Optimal. Leaf size=230 \[ \frac{B d^2 i^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac{d^2 i^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac{d i^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac{i^2 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac{B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac{B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2} \]
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Rubi [A] time = 0.589218, antiderivative size = 338, normalized size of antiderivative = 1.47, number of steps used = 19, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B d^2 i^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^3}+\frac{d^2 i^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac{2 d i^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3 (a+b x)}-\frac{i^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^3 g^3 (a+b x)^2}+\frac{B d^2 i^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{3 B d i^2 (b c-a d)}{2 b^3 g^3 (a+b x)}-\frac{B i^2 (b c-a d)^2}{4 b^3 g^3 (a+b x)^2}-\frac{B d^2 i^2 \log ^2(a+b x)}{2 b^3 g^3}-\frac{3 B d^2 i^2 \log (a+b x)}{2 b^3 g^3}+\frac{3 B d^2 i^2 \log (c+d x)}{2 b^3 g^3} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(16 c+16 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac{256 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^3}+\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^2}+\frac{256 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac{\left (256 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}+\frac{(512 d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac{\left (256 (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac{\left (256 B d^2\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}{b^3 g^3}+\frac{(512 B d (b c-a d)) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac{\left (128 B (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{\left (512 B d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac{\left (128 B (b c-a d)^3\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}-\frac{\left (256 B d^2\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}{b^3 e g^3}\\ &=-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{\left (512 B d (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^3}+\frac{\left (128 B (b c-a d)^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^3}-\frac{\left (256 B d^2\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}{b^3 e g^3}\\ &=-\frac{64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac{384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac{384 B d^2 \log (a+b x)}{b^3 g^3}-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{384 B d^2 \log (c+d x)}{b^3 g^3}-\frac{\left (256 B d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac{\left (256 B d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^3}\\ &=-\frac{64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac{384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac{384 B d^2 \log (a+b x)}{b^3 g^3}-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{384 B d^2 \log (c+d x)}{b^3 g^3}+\frac{256 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{\left (256 B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac{\left (256 B d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}\\ &=-\frac{64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac{384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac{384 B d^2 \log (a+b x)}{b^3 g^3}-\frac{128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{384 B d^2 \log (c+d x)}{b^3 g^3}+\frac{256 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac{\left (256 B d^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 )}{b^3 g^3}\\ &=-\frac{64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac{384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac{384 B d^2 \log (a+b x)}{b^3 g^3}-\frac{128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac{128 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac{512 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac{256 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac{384 B d^2 \log (c+d x)}{b^3 g^3}+\frac{256 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac{256 B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^3}\\ \end{align*}
Mathematica [A] time = 0.332654, size = 244, normalized size = 1.06 \[ \frac{i^2 \left (-2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+4 d^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{8 d (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac{2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+\frac{6 B d (a d-b c)}{a+b x}-\frac{B (b c-a d)^2}{(a+b x)^2}-6 B d^2 \log (a+b x)+6 B d^2 \log (c+d x)\right )}{4 b^3 g^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.07, size = 1495, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} +{\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{b^{3} g^{3} x^{3} + 3 \, a b^{2} g^{3} x^{2} + 3 \, a^{2} b g^{3} x + a^{3} g^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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