Optimal. Leaf size=188 \[ \frac{g^4 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d}-\frac{B g^4 n x (b c-a d)^4}{5 b^4}-\frac{B g^4 n (c+d x)^2 (b c-a d)^3}{10 b^3 d}-\frac{B g^4 n (c+d x)^3 (b c-a d)^2}{15 b^2 d}-\frac{B g^4 n (b c-a d)^5 \log (a+b x)}{5 b^5 d}-\frac{B g^4 n (c+d x)^4 (b c-a d)}{20 b d} \]
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Rubi [A] time = 0.127825, antiderivative size = 188, 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} \[ \frac{g^4 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d}-\frac{B g^4 n x (b c-a d)^4}{5 b^4}-\frac{B g^4 n (c+d x)^2 (b c-a d)^3}{10 b^3 d}-\frac{B g^4 n (c+d x)^3 (b c-a d)^2}{15 b^2 d}-\frac{B g^4 n (b c-a d)^5 \log (a+b x)}{5 b^5 d}-\frac{B g^4 n (c+d x)^4 (b c-a d)}{20 b d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac{(B n) \int \frac{(b c-a d) g^5 (c+d x)^4}{a+b x} \, dx}{5 d g}\\ &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac{\left (B (b c-a d) g^4 n\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d}\\ &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d}-\frac{\left (B (b c-a d) g^4 n\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d}\\ &=-\frac{B (b c-a d)^4 g^4 n x}{5 b^4}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2}{10 b^3 d}-\frac{B (b c-a d)^2 g^4 n (c+d x)^3}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4}{20 b d}-\frac{B (b c-a d)^5 g^4 n \log (a+b x)}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d}\\ \end{align*}
Mathematica [A] time = 0.105292, size = 146, normalized size = 0.78 \[ \frac{g^4 \left ((c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{B n (b c-a d) \left (6 b^2 (c+d x)^2 (b c-a d)^2+4 b^3 (c+d x)^3 (b c-a d)+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )}{12 b^5}\right )}{5 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.522, size = 0, normalized size = 0. \begin{align*} \int \left ( dgx+cg \right ) ^{4} \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 = 1.24272, size = 913, normalized size = 4.86 \begin{align*} \frac{1}{5} \, B d^{4} g^{4} x^{5} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + \frac{1}{5} \, A d^{4} g^{4} x^{5} + B c d^{3} g^{4} x^{4} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A c d^{3} g^{4} x^{4} + 2 \, B c^{2} d^{2} g^{4} x^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{2} d^{2} g^{4} x^{3} + 2 \, B c^{3} d g^{4} x^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{3} d g^{4} x^{2} + \frac{1}{60} \, B d^{4} g^{4} n{\left (\frac{12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac{12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac{3 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} - \frac{1}{6} \, B c d^{3} g^{4} 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 )} + B c^{2} d^{2} g^{4} 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 )} - 2 \, B c^{3} d g^{4} 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^{4} g^{4} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} + B c^{4} g^{4} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A c^{4} g^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.06928, size = 1188, normalized size = 6.32 \begin{align*} \frac{12 \, A b^{5} d^{5} g^{4} x^{5} - 12 \, B b^{5} c^{5} g^{4} n \log \left (d x + c\right ) + 12 \,{\left (5 \, B a b^{4} c^{4} d - 10 \, B a^{2} b^{3} c^{3} d^{2} + 10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{4} n \log \left (b x + a\right ) + 3 \,{\left (20 \, A b^{5} c d^{4} g^{4} -{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \,{\left (30 \, A b^{5} c^{2} d^{3} g^{4} -{\left (4 \, B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{4} n\right )} x^{3} + 6 \,{\left (20 \, A b^{5} c^{3} d^{2} g^{4} -{\left (6 \, B b^{5} c^{3} d^{2} - 10 \, B a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{4} n\right )} x^{2} + 12 \,{\left (5 \, A b^{5} c^{4} d g^{4} -{\left (4 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} + B a^{4} b d^{5}\right )} g^{4} n\right )} x + 12 \,{\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B b^{5} c d^{4} g^{4} x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} x^{2} + 5 \, B b^{5} c^{4} d g^{4} x\right )} \log \left (e\right ) + 12 \,{\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B b^{5} c d^{4} g^{4} n x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} n x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} n x^{2} + 5 \, B b^{5} c^{4} d g^{4} n x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{60 \, b^{5} 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 [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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