Optimal. Leaf size=129 \[ \frac {2 b d n \sqrt {d+e x^2}}{3 e^2}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^2}-\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^2} \]
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Rubi [A]
time = 0.11, 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 = {272, 45, 2392,
12, 457, 81, 52, 65, 214} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 52
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 )}{\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) \text {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) \text {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 ) \text {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 ) \text {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 = 0.09, size = 145, normalized size = 1.12 \begin {gather*} \frac {-6 a d \sqrt {d+e x^2}+5 b d n \sqrt {d+e x^2}+3 a e x^2 \sqrt {d+e x^2}-b e n x^2 \sqrt {d+e x^2}+6 b d^{3/2} n \log (x)+3 b \left (-2 d+e x^2\right ) \sqrt {d+e x^2} \log \left (c x^n\right )-6 b d^{3/2} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{9 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {e \,x^{2}+d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 151, normalized size = 1.17 \begin {gather*} \frac {1}{9} \, {\left (3 \, d^{\frac {3}{2}} e^{\left (-2\right )} \log \left (\frac {\sqrt {x^{2} e + d} - \sqrt {d}}{\sqrt {x^{2} e + d} + \sqrt {d}}\right ) - {\left ({\left (x^{2} e + d\right )}^{\frac {3}{2}} - 6 \, \sqrt {x^{2} e + d} d\right )} e^{\left (-2\right )}\right )} b n + \frac {1}{3} \, {\left (\sqrt {x^{2} e + d} x^{2} e^{\left (-1\right )} - 2 \, \sqrt {x^{2} e + d} d e^{\left (-2\right )}\right )} b \log \left (c x^{n}\right ) + \frac {1}{3} \, {\left (\sqrt {x^{2} e + d} x^{2} e^{\left (-1\right )} - 2 \, \sqrt {x^{2} e + d} d e^{\left (-2\right )}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 214, normalized size = 1.66 \begin {gather*} \left [\frac {1}{9} \, {\left (3 \, b d^{\frac {3}{2}} n \log \left (-\frac {x^{2} e - 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left ({\left (b n - 3 \, a\right )} x^{2} e - 5 \, b d n + 6 \, a d - 3 \, {\left (b x^{2} e - 2 \, b d\right )} \log \left (c\right ) - 3 \, {\left (b n x^{2} e - 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}, \frac {1}{9} \, {\left (6 \, b \sqrt {-d} d n \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - {\left ({\left (b n - 3 \, a\right )} x^{2} e - 5 \, b d n + 6 \, a d - 3 \, {\left (b x^{2} e - 2 \, b d\right )} \log \left (c\right ) - 3 \, {\left (b n x^{2} e - 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}\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^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d + e x^{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^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {e\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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