Optimal. Leaf size=223 \[ \frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A-B n\right )}{20 b^2}+\frac{g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}+\frac{B g^3 i n (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g^3 i n (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}-\frac{B g^3 i n (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac{B g^3 i n x (b c-a d)^4}{20 b d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.385909, antiderivative size = 243, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 43} \[ \frac{g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2}+\frac{d g^3 i (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^2}+\frac{B g^3 i n (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac{B g^3 i n (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}-\frac{B g^3 i n (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac{B g^3 i n (a+b x)^4 (b c-a d)}{20 b^2}-\frac{B g^3 i n x (b c-a d)^4}{20 b d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (108 c+108 d x) (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac{108 (b c-a d) (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{108 d (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g}\right ) \, dx\\ &=\frac{(108 (b c-a d)) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}+\frac{(108 d) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}\\ &=\frac{27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac{(108 B d n) \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b^2 g^2}-\frac{(27 B (b c-a d) n) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{b^2 g}\\ &=\frac{27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac{\left (108 B d (b c-a d) g^3 n\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{5 b^2}-\frac{\left (27 B (b c-a d)^2 g^3 n\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{b^2}\\ &=\frac{27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac{\left (108 B d (b c-a d) g^3 n\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^2}-\frac{\left (27 B (b c-a d)^2 g^3 n\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^2}\\ &=-\frac{27 B (b c-a d)^4 g^3 n x}{5 b d^3}+\frac{27 B (b c-a d)^3 g^3 n (a+b x)^2}{10 b^2 d^2}-\frac{9 B (b c-a d)^2 g^3 n (a+b x)^3}{5 b^2 d}-\frac{27 B (b c-a d) g^3 n (a+b x)^4}{5 b^2}+\frac{27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}+\frac{27 B (b c-a d)^5 g^3 n \log (c+d x)}{5 b^2 d^4}\\ \end{align*}
Mathematica [A] time = 0.250617, size = 269, normalized size = 1.21 \[ \frac{g^3 i \left (24 d (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+30 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{5 B n (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{d^4}+\frac{2 B n (b c-a d) \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )}{d^4}\right )}{120 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.531, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) \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 [B] time = 1.40488, size = 1509, normalized size = 6.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.847657, size = 1501, normalized size = 6.73 \begin{align*} \frac{24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \,{\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i n \log \left (b x + a\right ) + 6 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} i n \log \left (d x + c\right ) - 6 \,{\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{3} i n - 5 \,{\left (A b^{5} c d^{4} + 3 \, A a b^{4} d^{5}\right )} g^{3} i\right )} x^{4} - 2 \,{\left ({\left (B b^{5} c^{2} d^{3} + 10 \, B a b^{4} c d^{4} - 11 \, B a^{2} b^{3} d^{5}\right )} g^{3} i n - 60 \,{\left (A a b^{4} c d^{4} + A a^{2} b^{3} d^{5}\right )} g^{3} i\right )} x^{3} + 3 \,{\left ({\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + 9 \, B a^{3} b^{2} d^{5}\right )} g^{3} i n + 20 \,{\left (3 \, A a^{2} b^{3} c d^{4} + A a^{3} b^{2} d^{5}\right )} g^{3} i\right )} x^{2} + 6 \,{\left (20 \, A a^{3} b^{2} c d^{4} g^{3} i -{\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i n\right )} x + 6 \,{\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \,{\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \,{\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \,{\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (e\right ) + 6 \,{\left (4 \, B b^{5} d^{5} g^{3} i n x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i n x + 5 \,{\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i n x^{4} + 20 \,{\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i n x^{3} + 10 \,{\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i n x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{120 \, b^{2} d^{4}} \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 [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]