Optimal. Leaf size=352 \[ \frac{b^2 g^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}+\frac{g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3}-\frac{b g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3}-\frac{B g^2 i^2 n x (b c-a d)^4}{30 b^2 d^2}-\frac{B g^2 i^2 n (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{30 b^3 d^3}-\frac{B g^2 i^2 n (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac{B g^2 i^2 n (c+d x)^2 (b c-a d)^3}{60 b d^3}+\frac{B g^2 i^2 n (c+d x)^3 (b c-a d)^2}{10 d^3}-\frac{b B g^2 i^2 n (c+d x)^4 (b c-a d)}{20 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.543196, antiderivative size = 310, normalized size of antiderivative = 0.88, 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, 43} \[ \frac{d^2 g^2 i^2 (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3}+\frac{g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3}+\frac{d g^2 i^2 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^3}+\frac{B g^2 i^2 n x (b c-a d)^4}{30 b^2 d^2}-\frac{B g^2 i^2 n (b c-a d)^5 \log (c+d x)}{30 b^3 d^3}-\frac{B g^2 i^2 n (a+b x)^2 (b c-a d)^3}{60 b^3 d}-\frac{B g^2 i^2 n (a+b x)^3 (b c-a d)^2}{10 b^3}-\frac{B d g^2 i^2 n (a+b x)^4 (b c-a d)}{20 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2525
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (118 c+118 d x)^2 (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac{(-b c+a d)^2 g^2 (118 c+118 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}-\frac{b (b c-a d) g^2 (118 c+118 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{59 d^2}+\frac{b^2 g^2 (118 c+118 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{13924 d^2}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int (118 c+118 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{13924 d^2}-\frac{\left (b (b c-a d) g^2\right ) \int (118 c+118 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{59 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int (118 c+118 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}\\ &=\frac{13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac{13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{\left (b^2 B g^2 n\right ) \int \frac{22877577568 (b c-a d) (c+d x)^4}{a+b x} \, dx}{8215160 d^3}+\frac{\left (b B (b c-a d) g^2 n\right ) \int \frac{193877776 (b c-a d) (c+d x)^3}{a+b x} \, dx}{27848 d^3}-\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{1643032 (b c-a d) (c+d x)^2}{a+b x} \, dx}{354 d^3}\\ &=\frac{13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac{13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{\left (13924 b^2 B (b c-a d) g^2 n\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^3}+\frac{\left (6962 b B (b c-a d)^2 g^2 n\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^3}-\frac{\left (13924 B (b c-a d)^3 g^2 n\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 d^3}\\ &=\frac{13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac{13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}-\frac{\left (13924 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^3}+\frac{\left (6962 b B (b c-a d)^2 g^2 n\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{d^3}-\frac{\left (13924 B (b c-a d)^3 g^2 n\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 d^3}\\ &=-\frac{6962 B (b c-a d)^4 g^2 n x}{15 b^2 d^2}-\frac{3481 B (b c-a d)^3 g^2 n (c+d x)^2}{15 b d^3}+\frac{6962 B (b c-a d)^2 g^2 n (c+d x)^3}{5 d^3}-\frac{3481 b B (b c-a d) g^2 n (c+d x)^4}{5 d^3}-\frac{6962 B (b c-a d)^5 g^2 n \log (a+b x)}{15 b^3 d^3}+\frac{13924 (b c-a d)^2 g^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac{6962 b (b c-a d) g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac{13924 b^2 g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}\\ \end{align*}
Mathematica [A] time = 0.26903, size = 374, normalized size = 1.06 \[ \frac{g^2 i^2 \left (12 d^5 (a+b x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+30 d^4 (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+20 d^3 (a+b x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+10 B n (b c-a d)^3 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-5 B n (b c-a d)^2 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+B n (b c-a d) \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )\right )}{60 b^3 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.59, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \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 [B] time = 1.44025, size = 1804, normalized size = 5.12 \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 = 0.804151, size = 1573, normalized size = 4.47 \begin{align*} \frac{12 \, A b^{5} d^{5} g^{2} i^{2} x^{5} + 2 \,{\left (10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{2} i^{2} n \log \left (b x + a\right ) - 2 \,{\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} g^{2} i^{2} n \log \left (d x + c\right ) - 3 \,{\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{2} i^{2} n - 10 \,{\left (A b^{5} c d^{4} + A a b^{4} d^{5}\right )} g^{2} i^{2}\right )} x^{4} - 2 \,{\left (3 \,{\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} n - 10 \,{\left (A b^{5} c^{2} d^{3} + 4 \, A a b^{4} c d^{4} + A a^{2} b^{3} d^{5}\right )} g^{2} i^{2}\right )} x^{3} -{\left ({\left (B b^{5} c^{3} d^{2} + 15 \, B a b^{4} c^{2} d^{3} - 15 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{2} i^{2} n - 60 \,{\left (A a b^{4} c^{2} d^{3} + A a^{2} b^{3} c d^{4}\right )} g^{2} i^{2}\right )} x^{2} + 2 \,{\left (30 \, A a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} +{\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{2} i^{2} n\right )} x + 2 \,{\left (6 \, B b^{5} d^{5} g^{2} i^{2} x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} x + 15 \,{\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 10 \,{\left (B b^{5} c^{2} d^{3} + 4 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} + 30 \,{\left (B a b^{4} c^{2} d^{3} + B a^{2} b^{3} c d^{4}\right )} g^{2} i^{2} x^{2}\right )} \log \left (e\right ) + 2 \,{\left (6 \, B b^{5} d^{5} g^{2} i^{2} n x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} n x + 15 \,{\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} n x^{4} + 10 \,{\left (B b^{5} c^{2} d^{3} + 4 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} n x^{3} + 30 \,{\left (B a b^{4} c^{2} d^{3} + B a^{2} b^{3} c d^{4}\right )} g^{2} i^{2} n x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{60 \, b^{3} d^{3}} \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]