Optimal. Leaf size=422 \[ -\frac{2 B^2 g^3 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^4}+\frac{B g^3 (b c-a d)^4 \log \left (1-\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b d^4}+\frac{B g^3 (c+d x) (b c-a d)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{d^4}-\frac{B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{2 b d^2}+\frac{B g^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b d}+\frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{4 b}-\frac{5 B^2 g^3 x (b c-a d)^3}{3 d^3}+\frac{B^2 g^3 (a+b x)^2 (b c-a d)^2}{3 b d^2}+\frac{11 B^2 g^3 (b c-a d)^4 \log (a+b x)}{3 b d^4}+\frac{5 B^2 g^3 (b c-a d)^4 \log \left (\frac{c+d x}{a+b x}\right )}{3 b d^4} \]
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Rubi [A] time = 0.740716, antiderivative size = 469, normalized size of antiderivative = 1.11, number of steps used = 24, number of rules used = 13, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.382, Rules used = {2525, 12, 2528, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{2 B^2 g^3 (b c-a d)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d^4}-\frac{B g^3 (b c-a d)^4 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b d^4}-\frac{B g^3 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{2 b d^2}+\frac{A B g^3 x (b c-a d)^3}{d^3}+\frac{B g^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b d}+\frac{g^3 (a+b x)^4 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{4 b}+\frac{B^2 g^3 (a+b x) (b c-a d)^3 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{5 B^2 g^3 x (b c-a d)^3}{3 d^3}+\frac{B^2 g^3 (a+b x)^2 (b c-a d)^2}{3 b d^2}+\frac{B^2 g^3 (b c-a d)^4 \log ^2(c+d x)}{b d^4}+\frac{11 B^2 g^3 (b c-a d)^4 \log (c+d x)}{3 b d^4}-\frac{2 B^2 g^3 (b c-a d)^4 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d^4} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 43
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{B \int \frac{2 (b c-a d) g^4 (a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{2 b g}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \frac{(a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{b}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int \left (\frac{b (b c-a d)^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d}+\frac{(-b c+a d)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b}\\ &=\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{\left (B (b c-a d) g^3\right ) \int (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{d}+\frac{\left (B (b c-a d)^2 g^3\right ) \int (a+b x) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{d^2}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{d^3}+\frac{\left (B (b c-a d)^4 g^3\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{\left (B^2 (b c-a d) g^3\right ) \int \frac{2 (-b c+a d) (a+b x)^2}{c+d x} \, dx}{3 b d}+\frac{\left (B^2 (b c-a d)^2 g^3\right ) \int \frac{2 (b c-a d) (-a-b x)}{c+d x} \, dx}{2 b d^2}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \, dx}{d^3}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{e (c+d x)^2} \, dx}{b d^4}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}+\frac{\left (2 B^2 (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b d}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \frac{-a-b x}{c+d x} \, dx}{b d^2}+\frac{\left (2 B^2 (b c-a d)^4 g^3\right ) \int \frac{1}{c+d x} \, dx}{b d^3}+\frac{\left (B^2 (b c-a d)^4 g^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{(c+d x)^2} \, dx}{b d^4 e}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}+\frac{2 B^2 (b c-a d)^4 g^3 \log (c+d x)}{b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}+\frac{\left (2 B^2 (b c-a d)^2 g^3\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b d}+\frac{\left (B^2 (b c-a d)^3 g^3\right ) \int \left (-\frac{b}{d}+\frac{b c-a d}{d (c+d x)}\right ) \, dx}{b d^2}+\frac{\left (B^2 (b c-a d)^4 g^3\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^4 e}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{3 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{3 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{3 b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{\left (2 B^2 (b c-a d)^4 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^4}+\frac{\left (2 B^2 (b c-a d)^4 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{3 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{3 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{3 b d^4}-\frac{2 B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}+\frac{\left (2 B^2 (b c-a d)^4 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b d^4}+\frac{\left (2 B^2 (b c-a d)^4 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d^3}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{3 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{3 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{3 b d^4}-\frac{2 B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^4}+\frac{B^2 (b c-a d)^4 g^3 \log ^2(c+d x)}{b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}+\frac{\left (2 B^2 (b c-a d)^4 g^3\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^4}\\ &=\frac{A B (b c-a d)^3 g^3 x}{d^3}-\frac{5 B^2 (b c-a d)^3 g^3 x}{3 d^3}+\frac{B^2 (b c-a d)^2 g^3 (a+b x)^2}{3 b d^2}+\frac{11 B^2 (b c-a d)^4 g^3 \log (c+d x)}{3 b d^4}-\frac{2 B^2 (b c-a d)^4 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^4}+\frac{B^2 (b c-a d)^4 g^3 \log ^2(c+d x)}{b d^4}+\frac{B^2 (b c-a d)^3 g^3 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{b d^3}-\frac{B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{2 b d^2}+\frac{B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d)^4 g^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{b d^4}+\frac{g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b}-\frac{2 B^2 (b c-a d)^4 g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b d^4}\\ \end{align*}
Mathematica [A] time = 0.324396, size = 402, normalized size = 0.95 \[ \frac{g^3 \left (\frac{2 B (b c-a d) \left (-6 B (b c-a d)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 d^3 (a+b x)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-6 (b c-a d)^3 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+6 A b d x (b c-a d)^2-2 B (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+12 B (b c-a d)^3 \log (c+d x)-6 B (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{3 d^4}+(a+b x)^4 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2\right )}{4 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.569, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.95392, size = 2633, normalized size = 6.24 \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 (A^{2} b^{3} g^{3} x^{3} + 3 \, A^{2} a b^{2} g^{3} x^{2} + 3 \, A^{2} a^{2} b g^{3} x + A^{2} a^{3} g^{3} +{\left (B^{2} b^{3} g^{3} x^{3} + 3 \, B^{2} a b^{2} g^{3} x^{2} + 3 \, B^{2} a^{2} b g^{3} x + B^{2} a^{3} g^{3}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \,{\left (A B b^{3} g^{3} x^{3} + 3 \, A B a b^{2} g^{3} x^{2} + 3 \, A B a^{2} b g^{3} x + A B a^{3} g^{3}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ), 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{\left (b g x + a g\right )}^{3}{\left (B \log \left (\frac{{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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