Optimal. Leaf size=142 \[ \frac{(a+b x)^4 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{4 b}-\frac{B n x (b c-a d)^3}{4 d^3}+\frac{B n (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac{B n (b c-a d)^4 \log (c+d x)}{4 b d^4}-\frac{B n (a+b x)^3 (b c-a d)}{12 b d} \]
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Rubi [A] time = 0.135311, antiderivative size = 154, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {6742, 2492, 43} \[ \frac{A (a+b x)^4}{4 b}-\frac{B n x (b c-a d)^3}{4 d^3}+\frac{B n (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac{B n (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac{B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{B n (a+b x)^3 (b c-a d)}{12 b d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)^3+B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A (a+b x)^4}{4 b}+B \int (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A (a+b x)^4}{4 b}+\frac{B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{(B (b c-a d) n) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b}\\ &=\frac{A (a+b x)^4}{4 b}+\frac{B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}-\frac{(B (b c-a d) n) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac{B (b c-a d)^3 n x}{4 d^3}+\frac{B (b c-a d)^2 n (a+b x)^2}{8 b d^2}-\frac{B (b c-a d) n (a+b x)^3}{12 b d}+\frac{A (a+b x)^4}{4 b}+\frac{B (b c-a d)^4 n \log (c+d x)}{4 b d^4}+\frac{B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.472615, size = 273, normalized size = 1.92 \[ \frac{b d x \left (9 a^2 b d^2 (4 A d x-4 B c n+B d n x)+6 a^3 d^3 (4 A+3 B n)+2 a b^2 d \left (12 A d^2 x^2+B n \left (12 c^2-6 c d x+d^2 x^2\right )\right )+b^3 \left (6 A d^3 x^3+B c n \left (-6 c^2+3 c d x-2 d^2 x^2\right )\right )\right )+6 B n \left (6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+4 a^4 d^4-4 a b^3 c^3 d+b^4 c^4\right ) \log (c+d x)+6 B d^4 \left (6 a^2 b^2 x^2+4 a^3 b x+4 a^4+4 a b^3 x^3+b^4 x^4\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-18 a^4 B d^4 n \log (a+b x)}{24 b d^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.564, size = 1840, normalized size = 13. \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 [B] time = 1.24692, size = 630, normalized size = 4.44 \begin{align*} \frac{1}{4} \, B b^{3} x^{4} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{1}{4} \, A b^{3} x^{4} + B a b^{2} x^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a b^{2} x^{3} + \frac{3}{2} \, B a^{2} b x^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac{3}{2} \, A a^{2} b x^{2} + B a^{3} x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{3} x + \frac{{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} B a^{3}}{e} - \frac{3 \,{\left (\frac{a^{2} e n \log \left (b x + a\right )}{b^{2}} - \frac{c^{2} e n \log \left (d x + c\right )}{d^{2}} + \frac{{\left (b c e n - a d e n\right )} x}{b d}\right )} B a^{2} b}{2 \, e} + \frac{{\left (\frac{2 \, a^{3} e n \log \left (b x + a\right )}{b^{3}} - \frac{2 \, c^{3} e n \log \left (d x + c\right )}{d^{3}} - \frac{{\left (b^{2} c d e n - a b d^{2} e n\right )} x^{2} - 2 \,{\left (b^{2} c^{2} e n - a^{2} d^{2} e n\right )} x}{b^{2} d^{2}}\right )} B a b^{2}}{2 \, e} - \frac{{\left (\frac{6 \, a^{4} e n \log \left (b x + a\right )}{b^{4}} - \frac{6 \, c^{4} e n \log \left (d x + c\right )}{d^{4}} + \frac{2 \,{\left (b^{3} c d^{2} e n - a b^{2} d^{3} e n\right )} x^{3} - 3 \,{\left (b^{3} c^{2} d e n - a^{2} b d^{3} e n\right )} x^{2} + 6 \,{\left (b^{3} c^{3} e n - a^{3} d^{3} e n\right )} x}{b^{3} d^{3}}\right )} B b^{3}}{24 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.06623, size = 871, normalized size = 6.13 \begin{align*} \frac{6 \, A b^{4} d^{4} x^{4} + 2 \,{\left (12 \, A a b^{3} d^{4} -{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n\right )} x^{3} + 3 \,{\left (12 \, A a^{2} b^{2} d^{4} +{\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + 3 \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + 6 \,{\left (4 \, A a^{3} b d^{4} -{\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - 3 \, B a^{3} b d^{4}\right )} n\right )} x + 6 \,{\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x + B a^{4} d^{4} n\right )} \log \left (b x + a\right ) - 6 \,{\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x -{\left (B b^{4} c^{4} - 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}\right )} n\right )} \log \left (d x + c\right ) + 6 \,{\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x\right )} \log \left (e\right )}{24 \, b 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 [B] time = 5.42466, size = 479, normalized size = 3.37 \begin{align*} \frac{B a^{4} n \log \left (b x + a\right )}{4 \, b} + \frac{1}{4} \,{\left (A b^{3} + B b^{3}\right )} x^{4} - \frac{{\left (B b^{3} c n - B a b^{2} d n - 12 \, A a b^{2} d - 12 \, B a b^{2} d\right )} x^{3}}{12 \, d} + \frac{1}{4} \,{\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (b x + a\right ) - \frac{1}{4} \,{\left (B b^{3} n x^{4} + 4 \, B a b^{2} n x^{3} + 6 \, B a^{2} b n x^{2} + 4 \, B a^{3} n x\right )} \log \left (d x + c\right ) + \frac{{\left (B b^{3} c^{2} n - 4 \, B a b^{2} c d n + 3 \, B a^{2} b d^{2} n + 12 \, A a^{2} b d^{2} + 12 \, B a^{2} b d^{2}\right )} x^{2}}{8 \, d^{2}} - \frac{{\left (B b^{3} c^{3} n - 4 \, B a b^{2} c^{2} d n + 6 \, B a^{2} b c d^{2} n - 3 \, B a^{3} d^{3} n - 4 \, A a^{3} d^{3} - 4 \, B a^{3} d^{3}\right )} x}{4 \, d^{3}} + \frac{{\left (B b^{3} c^{4} n - 4 \, B a b^{2} c^{3} d n + 6 \, B a^{2} b c^{2} d^{2} n - 4 \, B a^{3} c d^{3} n\right )} \log \left (d x + c\right )}{4 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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