Optimal. Leaf size=100 \[ -\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {272, 45, 2392,
12, 457, 81, 65, 214} \begin {gather*} \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 65
Rule 81
Rule 214
Rule 272
Rule 457
Rule 2392
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-(b n) \int \frac {2 d+e x^2}{e^2 x \sqrt {d+e x^2}} \, dx\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b n) \int \frac {2 d+e x^2}{x \sqrt {d+e x^2}} \, dx}{e^2}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b n) \text {Subst}\left (\int \frac {2 d+e x}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(b d n) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{e^2}\\ &=-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {(2 b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e^3}\\ &=-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 a d-b d n+a e x^2-b e n x^2-2 b \sqrt {d} n \sqrt {d+e x^2} \log (x)+b \left (2 d+e x^2\right ) \log \left (c x^n\right )+2 b \sqrt {d} n \sqrt {d+e x^2} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{e^2 \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 133, normalized size = 1.33 \begin {gather*} -{\left (\sqrt {d} e^{\left (-2\right )} \log \left (\frac {\sqrt {x^{2} e + d} - \sqrt {d}}{\sqrt {x^{2} e + d} + \sqrt {d}}\right ) + \sqrt {x^{2} e + d} e^{\left (-2\right )}\right )} b n + {\left (\frac {x^{2} e^{\left (-1\right )}}{\sqrt {x^{2} e + d}} + \frac {2 \, d e^{\left (-2\right )}}{\sqrt {x^{2} e + d}}\right )} b \log \left (c x^{n}\right ) + {\left (\frac {x^{2} e^{\left (-1\right )}}{\sqrt {x^{2} e + d}} + \frac {2 \, d e^{\left (-2\right )}}{\sqrt {x^{2} e + d}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 252, normalized size = 2.52 \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 ) - {\left ({\left (b n - a\right )} x^{2} e + b d n - 2 \, a d - {\left (b x^{2} e + 2 \, b d\right )} \log \left (c\right ) - {\left (b n x^{2} e + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{x^{2} e^{3} + d e^{2}}, -\frac {2 \, {\left (b n x^{2} e + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left ({\left (b n - a\right )} x^{2} e + b d n - 2 \, a d - {\left (b x^{2} e + 2 \, b d\right )} \log \left (c\right ) - {\left (b n x^{2} e + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{x^{2} e^{3} + d e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 24.83, size = 163, normalized size = 1.63 \begin {gather*} a \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {x^{4}}{16 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\- \frac {2 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{e^{2}} + \frac {d}{e^{\frac {5}{2}} x \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {x}{e^{\frac {3}{2}} \sqrt {\frac {d}{e x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{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 = -0.01 \begin {gather*} \int \frac {x^3\,\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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