Optimal. Leaf size=156 \[ \frac{g^3 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d}-\frac{B g^3 n x (b c-a d)^3}{4 b^3}-\frac{B g^3 n (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac{B g^3 n (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac{B g^3 n (c+d x)^3 (b c-a d)}{12 b d} \]
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Rubi [A] time = 0.1021, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 43} \[ \frac{g^3 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d}-\frac{B g^3 n x (b c-a d)^3}{4 b^3}-\frac{B g^3 n (c+d x)^2 (b c-a d)^2}{8 b^2 d}-\frac{B g^3 n (b c-a d)^4 \log (a+b x)}{4 b^4 d}-\frac{B g^3 n (c+d x)^3 (b c-a d)}{12 b d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (c g+d g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac{(B n) \int \frac{(b c-a d) g^4 (c+d x)^3}{a+b x} \, dx}{4 d g}\\ &=\frac{g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac{\left (B (b c-a d) g^3 n\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d}\\ &=\frac{g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d}-\frac{\left (B (b c-a d) g^3 n\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{4 d}\\ &=-\frac{B (b c-a d)^3 g^3 n x}{4 b^3}-\frac{B (b c-a d)^2 g^3 n (c+d x)^2}{8 b^2 d}-\frac{B (b c-a d) g^3 n (c+d x)^3}{12 b d}-\frac{B (b c-a d)^4 g^3 n \log (a+b x)}{4 b^4 d}+\frac{g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0937343, size = 124, normalized size = 0.79 \[ \frac{g^3 \left ((c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{6 b^4}\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int \left ( dgx+cg \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 [B] time = 1.25501, size = 647, normalized size = 4.15 \begin{align*} \frac{1}{4} \, B d^{3} 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 d^{3} g^{3} x^{4} + B c d^{2} g^{3} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A c d^{2} g^{3} x^{3} + \frac{3}{2} \, B c^{2} d g^{3} x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{3}{2} \, A c^{2} d g^{3} x^{2} - \frac{1}{24} \, B d^{3} 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 c d^{2} g^{3} 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 c^{2} d g^{3} 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 c^{3} g^{3} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B c^{3} g^{3} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A c^{3} g^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.07087, size = 883, normalized size = 5.66 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{3} x^{4} - 6 \, B b^{4} c^{4} g^{3} n \log \left (d x + c\right ) + 6 \,{\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{3} n \log \left (b x + a\right ) + 2 \,{\left (12 \, A b^{4} c d^{3} g^{3} -{\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} c^{2} d^{2} g^{3} -{\left (3 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{3} n\right )} x^{2} + 6 \,{\left (4 \, A b^{4} c^{3} d g^{3} -{\left (3 \, B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 4 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} g^{3} n\right )} x + 6 \,{\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B b^{4} c d^{3} g^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} g^{3} x^{2} + 4 \, B b^{4} c^{3} d g^{3} x\right )} \log \left (e\right ) + 6 \,{\left (B b^{4} d^{4} g^{3} n x^{4} + 4 \, B b^{4} c d^{3} g^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} g^{3} n x^{2} + 4 \, B b^{4} c^{3} d g^{3} n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{24 \, b^{4} d} \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 [B] time = 143.004, size = 537, normalized size = 3.44 \begin{align*} -\frac{B c^{4} g^{3} n \log \left (-d x - c\right )}{4 \, d} + \frac{1}{4} \,{\left (A d^{3} g^{3} + B d^{3} g^{3}\right )} x^{4} - \frac{{\left (B b c d^{2} g^{3} n - B a d^{3} g^{3} n - 12 \, A b c d^{2} g^{3} - 12 \, B b c d^{2} g^{3}\right )} x^{3}}{12 \, b} + \frac{1}{4} \,{\left (B d^{3} g^{3} n x^{4} + 4 \, B c d^{2} g^{3} n x^{3} + 6 \, B c^{2} d g^{3} n x^{2} + 4 \, B c^{3} g^{3} n x\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (3 \, B b^{2} c^{2} d g^{3} n - 4 \, B a b c d^{2} g^{3} n + B a^{2} d^{3} g^{3} n - 12 \, A b^{2} c^{2} d g^{3} - 12 \, B b^{2} c^{2} d g^{3}\right )} x^{2}}{8 \, b^{2}} - \frac{{\left (3 \, B b^{3} c^{3} g^{3} n - 6 \, B a b^{2} c^{2} d g^{3} n + 4 \, B a^{2} b c d^{2} g^{3} n - B a^{3} d^{3} g^{3} n - 4 \, A b^{3} c^{3} g^{3} - 4 \, B b^{3} c^{3} g^{3}\right )} x}{4 \, b^{3}} + \frac{{\left (4 \, B a b^{3} c^{3} g^{3} n - 6 \, B a^{2} b^{2} c^{2} d g^{3} n + 4 \, B a^{3} b c d^{2} g^{3} n - B a^{4} d^{3} g^{3} n\right )} \log \left (b x + a\right )}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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