Optimal. Leaf size=102 \[ \frac {A (a+b x)}{i^2 (c+d x) (b c-a d)}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{i^2 (c+d x) (b c-a d)}-\frac {B n (a+b x)}{i^2 (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.05, 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, 44} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{d i^2 (c+d x)}+\frac {b B n \log (a+b x)}{d i^2 (b c-a d)}-\frac {b B n \log (c+d x)}{d i^2 (b c-a d)}+\frac {B n}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(146 c+146 d x)^2} \, dx &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{21316 d (c+d x)}+\frac {(B n) \int \frac {b c-a d}{146 (a+b x) (c+d x)^2} \, dx}{146 d}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{21316 d (c+d x)}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{21316 d}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{21316 d (c+d x)}+\frac {(B (b c-a d) n) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{21316 d}\\ &=\frac {B n}{21316 d (c+d x)}+\frac {b B n \log (a+b x)}{21316 d (b c-a d)}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{21316 d (c+d x)}-\frac {b B n \log (c+d x)}{21316 d (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 114, normalized size = 1.12 \[ \frac {B n (b c-a d) \left (\frac {1}{(c+d x) (b c-a d)}+\frac {b \log (a+b x)}{(b c-a d)^2}-\frac {b \log (c+d x)}{(b c-a d)^2}\right )}{d i^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{d i (c i+d i x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 105, normalized size = 1.03 \[ -\frac {A b c - A a d - {\left (B b c - B a d\right )} n + {\left (B b c - B a d\right )} \log \relax (e) - {\left (B b d n x + B a d n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.71, size = 84, normalized size = 0.82 \[ -{\left (\frac {{\left (b x + a\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{d x + c} - \frac {{\left (B n - A - B\right )} {\left (b x + a\right )}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (d i x +c i \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 136, normalized size = 1.33 \[ B n {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {A}{d^{2} i^{2} x + c d i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.84, size = 113, normalized size = 1.11 \[ -\frac {A-B\,n}{x\,d^2\,i^2+c\,d\,i^2}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{d\,\left (c\,i^2+d\,i^2\,x\right )}+\frac {B\,b\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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