Optimal. Leaf size=235 \[ -\frac{B g n x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac{(f+g x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 g}-\frac{B g^2 n x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac{B n (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac{B g^3 n x^3 (b c-a d)}{12 b d}+\frac{B n (d f-c g)^4 \log (c+d x)}{4 d^4 g} \]
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Rubi [A] time = 0.359478, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 72} \[ -\frac{B g n x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac{(f+g x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 g}-\frac{B g^2 n x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac{B n (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac{B g^3 n x^3 (b c-a d)}{12 b d}+\frac{B n (d f-c g)^4 \log (c+d x)}{4 d^4 g} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 72
Rubi steps
\begin{align*} \int (f+g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{(f+g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac{(B n) \int \frac{(b c-a d) (f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac{(f+g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac{(B (b c-a d) n) \int \frac{(f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac{(f+g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac{(B (b c-a d) n) \int \left (\frac{g^2 \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )}{b^3 d^3}+\frac{g^3 (4 b d f-b c g-a d g) x}{b^2 d^2}+\frac{g^4 x^2}{b d}+\frac{(b f-a g)^4}{b^3 (b c-a d) (a+b x)}+\frac{(d f-c g)^4}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{4 g}\\ &=-\frac{B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n x}{4 b^3 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) n x^2}{8 b^2 d^2}-\frac{B (b c-a d) g^3 n x^3}{12 b d}-\frac{B (b f-a g)^4 n \log (a+b x)}{4 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 g}+\frac{B (d f-c g)^4 n \log (c+d x)}{4 d^4 g}\\ \end{align*}
Mathematica [A] time = 0.28813, size = 219, normalized size = 0.93 \[ \frac{(f+g x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n \left (6 b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f)+2 b^3 d^3 g^4 x^3 (b c-a d)+6 d^4 (b f-a g)^4 \log (a+b x)-6 b^4 (d f-c g)^4 \log (c+d x)\right )}{6 b^4 d^4}}{4 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20435, size = 598, normalized size = 2.54 \begin{align*} \frac{1}{4} \, B g^{3} x^{4} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{1}{4} \, A g^{3} x^{4} + B f g^{2} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f g^{2} x^{3} + \frac{3}{2} \, B f^{2} g x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{3}{2} \, A f^{2} g x^{2} - \frac{1}{24} \, B g^{3} n{\left (\frac{6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac{6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac{2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac{1}{2} \, B f g^{2} n{\left (\frac{2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac{3}{2} \, B f^{2} g n{\left (\frac{a^{2} \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c - a d\right )} x}{b d}\right )} + B f^{3} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B f^{3} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50081, size = 1053, normalized size = 4.48 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{3} x^{4} + 2 \,{\left (12 \, A b^{4} d^{4} f g^{2} -{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3} n\right )} x^{3} + 3 \,{\left (12 \, A b^{4} d^{4} f^{2} g -{\left (4 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f g^{2} -{\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} g^{3}\right )} n\right )} x^{2} + 6 \,{\left (4 \, B a b^{3} d^{4} f^{3} - 6 \, B a^{2} b^{2} d^{4} f^{2} g + 4 \, B a^{3} b d^{4} f g^{2} - B a^{4} d^{4} g^{3}\right )} n \log \left (b x + a\right ) - 6 \,{\left (4 \, B b^{4} c d^{3} f^{3} - 6 \, B b^{4} c^{2} d^{2} f^{2} g + 4 \, B b^{4} c^{3} d f g^{2} - B b^{4} c^{4} g^{3}\right )} n \log \left (d x + c\right ) + 6 \,{\left (4 \, A b^{4} d^{4} f^{3} -{\left (6 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f^{2} g - 4 \,{\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} f g^{2} +{\left (B b^{4} c^{3} d - B a^{3} b d^{4}\right )} g^{3}\right )} n\right )} x + 6 \,{\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B b^{4} d^{4} f g^{2} x^{3} + 6 \, B b^{4} d^{4} f^{2} g x^{2} + 4 \, B b^{4} d^{4} f^{3} x\right )} \log \left (e\right ) + 6 \,{\left (B b^{4} d^{4} g^{3} n x^{4} + 4 \, B b^{4} d^{4} f g^{2} n x^{3} + 6 \, B b^{4} d^{4} f^{2} g n x^{2} + 4 \, B b^{4} d^{4} f^{3} n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{24 \, b^{4} d^{4}} \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(-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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