Optimal. Leaf size=102 \[ \frac{A (a+b x)}{i^2 (c+d x) (b c-a d)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{i^2 (c+d x) (b c-a d)}-\frac{B n (a+b x)}{i^2 (c+d x) (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.078694, 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.
[In]
[Out]
Rule 2525
Rule 12
Rule 44
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*}
Mathematica [A] time = 0.0528881, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.518, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dix+ci \right ) ^{2}} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16073, size = 184, normalized size = 1.8 \begin{align*} 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.507928, size = 221, normalized size = 2.17 \begin{align*} -\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 \left (e\right ) -{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.79707, size = 134, normalized size = 1.31 \begin{align*} -\frac{B b n \log \left (b x + a\right )}{b c d - a d^{2}} + \frac{B b n \log \left (d x + c\right )}{b c d - a d^{2}} + \frac{B n \log \left (\frac{b x + a}{d x + c}\right )}{d^{2} x + c d} - \frac{B n - A - B}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]