Optimal. Leaf size=58 \[ -\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2351, 223, 212}
\begin {gather*} \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 2351
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {(b n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {(b n) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{d}\\ &=-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 70, normalized size = 1.21 \begin {gather*} \frac {\frac {a x}{\sqrt {d+e x^2}}+\frac {b x \log \left (c x^n\right )}{\sqrt {d+e x^2}}-\frac {b n \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{\sqrt {e}}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 56, normalized size = 0.97 \begin {gather*} -\frac {b n \operatorname {arsinh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{d} + \frac {b x \log \left (c x^{n}\right )}{\sqrt {x^{2} e + d} d} + \frac {a x}{\sqrt {x^{2} e + d} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 95, normalized size = 1.64 \begin {gather*} \frac {{\left (b n x^{2} e + b d n\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + 2 \, {\left (b n x e \log \left (x\right ) + b x e \log \left (c\right ) + a x e\right )} \sqrt {x^{2} e + d}}{2 \, {\left (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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \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 {a+b\,\ln \left (c\,x^n\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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