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{10368 d^3}-\frac {b B (b c-a d) g^2 n \log ^2(c+d x)}{20736 d^3}+\frac {b B (b c-a d) g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{10368 d^3}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 252, normalized size = 0.92 \[ \frac {g^2 \left (-2 b (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {(b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+b B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b B n (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+\frac {B n (b c-a d)^2}{c+d x}+b B n (b c-a d) \log (a+b x)-2 b B n (b c-a d) \log (c+d x)+A b^2 d x\right )}{d^3 i^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} + {\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (d i x +c i \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.92, size = 1273, normalized size = 4.63 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \]
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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