Optimal. Leaf size=57 \[ -\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.05, 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 = {2376, 272, 65,
214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 2376
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) \text {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) \text {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 = 0.09, size = 77, normalized size = 1.35 \begin {gather*} -\frac {\frac {a}{\sqrt {d+e x^2}}-\frac {b n \log (x)}{\sqrt {d}}+\frac {b \log \left (c x^n\right )}{\sqrt {d+e x^2}}+\frac {b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{\sqrt {d}}}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 57, normalized size = 1.00 \begin {gather*} -\frac {b n \operatorname {arsinh}\left (\frac {\sqrt {d} e^{\left (-\frac {1}{2}\right )}}{{\left | x \right |}}\right ) e^{\left (-1\right )}}{\sqrt {d}} - \frac {b e^{\left (-1\right )} \log \left (c x^{n}\right )}{\sqrt {x^{2} e + d}} - \frac {a e^{\left (-1\right )}}{\sqrt {x^{2} e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 176, normalized size = 3.09 \begin {gather*} \left [\frac {{\left (b n x^{2} e + b d n\right )} \sqrt {d} \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt {x^{2} e + d}}{2 \, {\left (d x^{2} e^{2} + d^{2} e\right )}}, \frac {{\left (b n x^{2} e + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - {\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt {x^{2} e + d}}{d x^{2} e^{2} + d^{2} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.14, size = 80, normalized size = 1.40 \begin {gather*} - \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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