Integrand size = 25, antiderivative size = 155 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}+\frac {8 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3} \]
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Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {272, 45, 2392, 12, 1265, 911, 1275, 212} \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {8 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}+\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3} \]
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Rule 12
Rule 45
Rule 212
Rule 272
Rule 911
Rule 1265
Rule 1275
Rule 2392
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-(b n) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{3 e^3 x \left (d+e x^2\right )^{3/2}} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^3} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \text {Subst}\left (\int \frac {8 d^2+12 d e x+3 e^2 x^2}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^3} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \text {Subst}\left (\int \frac {-d^2+6 d x^2+3 x^4}{x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \text {Subst}\left (\int \left (3 e+\frac {d e}{x^2}-\frac {8 d e}{d-x^2}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4} \\ & = \frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {(8 b d n) \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^3} \\ & = \frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.32 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {8 b \sqrt {d} n \log (x)}{3 e^3}+\frac {b n \left (8 d^2+12 d e x^2+3 e^2 x^4\right ) \log (x)}{3 e^3 \left (d+e x^2\right )^{3/2}}+\sqrt {d+e x^2} \left (-\frac {d^2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{3 e^3 \left (d+e x^2\right )^2}+\frac {a-b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )}{e^3}+\frac {d \left (6 a+b n+6 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{3 e^3 \left (d+e x^2\right )}\right )+\frac {8 b \sqrt {d} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{3 e^3} \]
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\[\int \frac {x^{5} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Time = 0.36 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.59 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {4 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (3 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + 2 \, b d^{2} n - 8 \, a d^{2} + {\left (5 \, b d e n - 12 \, a d e\right )} x^{2} - {\left (3 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 8 \, b d^{2}\right )} \log \left (c\right ) - {\left (3 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3 \, {\left (e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}\right )}}, -\frac {8 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (3 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + 2 \, b d^{2} n - 8 \, a d^{2} + {\left (5 \, b d e n - 12 \, a d e\right )} x^{2} - {\left (3 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 8 \, b d^{2}\right )} \log \left (c\right ) - {\left (3 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{3 \, {\left (e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}\right )}}\right ] \]
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Time = 65.07 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.68 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=a \left (\begin {cases} - \frac {d^{2}}{3 e^{3} \left (d + e x^{2}\right )^{\frac {3}{2}}} + \frac {2 d}{e^{3} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{3}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} - \frac {3 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{e^{3}} - \frac {2 d^{5} \sqrt {1 + \frac {e x^{2}}{d}}}{6 d^{\frac {9}{2}} e^{3} + 6 d^{\frac {7}{2}} e^{4} x^{2}} - \frac {d^{5} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e^{3} + 6 d^{\frac {7}{2}} e^{4} x^{2}} + \frac {2 d^{5} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e^{3} + 6 d^{\frac {7}{2}} e^{4} x^{2}} - \frac {d^{4} x^{2} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e^{2} + 6 d^{\frac {7}{2}} e^{3} x^{2}} + \frac {2 d^{4} x^{2} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e^{2} + 6 d^{\frac {7}{2}} e^{3} x^{2}} + \frac {d}{e^{\frac {7}{2}} x \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {x}{e^{\frac {5}{2}} \sqrt {\frac {d}{e x^{2}} + 1}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x^{6}}{36 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} - \frac {d^{2}}{3 e^{3} \left (d + e x^{2}\right )^{\frac {3}{2}}} + \frac {2 d}{e^{3} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{3}} & \text {for}\: e \neq 0 \\\frac {x^{6}}{6 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
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Exception generated. \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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