Integrand size = 31, antiderivative size = 406 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=-\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}-\frac {2 b e^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}} \]
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Time = 0.90 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 53} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{5/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {f+g x} (e f-d g)^2}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (f+g x)^{3/2} (e f-d g)}+\frac {2 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {16 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}-\frac {2 b e^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}}-\frac {4 b e n}{3 \sqrt {f+g x} (e f-d g)^2} \]
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Rule 12
Rule 53
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2389
Rule 2390
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \frac {\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 )^{5/2}} \, dx,x,d+e x\right )}{e} \\ & = \frac {\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 )}{e f-d g}-\frac {g \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 )^{5/2}} \, dx,x,d+e x\right )}{e (e f-d g)} \\ & = \frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {e \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 )}{(e f-d g)^2}-\frac {g \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 )}{(e f-d g)^2}-\frac {(2 b n) \text {Subst}\left (\int \frac {1}{x \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}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}-\frac {(2 b e n) \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 (e f-d g)^2}-\frac {(b e n) \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 )}{(e f-d g)^2}-\frac {(2 b e n) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{(e f-d g)^2} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}+\frac {\left (2 b e^{3/2} n\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 )}{(e f-d g)^{5/2}}-\frac {\left (4 b e^2 n\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 (e f-d g)^2}-\frac {\left (4 b e^2 n\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 )}{g (e f-d g)^2} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}+\frac {\left (4 b e^{5/2} n\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 )}{(e f-d g)^{5/2}} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}+\frac {\left (4 b e^{5/2} n\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 )}{(e f-d g)^{5/2}} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}-\frac {\left (4 b e^2 n\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 )}{(e f-d g)^3} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \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 )}{(e f-d g)^{5/2}}+\frac {\left (4 b e^2 n\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 )}{(e f-d g)^3} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \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 )}{(e f-d g)^{5/2}}-\frac {\left (4 b e^{3/2} n\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 )}{(e f-d g)^{5/2}} \\ & = -\frac {4 b e n}{3 (e f-d g)^2 \sqrt {f+g x}}+\frac {16 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 (e f-d g)^{5/2}}+\frac {2 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{5/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 (e f-d g) (f+g x)^{3/2}}+\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^2 \sqrt {f+g x}}-\frac {2 e^{3/2} \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 )}{(e f-d g)^{5/2}}-\frac {4 b e^{3/2} n \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 )}{(e f-d g)^{5/2}}-\frac {2 b e^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.51 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.50 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\frac {24 b e^{3/2} n (f+g x)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-8 b e \sqrt {e f-d g} n (f+g x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {e (f+g x)}{e f-d g}\right )+4 (e f-d g)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+12 e \sqrt {e f-d g} (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )+6 e^{3/2} (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-6 e^{3/2} (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-3 b e^{3/2} n (f+g x)^{3/2} \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+3 b e^{3/2} n (f+g x)^{3/2} \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )}{6 (e f-d g)^{5/2} (f+g x)^{3/2}} \]
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\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (e x +d \right ) \left (g x +f \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{5/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]
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