Optimal. Leaf size=341 \[ -\frac{3 b B g^3 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac{g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac{e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d^2 i^2 (c+d x)}-\frac{b g^3 (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac{e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{2 d^4 i^2}-\frac{g^3 (6 A+5 B) (a+b x) (b c-a d)^2}{2 d^3 i^2 (c+d x)}+\frac{g^3 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d i^2 (c+d x)}-\frac{3 B g^3 (a+b x) (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}+\frac{3 B g^3 (a+b x) (b c-a d)^2}{d^3 i^2 (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.727061, antiderivative size = 519, normalized size of antiderivative = 1.52, number of steps used = 22, number of rules used = 14, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2528, 2486, 31, 2525, 12, 72, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{3 b B g^3 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 i^2}-\frac{a^2 b B g^3 \log (a+b x)}{2 d^2 i^2}+\frac{b^3 g^3 x^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^2}-\frac{A b^2 g^3 x (2 b c-3 a d)}{d^3 i^2}+\frac{g^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^2 (c+d x)}+\frac{3 b g^3 (b c-a d)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^2}-\frac{b^2 B g^3 x (b c-a d)}{2 d^3 i^2}-\frac{b B g^3 (a+b x) (2 b c-3 a d) \log \left (\frac{e (a+b x)}{c+d x}\right )}{d^3 i^2}-\frac{B g^3 (b c-a d)^3}{d^4 i^2 (c+d x)}+\frac{3 b B g^3 (b c-a d)^2 \log ^2(c+d x)}{2 d^4 i^2}-\frac{b B g^3 (b c-a d)^2 \log (a+b x)}{d^4 i^2}-\frac{3 b B g^3 (b c-a d)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^4 i^2}+\frac{b B g^3 (b c-a d)^2 \log (c+d x)}{d^4 i^2}+\frac{b B g^3 (2 b c-3 a d) (b c-a d) \log (c+d x)}{d^4 i^2}+\frac{b^3 B c^2 g^3 \log (c+d x)}{2 d^4 i^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 44
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int \frac{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(39 c+39 d x)^2} \, dx &=\int \left (-\frac{b^2 (2 b c-3 a d) g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^3}+\frac{b^3 g^3 x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^2}+\frac{(-b c+a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^3 (c+d x)^2}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{507 d^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{1521 d^2}-\frac{\left (b^2 (2 b c-3 a d) g^3\right ) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{1521 d^3}+\frac{\left (b (b c-a d)^2 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{507 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1521 d^3}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac{\left (b^3 B g^3\right ) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{3042 d^2}-\frac{\left (b^2 B (2 b c-3 a d) g^3\right ) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx}{1521 d^3}-\frac{\left (b B (b c-a d)^2 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{507 d^4}-\frac{\left (B (b c-a d)^3 g^3\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1521 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac{\left (b^3 B (b c-a d) g^3\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{3042 d^2}+\frac{\left (b B (2 b c-3 a d) (b c-a d) g^3\right ) \int \frac{1}{c+d x} \, dx}{1521 d^3}-\frac{\left (B (b c-a d)^4 g^3\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1521 d^4}-\frac{\left (b B (b c-a d)^2 g^3\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{507 d^4 e}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac{\left (b^3 B (b c-a d) g^3\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{3042 d^2}-\frac{\left (B (b c-a d)^4 g^3\right ) \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}{1521 d^4}-\frac{\left (b B (b c-a d)^2 g^3\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{507 d^4 e}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac{B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac{a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac{b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac{b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}-\frac{\left (b^2 B (b c-a d)^2 g^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{507 d^4}+\frac{\left (b B (b c-a d)^2 g^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{507 d^3}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac{B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac{a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac{b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac{b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac{b B (b c-a d)^2 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac{\left (b B (b c-a d)^2 g^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{507 d^4}+\frac{\left (b B (b c-a d)^2 g^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{507 d^3}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac{B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac{a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac{b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac{b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac{b B (b c-a d)^2 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac{b B (b c-a d)^2 g^3 \log ^2(c+d x)}{1014 d^4}+\frac{\left (b B (b c-a d)^2 g^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{507 d^4}\\ &=-\frac{A b^2 (2 b c-3 a d) g^3 x}{1521 d^3}-\frac{b^2 B (b c-a d) g^3 x}{3042 d^3}-\frac{B (b c-a d)^3 g^3}{1521 d^4 (c+d x)}-\frac{a^2 b B g^3 \log (a+b x)}{3042 d^2}-\frac{b B (b c-a d)^2 g^3 \log (a+b x)}{1521 d^4}-\frac{b B (2 b c-3 a d) g^3 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{1521 d^3}+\frac{b^3 g^3 x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3042 d^2}+\frac{(b c-a d)^3 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1521 d^4 (c+d x)}+\frac{b^3 B c^2 g^3 \log (c+d x)}{3042 d^4}+\frac{b B (2 b c-3 a d) (b c-a d) g^3 \log (c+d x)}{1521 d^4}+\frac{b B (b c-a d)^2 g^3 \log (c+d x)}{1521 d^4}-\frac{b B (b c-a d)^2 g^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{507 d^4}+\frac{b (b c-a d)^2 g^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{507 d^4}+\frac{b B (b c-a d)^2 g^3 \log ^2(c+d x)}{1014 d^4}-\frac{b B (b c-a d)^2 g^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{507 d^4}\\ \end{align*}
Mathematica [A] time = 0.42726, size = 359, normalized size = 1.05 \[ \frac{g^3 \left (-3 b B (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+b B \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )+b^3 d^2 x^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 A b^2 d x (2 b c-3 a d)+6 b (b c-a d)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{2 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-2 b B d (a+b x) (2 b c-3 a d) \log \left (\frac{e (a+b x)}{c+d x}\right )+2 b B (2 b c-3 a d) (b c-a d) \log (c+d x)-2 B (b c-a d)^2 \left (\frac{b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )\right )}{2 d^4 i^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.174, size = 2973, normalized size = 8.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.5938, size = 1810, normalized size = 5.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A b^{3} g^{3} x^{3} + 3 \, A a b^{2} g^{3} x^{2} + 3 \, A a^{2} b g^{3} x + A a^{3} g^{3} +{\left (B b^{3} g^{3} x^{3} + 3 \, B a b^{2} g^{3} x^{2} + 3 \, B a^{2} b g^{3} x + B a^{3} g^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{d^{2} i^{2} x^{2} + 2 \, c d i^{2} x + c^{2} i^{2}}, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b g x + a g\right )}^{3}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]