Optimal. Leaf size=89 \[ \frac {b n x}{3 d e \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d e^{3/2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2373, 294, 223,
212} \begin {gather*} \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d e^{3/2}}+\frac {b n x}{3 d e \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 294
Rule 2373
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b n) \int \frac {x^2}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac {b n x}{3 d e \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d e}\\ &=\frac {b n x}{3 d e \sqrt {d+e x^2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {(b n) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d e}\\ &=\frac {b n x}{3 d e \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d e^{3/2}}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d \left (d+e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 101, normalized size = 1.13 \begin {gather*} \frac {\sqrt {e} x \left (a e x^2+b n \left (d+e x^2\right )\right )+b e^{3/2} x^3 \log \left (c x^n\right )-b n \left (d+e x^2\right )^{3/2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{3 d e^{3/2} \left (d+e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 134, normalized size = 1.51 \begin {gather*} \frac {{\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} 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^{3} e^{2} \log \left (x\right ) + b x^{3} e^{2} \log \left (c\right ) + {\left (b n + a\right )} x^{3} e^{2} + b d n x e\right )} \sqrt {x^{2} e + d}}{6 \, {\left (d x^{4} e^{4} + 2 \, d^{2} x^{2} e^{3} + d^{3} e^{2}\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 {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{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 = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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