Optimal. Leaf size=113 \[ \frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \]
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Rubi [A] time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2323, 2314, 217, 206, 191} \[ \frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 206
Rule 217
Rule 2314
Rule 2323
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}-\frac {(b n) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^2}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^2}\\ &=-\frac {b n x}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 116, normalized size = 1.03 \[ \frac {\sqrt {e} x \left (a \left (3 d+2 e x^2\right )-b n \left (d+e x^2\right )\right )+b \sqrt {e} x \left (3 d+2 e x^2\right ) \log \left (c x^n\right )-2 b n \left (d+e x^2\right )^{3/2} \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{3 d^2 \sqrt {e} \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 337, normalized size = 2.98 \[ \left [\frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left ({\left (b e^{2} n - 2 \, a e^{2}\right )} x^{3} + {\left (b d e n - 3 \, a d e\right )} x - {\left (2 \, b e^{2} x^{3} + 3 \, b d e x\right )} \log \relax (c) - {\left (2 \, b e^{2} n x^{3} + 3 \, b d e n x\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {2 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left ({\left (b e^{2} n - 2 \, a e^{2}\right )} x^{3} + {\left (b d e n - 3 \, a d e\right )} x - {\left (2 \, b e^{2} x^{3} + 3 \, b d e x\right )} \log \relax (c) - {\left (2 \, b e^{2} n x^{3} + 3 \, b d e n x\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a {\left (\frac {2 \, x}{\sqrt {e x^{2} + d} d^{2}} + \frac {x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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