Optimal. Leaf size=137 \[ -\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{2 b (a+b x)^2}+\frac{B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{B d^2 n \log (c+d x)}{2 b (b c-a d)^2}+\frac{B d n}{2 b (a+b x) (b c-a d)}-\frac{B n}{4 b (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.154141, antiderivative size = 149, normalized size of antiderivative = 1.09, 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, 44} \[ -\frac{A}{2 b (a+b x)^2}+\frac{B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{B d n}{2 b (a+b x) (b c-a d)}-\frac{B n}{4 b (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2492
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx &=\int \left (\frac{A}{(a+b x)^3}+\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}\right ) \, dx\\ &=-\frac{A}{2 b (a+b x)^2}+B \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx\\ &=-\frac{A}{2 b (a+b x)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b}\\ &=-\frac{A}{2 b (a+b x)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}+\frac{(B (b c-a d) n) \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}\\ &=-\frac{A}{2 b (a+b x)^2}-\frac{B n}{4 b (a+b x)^2}+\frac{B d n}{2 b (b c-a d) (a+b x)}+\frac{B d^2 n \log (a+b x)}{2 b (b c-a d)^2}-\frac{B d^2 n \log (c+d x)}{2 b (b c-a d)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.338144, size = 121, normalized size = 0.88 \[ -\frac{\frac{2 A}{(a+b x)^2}+B n \left (-\frac{2 d^2 \log (a+b x)}{(b c-a d)^2}+\frac{2 d^2 \log (c+d x)}{(b c-a d)^2}+\frac{\frac{2 d (a+b x)}{a d-b c}+1}{(a+b x)^2}\right )+\frac{2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.425, size = 1379, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14743, size = 311, normalized size = 2.27 \begin{align*} \frac{{\left (\frac{2 \, d^{2} e n \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{2 \, d^{2} e n \log \left (d x + c\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac{2 \, b d e n x - b c e n + 3 \, a d e n}{a^{2} b^{2} c - a^{3} b d +{\left (b^{4} c - a b^{3} d\right )} x^{2} + 2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x}\right )} B}{4 \, e} - \frac{B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} - \frac{A}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.04422, size = 639, normalized size = 4.66 \begin{align*} -\frac{2 \, A b^{2} c^{2} - 4 \, A a b c d + 2 \, A a^{2} d^{2} - 2 \,{\left (B b^{2} c d - B a b d^{2}\right )} n x +{\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2}\right )} n - 2 \,{\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x -{\left (B b^{2} c^{2} - 2 \, B a b c d\right )} n\right )} \log \left (b x + a\right ) + 2 \,{\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x -{\left (B b^{2} c^{2} - 2 \, B a b c d\right )} n\right )} \log \left (d x + c\right ) + 2 \,{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} \log \left (e\right )}{4 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2} + 2 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x\right )}} \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 = 1.27317, size = 323, normalized size = 2.36 \begin{align*} \frac{B d^{2} n \log \left (b x + a\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac{B d^{2} n \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac{B n \log \left (b x + a\right )}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac{B n \log \left (d x + c\right )}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} + \frac{2 \, B b d n x - B b c n + 3 \, B a d n - 2 \, A b c - 2 \, B b c + 2 \, A a d + 2 \, B a d}{4 \,{\left (b^{4} c x^{2} - a b^{3} d x^{2} + 2 \, a b^{3} c x - 2 \, a^{2} b^{2} d x + a^{2} b^{2} c - a^{3} b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]