Integrand size = 25, antiderivative size = 100 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2} \]
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Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {272, 45, 2392, 12, 457, 81, 65, 214} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {2 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2}-\frac {b n \sqrt {d+e x^2}}{e^2} \]
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Rule 12
Rule 45
Rule 65
Rule 81
Rule 214
Rule 272
Rule 457
Rule 2392
Rubi steps \begin{align*} \text {integral}& = \frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-(b n) \int \frac {2 d+e x^2}{e^2 x \sqrt {d+e x^2}} \, dx \\ & = \frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b n) \int \frac {2 d+e x^2}{x \sqrt {d+e x^2}} \, dx}{e^2} \\ & = \frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b n) \text {Subst}\left (\int \frac {2 d+e x}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2} \\ & = -\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b d n) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{e^2} \\ & = -\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(2 b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e^3} \\ & = -\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {2 a d-b d n+a e x^2-b e n x^2-2 b \sqrt {d} n \sqrt {d+e x^2} \log (x)+b \left (2 d+e x^2\right ) \log \left (c x^n\right )+2 b \sqrt {d} n \sqrt {d+e x^2} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{e^2 \sqrt {d+e x^2}} \]
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\[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.45 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {d} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (b d n + {\left (b e n - a e\right )} x^{2} - 2 \, a d - {\left (b e x^{2} + 2 \, b d\right )} \log \left (c\right ) - {\left (b e n x^{2} + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{e^{3} x^{2} + d e^{2}}, -\frac {2 \, {\left (b e n x^{2} + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (b d n + {\left (b e n - a e\right )} x^{2} - 2 \, a d - {\left (b e x^{2} + 2 \, b d\right )} \log \left (c\right ) - {\left (b e n x^{2} + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{e^{3} x^{2} + d e^{2}}\right ] \]
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Time = 24.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=a \left (\begin {cases} \frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} - \frac {2 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{e^{2}} + \frac {d}{e^{\frac {5}{2}} x \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {x}{e^{\frac {3}{2}} \sqrt {\frac {d}{e x^{2}} + 1}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x^{4}}{16 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
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Exception generated. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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