Optimal. Leaf size=129 \[ -\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^2}+\frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2} \]
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Rubi [A] time = 0.16, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {266, 43, 2350, 12, 446, 80, 50, 63, 208} \[ -\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^2}+\frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 50
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 )}{\sqrt {d+e x^2}} \, dx &=-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-(b n) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{3 e^2 x} \, dx\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {(b n) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{x} \, dx}{3 e^2}\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {(b n) \operatorname {Subst}\left (\int \frac {(-2 d+e x) \sqrt {d+e x}}{x} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {(b d n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )}{3 e^2}\\ &=\frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {\left (b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2}\\ &=\frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^3}\\ &=\frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2}-\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 145, normalized size = 1.12 \[ \frac {3 a e x^2 \sqrt {d+e x^2}-6 a d \sqrt {d+e x^2}+3 b \left (e x^2-2 d\right ) \sqrt {d+e x^2} \log \left (c x^n\right )-6 b d^{3/2} n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )+6 b d^{3/2} n \log (x)-b e n x^2 \sqrt {d+e x^2}+5 b d n \sqrt {d+e x^2}}{9 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 207, normalized size = 1.60 \[ \left [\frac {3 \, b d^{\frac {3}{2}} n \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + {\left (5 \, b d n - {\left (b e n - 3 \, a e\right )} x^{2} - 6 \, a d + 3 \, {\left (b e x^{2} - 2 \, b d\right )} \log \relax (c) + 3 \, {\left (b e n x^{2} - 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{9 \, e^{2}}, \frac {6 \, b \sqrt {-d} d n \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (5 \, b d n - {\left (b e n - 3 \, a e\right )} x^{2} - 6 \, a d + 3 \, {\left (b e x^{2} - 2 \, b d\right )} \log \relax (c) + 3 \, {\left (b e n x^{2} - 2 \, b d n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{9 \, e^{2}}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{3}}{\sqrt {e \,x^{2}+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 149, normalized size = 1.16 \[ \frac {1}{9} \, b n {\left (\frac {3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{e^{2}} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} - 6 \, \sqrt {e x^{2} + d} d}{e^{2}}\right )} + \frac {1}{3} \, {\left (\frac {\sqrt {e x^{2} + d} x^{2}}{e} - \frac {2 \, \sqrt {e x^{2} + d} d}{e^{2}}\right )} b \log \left (c x^{n}\right ) + \frac {1}{3} \, {\left (\frac {\sqrt {e x^{2} + d} x^{2}}{e} - \frac {2 \, \sqrt {e x^{2} + d} d}{e^{2}}\right )} a \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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