Optimal. Leaf size=189 \[ -\frac{b i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^2}+\frac{d i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^2}-\frac{b B i^2 n (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^2}+\frac{B d i^2 n (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^2} \]
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Rubi [A] time = 0.60018, antiderivative size = 340, normalized size of antiderivative = 1.8, number of steps used = 14, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^2 i^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^5 (a+b x)^2}-\frac{2 d i^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^5 (a+b x)^3}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^3 g^5 (a+b x)^4}+\frac{B d^3 i^2 n}{12 b^3 g^5 (a+b x) (b c-a d)}+\frac{B d^4 i^2 n \log (a+b x)}{12 b^3 g^5 (b c-a d)^2}-\frac{B d^4 i^2 n \log (c+d x)}{12 b^3 g^5 (b c-a d)^2}-\frac{5 B d i^2 n (b c-a d)}{36 b^3 g^5 (a+b x)^3}-\frac{B i^2 n (b c-a d)^2}{16 b^3 g^5 (a+b x)^4}-\frac{B d^2 i^2 n}{24 b^3 g^5 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{(125 c+125 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac{15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^5}+\frac{31250 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^4}+\frac{15625 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac{\left (15625 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac{(31250 d (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{\left (15625 (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac{15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac{31250 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac{15625 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac{\left (15625 B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^5}+\frac{(31250 B d (b c-a d) n) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (15625 B (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^5}\\ &=-\frac{15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac{31250 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac{15625 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac{\left (15625 B d^2 (b c-a d) n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^5}+\frac{\left (31250 B d (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac{\left (15625 B (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^5}\\ &=-\frac{15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac{31250 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac{15625 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}+\frac{\left (15625 B d^2 (b c-a d) n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^3 g^5}+\frac{\left (31250 B d (b c-a d)^2 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^5}+\frac{\left (15625 B (b c-a d)^3 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b^3 g^5}\\ &=-\frac{15625 B (b c-a d)^2 n}{16 b^3 g^5 (a+b x)^4}-\frac{78125 B d (b c-a d) n}{36 b^3 g^5 (a+b x)^3}-\frac{15625 B d^2 n}{24 b^3 g^5 (a+b x)^2}+\frac{15625 B d^3 n}{12 b^3 (b c-a d) g^5 (a+b x)}+\frac{15625 B d^4 n \log (a+b x)}{12 b^3 (b c-a d)^2 g^5}-\frac{15625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac{31250 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac{15625 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}-\frac{15625 B d^4 n \log (c+d x)}{12 b^3 (b c-a d)^2 g^5}\\ \end{align*}
Mathematica [B] time = 0.432689, size = 474, normalized size = 2.51 \[ -\frac{d^2 i^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3 g^5 (a+b x)^2}-\frac{2 d i^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^5 (a+b x)^3}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^3 g^5 (a+b x)^4}-\frac{B d^2 i^2 n \left (-\frac{2 d^2 \log (a+b x)}{(b c-a d)^2}+\frac{2 d^2 \log (c+d x)}{(b c-a d)^2}-\frac{2 d}{(a+b x) (b c-a d)}+\frac{1}{(a+b x)^2}\right )}{4 b^3 g^5}-\frac{B d i^2 n \left (\frac{6 d^2}{(a+b x) (b c-a d)}+\frac{6 d^3 \log (a+b x)}{(b c-a d)^2}-\frac{6 d^3 \log (c+d x)}{(b c-a d)^2}+\frac{2 (b