Integrand size = 34, antiderivative size = 319 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+2 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{6 b d^2}-\frac {B (b c-a d)^3 g^3 (a+b x) \left (3 A+5 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d^3}-\frac {B (b c-a d)^4 g^3 \left (3 A+11 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b d^4}-\frac {2 B^2 (b c-a d)^4 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^4} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2550, 2381, 2384, 2354, 2438} \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=-\frac {B g^3 (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+3 A+11 B\right )}{3 b d^4}-\frac {B g^3 (a+b x) (b c-a d)^3 \left (3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+3 A+5 B\right )}{3 b d^3}+\frac {B g^3 (a+b x)^2 (b c-a d)^2 \left (3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+3 A+2 B\right )}{6 b d^2}-\frac {B g^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 b}-\frac {2 B^2 g^3 (b c-a d)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^4} \]
[In]
[Out]
Rule 2354
Rule 2381
Rule 2384
Rule 2438
Rule 2550
Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {x^3 \left (A+B \log \left (e x^2\right )\right )^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}-\frac {\left (B (b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {x^3 \left (A+B \log \left (e x^2\right )\right )}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{b} \\ & = -\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {\left (B (b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {x^2 \left (3 A+2 B+3 B \log \left (e x^2\right )\right )}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b d} \\ & = -\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+2 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{6 b d^2}-\frac {\left (B (b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {x \left (6 B+2 (3 A+2 B)+6 B \log \left (e x^2\right )\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b d^2} \\ & = -\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+2 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{6 b d^2}-\frac {B (b c-a d)^3 g^3 (a+b x) \left (3 A+5 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d^3}+\frac {\left (B (b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {18 B+2 (3 A+2 B)+6 B \log \left (e x^2\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b d^3} \\ & = -\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+2 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{6 b d^2}-\frac {B (b c-a d)^3 g^3 (a+b x) \left (3 A+5 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d^3}-\frac {B (b c-a d)^4 g^3 \left (3 A+11 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b d^4}+\frac {\left (2 B^2 (b c-a d)^4 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b d^4} \\ & = -\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+2 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{6 b d^2}-\frac {B (b c-a d)^3 g^3 (a+b x) \left (3 A+5 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d^3}-\frac {B (b c-a d)^4 g^3 \left (3 A+11 B+3 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b d^4}-\frac {2 B^2 (b c-a d)^4 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d^4} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.26 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\frac {g^3 \left ((a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2-\frac {2 B (b c-a d) \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-12 B (b c-a d)^3 \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)+2 B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+6 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+6 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^4}\right )}{4 b} \]
[In]
[Out]
\[\int \left (b g x +a g \right )^{3} {\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2}d x\]
[In]
[Out]
\[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (308) = 616\).
Time = 0.33 (sec) , antiderivative size = 1948, normalized size of antiderivative = 6.11 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2 \,d x \]
[In]
[Out]