Optimal. Leaf size=93 \[ -\frac {i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {B i^2 n (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)} \]
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Rubi [B] time = 0.52, antiderivative size = 301, normalized size of antiderivative = 3.24, 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} \[ -\frac {d^2 i^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)}-\frac {d i^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)^2}-\frac {i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 g^4 (a+b x)^3}-\frac {B d^3 i^2 n \log (a+b x)}{3 b^3 g^4 (b c-a d)}+\frac {B d^3 i^2 n \log (c+d x)}{3 b^3 g^4 (b c-a d)}-\frac {B d i^2 n (b c-a d)}{3 b^3 g^4 (a+b x)^2}-\frac {B i^2 n (b c-a d)^2}{9 b^3 g^4 (a+b x)^3}-\frac {B d^2 i^2 n}{3 b^3 g^4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(124 c+124 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)^4} \, dx &=\int \left (\frac {15376 (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^4 (a+b x)^4}+\frac {30752 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^4 (a+b x)^3}+\frac {15376 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (15376 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 g^4}+\frac {(30752 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)^3} \, dx}{b^2 g^4}+\frac {\left (15376 (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)^4} \, dx}{b^2 g^4}\\ &=-\frac {15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {15376 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {15376 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (15376 B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {(15376 B d (b c-a d) n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (15376 B (b c-a d)^2 n\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {15376 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {15376 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (15376 B d^2 (b c-a d) n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (15376 B d (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (15376 B (b c-a d)^3 n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {15376 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {15376 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac {\left (15376 B d^2 (b c-a d) n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac {\left (15376 B d (b c-a d)^2 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}{b^3 g^4}+\frac {\left (15376 B (b c-a d)^3 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^4}\\ &=-\frac {15376 B (b c-a d)^2 n}{9 b^3 g^4 (a+b x)^3}-\frac {15376 B d (b c-a d) n}{3 b^3 g^4 (a+b x)^2}-\frac {15376 B d^2 n}{3 b^3 g^4 (a+b x)}-\frac {15376 B d^3 n \log (a+b x)}{3 b^3 (b c-a d) g^4}-\frac {15376 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^4 (a+b x)^3}-\frac {15376 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}-\frac {15376 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}+\frac {15376 B d^3 n \log (c+d x)}{3 b^3 (b c-a d) g^4}\\ \end {align*}
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Mathematica [B] time = 0.32, size = 329, normalized size = 3.54 \[ -\frac {i^2 \left (-3 a^3 A d^3-3 a^3 B d^3 n \log (c+d x)-a^3 B d^3 n-9 a^2 A b d^3 x+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-9 a^2 b B d^3 n x \log (c+d x)-3 a^2 b B d^3 n x-9 a A b^2 d^3 x^2-9 a b^2 B d^3 n x^2 \log (c+d x)-3 a b^2 B d^3 n x^2+3 B d^3 n (a+b x)^3 \log (a+b x)+3 A b^3 c^3+9 A b^3 c^2 d x+9 A b^3 c d^2 x^2+b^3 B c^3 n+3 b^3 B c^2 d n x-3 b^3 B d^3 n x^3 \log (c+d x)+3 b^3 B c d^2 n x^2\right )}{9 b^3 g^4 (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 409, normalized size = 4.40 \[ -\frac {{\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c^{3} - A a^{3} d^{3}\right )} i^{2} + 3 \, {\left ({\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c d^{2} - A a b^{2} d^{3}\right )} i^{2}\right )} x^{2} + 3 \, {\left ({\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c^{2} d - A a^{2} b d^{3}\right )} i^{2}\right )} x + 3 \, {\left (3 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} x + {\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2}\right )} \log \relax (e) + 3 \, {\left (B b^{3} d^{3} i^{2} n x^{3} + 3 \, B b^{3} c d^{2} i^{2} n x^{2} + 3 \, B b^{3} c^{2} d i^{2} n x + B b^{3} c^{3} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{9 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 53.75, size = 94, normalized size = 1.01 \[ \frac {1}{9} \, {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} {\left (\frac {3 \, {\left (d x + c\right )}^{3} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{3} g^{4}} + \frac {{\left (B n + 3 \, A + 3 \, B\right )} {\left (d x + c\right )}^{3}}{{\left (b x + a\right )}^{3} g^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.94, size = 1544, normalized size = 16.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.62, size = 421, normalized size = 4.53 \[ -\frac {x\,\left (3\,A\,a\,b\,d^2\,i^2+3\,A\,b^2\,c\,d\,i^2+B\,a\,b\,d^2\,i^2\,n+B\,b^2\,c\,d\,i^2\,n\right )+x^2\,\left (3\,A\,b^2\,d^2\,i^2+B\,b^2\,d^2\,i^2\,n\right )+A\,a^2\,d^2\,i^2+A\,b^2\,c^2\,i^2+\frac {B\,a^2\,d^2\,i^2\,n}{3}+\frac {B\,b^2\,c^2\,i^2\,n}{3}+A\,a\,b\,c\,d\,i^2+\frac {B\,a\,b\,c\,d\,i^2\,n}{3}}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^3}+\frac {B\,c\,d\,i^2}{3\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^3}+\frac {B\,c\,d\,i^2}{3\,b^2}\right )+\frac {2\,B\,a\,d^2\,i^2}{3\,b^2}+\frac {2\,B\,c\,d\,i^2}{3\,b}\right )+\frac {B\,c^2\,i^2}{3\,b}+\frac {B\,d^2\,i^2\,x^2}{b}\right )}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B\,d^3\,i^2\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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