Integrand size = 25, antiderivative size = 443 \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b d n x}{3 e^3 \sqrt {d+e x^2}}-\frac {b n x \sqrt {d+e x^2}}{4 e^3}-\frac {31 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{12 e^{7/2} \sqrt {d+e x^2}}-\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 e^{7/2} \sqrt {d+e x^2}} \]
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Time = 0.38 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2386, 294, 327, 221, 2392, 21, 1171, 396, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {5 d^{3/2} \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {5 b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 e^{7/2} \sqrt {d+e x^2}}-\frac {5 b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}-\frac {31 b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{12 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {\frac {e x^2}{d}+1} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {b n x \sqrt {d+e x^2}}{4 e^3}+\frac {b d n x}{3 e^3 \sqrt {d+e x^2}} \]
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Rule 21
Rule 221
Rule 294
Rule 327
Rule 396
Rule 1171
Rule 2221
Rule 2317
Rule 2386
Rule 2392
Rule 2438
Rule 3797
Rule 5775
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {e x^2}{d}} \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (1+\frac {e x^2}{d}\right )^{5/2}} \, dx}{d^2 \sqrt {d+e x^2}} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \left (\frac {d^3 \sqrt {1+\frac {e x^2}{d}} \left (15 d^2+20 d e x^2+3 e^2 x^4\right )}{6 e^3 \left (d+e x^2\right )^2}-\frac {5 d^{7/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{7/2} x}\right ) \, dx}{d^2 \sqrt {d+e x^2}} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}+\frac {\left (5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (b d n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}} \left (15 d^2+20 d e x^2+3 e^2 x^4\right )}{\left (d+e x^2\right )^2} \, dx}{6 e^3 \sqrt {d+e x^2}} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}+\frac {\left (5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {15 d^2+20 d e x^2+3 e^2 x^4}{\left (1+\frac {e x^2}{d}\right )^{3/2}} \, dx}{6 d e^3 \sqrt {d+e x^2}} \\ & = \frac {b d n x}{3 e^3 \sqrt {d+e x^2}}-\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{7/2} \sqrt {d+e x^2}}+\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {-17 d^2-3 d e x^2}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{6 d e^3 \sqrt {d+e x^2}} \\ & = \frac {b d n x}{3 e^3 \sqrt {d+e x^2}}-\frac {b n x \sqrt {d+e x^2}}{4 e^3}-\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (31 b d n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{12 e^3 \sqrt {d+e x^2}} \\ & = \frac {b d n x}{3 e^3 \sqrt {d+e x^2}}-\frac {b n x \sqrt {d+e x^2}}{4 e^3}-\frac {31 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{12 e^{7/2} \sqrt {d+e x^2}}-\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {\left (5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 e^{7/2} \sqrt {d+e x^2}} \\ & = \frac {b d n x}{3 e^3 \sqrt {d+e x^2}}-\frac {b n x \sqrt {d+e x^2}}{4 e^3}-\frac {31 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{12 e^{7/2} \sqrt {d+e x^2}}-\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{4 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{7/2} \sqrt {d+e x^2}}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {d+e x^2}}+\frac {5 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e^3}-\frac {5 d^{3/2} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{7/2} \sqrt {d+e x^2}}+\frac {5 b d^{3/2} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{4 e^{7/2} \sqrt {d+e x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.21 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.45 \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b n x^7 \sqrt {1+\frac {e x^2}{d}} \left (5 \, _3F_2\left (\frac {7}{2},\frac {7}{2},\frac {7}{2};\frac {9}{2},\frac {9}{2};-\frac {e x^2}{d}\right )+7 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7}{2},\frac {9}{2},-\frac {e x^2}{d}\right ) (-1+2 \log (x))\right )}{98 d^2 \sqrt {d+e x^2}}+\frac {x \left (15 d^2+20 d e x^2+3 e^2 x^4\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{6 e^3 \left (d+e x^2\right )^{3/2}}-\frac {5 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{2 e^{7/2}} \]
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\[\int \frac {x^{6} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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