Optimal. Leaf size=100 \[ \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2} \]
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Rubi [A] time = 0.16, 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 = {266, 43, 2350, 12, 446, 80, 63, 208} \[ \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^2}+\frac {2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 80
Rule 208
Rule 266
Rule 446
Rule 2350
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) \operatorname {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) \operatorname {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) \operatorname {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 = 0.15, size = 118, normalized size = 1.18 \[ \frac {2 a d+a e x^2+b \left (2 d+e x^2\right ) \log \left (c x^n\right )-2 b \sqrt {d} n \log (x) \sqrt {d+e x^2}+2 b \sqrt {d} n \sqrt {d+e x^2} \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )-b d n-b e n x^2}{e^2 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 245, normalized size = 2.45 \[ \left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {d} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (b d n + {\left (b e n - a e\right )} x^{2} - 2 \, a d - {\left (b e x^{2} + 2 \, b d\right )} \log \relax (c) - {\left (b e n x^{2} + 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{e^{3} x^{2} + d e^{2}}, -\frac {2 \, {\left (b e n x^{2} + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (b d n + {\left (b e n - a e\right )} x^{2} - 2 \, a d - {\left (b e x^{2} + 2 \, b d\right )} \log \relax (c) - {\left (b e n x^{2} + 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{e^{3} x^{2} + d e^{2}}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{3}}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.68, size = 132, normalized size = 1.32 \[ -b n {\left (\frac {\sqrt {d} \log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{e^{2}} + \frac {\sqrt {e x^{2} + d}}{e^{2}}\right )} + b {\left (\frac {x^{2}}{\sqrt {e x^{2} + d} e} + \frac {2 \, d}{\sqrt {e x^{2} + d} e^{2}}\right )} \log \left (c x^{n}\right ) + a {\left (\frac {x^{2}}{\sqrt {e x^{2} + d} e} + \frac {2 \, d}{\sqrt {e x^{2} + d} e^{2}}\right )} \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 47.87, size = 163, normalized size = 1.63 \[ a \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {x^{4}}{16 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\- \frac {2 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{e^{2}} + \frac {d}{e^{\frac {5}{2}} x \sqrt {\frac {d}{e x^{2}} + 1}} + \frac {x}{e^{\frac {3}{2}} \sqrt {\frac {d}{e x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{4}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {d}{e^{2} \sqrt {d + e x^{2}}} + \frac {\sqrt {d + e x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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