Integrand size = 26, antiderivative size = 423 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\frac {16 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}} \]
[Out]
Time = 0.71 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2445, 2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=-\frac {8 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \sqrt {f+g x} (e f-d g)}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {16 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}} \]
[In]
[Out]
Rule 12
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2389
Rule 2390
Rule 2445
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {(4 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx}{3 g} \\ & = -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {(4 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 )^{3/2}} \, dx,x,d+e x\right )}{3 g} \\ & = -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {(4 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 )^{3/2}} \, dx,x,d+e x\right )}{3 (e f-d g)}+\frac {(4 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 g (e f-d g)} \\ & = \frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {\left (4 b^2 e n^2\right ) \text {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{3 g (e f-d g)}-\frac {\left (8 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 g (e f-d g)} \\ & = \frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (8 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{3 g (e f-d g)^{3/2}}-\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 g^2 (e f-d g)} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (16 b^2 e^{5/2} n^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^{3/2}} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (16 b^2 e^{5/2} n^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^{3/2}} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^2} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}+\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^2} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {\left (16 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}} \\ & = \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\frac {2 \left (-\left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {b e n (f+g x) \left (8 b \sqrt {e} n \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+4 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )+2 \sqrt {e} \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-2 \sqrt {e} \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-b \sqrt {e} n \sqrt {f+g x} \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+b \sqrt {e} n \sqrt {f+g x} \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )\right )}{(e f-d g)^{3/2}}\right )}{3 g (f+g x)^{3/2}} \]
[In]
[Out]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g x +f \right )^{\frac {5}{2}}}d x\]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^{5/2}} \,d x \]
[In]
[Out]