Optimal. Leaf size=202 \[ -\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {b^2 e^2 n^2 \log (f+g x)}{g (e f-d g)^2}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2445, 2458,
2389, 2379, 2438, 2351, 31} \begin {gather*} \frac {b^2 e^2 n^2 \text {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2}-\frac {b e^2 n \log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (e f-d g)^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {b^2 e^2 n^2 \log (f+g x)}{g (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^3} \, dx &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {(b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^2} \, dx}{g}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{g}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}-\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^2} \, dx,x,d+e x\right )}{e f-d g}+\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )} \, dx,x,d+e x\right )}{g (e f-d g)}\\ &=-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}-\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\frac {e f-d g}{e}+\frac {g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^2}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^2}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{\frac {e f-d g}{e}+\frac {g x}{e}} \, dx,x,d+e x\right )}{(e f-d g)^2}\\ &=-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 (f+g x)}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {b^2 e^2 n^2 \log (f+g x)}{g (e f-d g)^2}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g (e f-d g)^2}\\ &=-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 (f+g x)}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g)^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (f+g x)^2}+\frac {b^2 e^2 n^2 \log (f+g x)}{g (e f-d g)^2}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)^2}-\frac {b^2 e^2 n^2 \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 204, normalized size = 1.01 \begin {gather*} \frac {-\left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {e (f+g x) \left (2 b (e f-d g) n \left (a+b \log \left (c (d+e x)^n\right )\right )+e (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b^2 e n^2 (f+g x) (\log (d+e x)-\log (f+g x))-2 b e n (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 b^2 e n^2 (f+g x) \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )\right )}{(e f-d g)^2}}{2 g (f+g x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.46, size = 1473, normalized size = 7.29
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1473\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}{\left (f + g x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________