c-a d)}{(a+b x)^3}-\frac{3 d}{(a+b x)^2}\right )}{9 b^3 g^5}-\frac{B i^2 n \left (-\frac{12 d^3}{(a+b x) (b c-a d)}-\frac{12 d^4 \log (a+b x)}{(b c-a d)^2}+\frac{12 d^4 \log (c+d x)}{(b c-a d)^2}-\frac{4 d (b c-a d)}{(a+b x)^3}+\frac{3 (b c-a d)^2}{(a+b x)^4}+\frac{6 d^2}{(a+b x)^2}\right )}{48 b^3 g^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.688, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \left ( bgx+ag \right ) ^{5}} \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 = 1.95532, size = 3033, normalized size = 16.05 \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 [B] time = 0.558484, size = 1446, normalized size = 7.65 \begin{align*} \frac{12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{2} n x^{3} -{\left (9 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 7 \, B a^{4} d^{4}\right )} i^{2} n - 12 \,{\left (3 \, A b^{4} c^{4} - 4 \, A a b^{3} c^{3} d + A a^{4} d^{4}\right )} i^{2} - 6 \,{\left ({\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i^{2} n + 12 \,{\left (A b^{4} c^{2} d^{2} - 2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} i^{2}\right )} x^{2} - 4 \,{\left ({\left (5 \, B b^{4} c^{3} d - 12 \, B a b^{3} c^{2} d^{2} + 7 \, B a^{3} b d^{4}\right )} i^{2} n + 12 \,{\left (2 \, A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + A a^{3} b d^{4}\right )} i^{2}\right )} x - 12 \,{\left (6 \,{\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} i^{2} x^{2} + 4 \,{\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + B a^{3} b d^{4}\right )} i^{2} x +{\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + B a^{4} d^{4}\right )} i^{2}\right )} \log \left (e\right ) + 12 \,{\left (B b^{4} d^{4} i^{2} n x^{4} + 4 \, B a b^{3} d^{4} i^{2} n x^{3} - 6 \,{\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3}\right )} i^{2} n x^{2} - 4 \,{\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2}\right )} i^{2} n x -{\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} i^{2} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{144 \,{\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x +{\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\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 [B] time = 1.31058, size = 942, normalized size = 4.98 \begin{align*} -\frac{B d^{4} n \log \left (b x + a\right )}{12 \,{\left (b^{5} c^{2} g^{5} - 2 \, a b^{4} c d g^{5} + a^{2} b^{3} d^{2} g^{5}\right )}} + \frac{B d^{4} n \log \left (d x + c\right )}{12 \,{\left (b^{5} c^{2} g^{5} - 2 \, a b^{4} c d g^{5} + a^{2} b^{3} d^{2} g^{5}\right )}} + \frac{{\left (6 \, B b^{2} d^{2} n x^{2} + 8 \, B b^{2} c d n x + 4 \, B a b d^{2} n x + 3 \, B b^{2} c^{2} n + 2 \, B a b c d n + B a^{2} d^{2} n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{12 \,{\left (b^{7} g^{5} x^{4} + 4 \, a b^{6} g^{5} x^{3} + 6 \, a^{2} b^{5} g^{5} x^{2} + 4 \, a^{3} b^{4} g^{5} x + a^{4} b^{3} g^{5}\right )}} - \frac{12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} - 20 \, B b^{3} c^{2} d n x + 28 \, B a b^{2} c d^{2} n x + 28 \, B a^{2} b d^{3} n x - 72 \, A b^{3} c d^{2} x^{2} - 72 \, B b^{3} c d^{2} x^{2} + 72 \, A a b^{2} d^{3} x^{2} + 72 \, B a b^{2} d^{3} x^{2} - 9 \, B b^{3} c^{3} n + 7 \, B a b^{2} c^{2} d n + 7 \, B a^{2} b c d^{2} n + 7 \, B a^{3} d^{3} n - 96 \, A b^{3} c^{2} d x - 96 \, B b^{3} c^{2} d x + 48 \, A a b^{2} c d^{2} x + 48 \, B a b^{2} c d^{2} x + 48 \, A a^{2} b d^{3} x + 48 \, B a^{2} b d^{3} x - 36 \, A b^{3} c^{3} - 36 \, B b^{3} c^{3} + 12 \, A a b^{2} c^{2} d + 12 \, B a b^{2} c^{2} d + 12 \, A a^{2} b c d^{2} + 12 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 12 \, B a^{3} d^{3}}{144 \,{\left (b^{8} c g^{5} x^{4} - a b^{7} d g^{5} x^{4} + 4 \, a b^{7} c g^{5} x^{3} - 4 \, a^{2} b^{6} d g^{5} x^{3} + 6 \, a^{2} b^{6} c g^{5} x^{2} - 6 \, a^{3} b^{5} d g^{5} x^{2} + 4 \, a^{3} b^{5} c g^{5} x - 4 \, a^{4} b^{4} d g^{5} x + a^{4} b^{4} c g^{5} - a^{5} b^{3} d g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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