Optimal. Leaf size=57 \[ -\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e} \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2338, 266, 63, 208} \[ -\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2338
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}+\frac {(b n) \int \frac {1}{x \sqrt {d+e x^2}} \, dx}{e}\\ &=-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e^2}\\ &=-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 77, normalized size = 1.35 \[ -\frac {\frac {a}{\sqrt {d+e x^2}}+\frac {b \log \left (c x^n\right )}{\sqrt {d+e x^2}}+\frac {b n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{\sqrt {d}}-\frac {b n \log (x)}{\sqrt {d}}}{e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 169, normalized size = 2.96 \[ \left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \sqrt {e x^{2} + d}}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {{\left (b e n x^{2} + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) - {\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )} \sqrt {e x^{2} + d}}{d e^{2} x^{2} + d^{2} e}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 59, normalized size = 1.04 \[ -\frac {b n \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{\sqrt {d} e} - \frac {b \log \left (c x^{n}\right )}{\sqrt {e x^{2} + d} e} - \frac {a}{\sqrt {e x^{2} + d} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.77, size = 80, normalized size = 1.40 \[ - \frac {a}{e \sqrt {d + e x^{2}}} - b n \left (\begin {cases} \frac {x^{2}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {\operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{\sqrt {d} e} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{2}}{2 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\- \frac {1}{e \sqrt {d + e x^{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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