Optimal. Leaf size=125 \[ -\frac {b d^2 n \sqrt {d+e x^2}}{5 e}-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{5 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e} \]
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Rubi [A]
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2376, 272, 52,
65, 214} \begin {gather*} \frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}+\frac {b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{5 e}-\frac {b d^2 n \sqrt {d+e x^2}}{5 e}-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2}}{x} \, dx}{5 e}\\ &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {(b n) \text {Subst}\left (\int \frac {(d+e x)^{5/2}}{x} \, dx,x,x^2\right )}{10 e}\\ &=-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {(b d n) \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^2\right )}{10 e}\\ &=-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (b d^2 n\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )}{10 e}\\ &=-\frac {b d^2 n \sqrt {d+e x^2}}{5 e}-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (b d^3 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{10 e}\\ &=-\frac {b d^2 n \sqrt {d+e x^2}}{5 e}-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (b d^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{5 e^2}\\ &=-\frac {b d^2 n \sqrt {d+e x^2}}{5 e}-\frac {b d n \left (d+e x^2\right )^{3/2}}{15 e}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 e}+\frac {b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{5 e}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 181, normalized size = 1.45 \begin {gather*} -\frac {b d^{5/2} n \log (x)}{5 e}+\frac {b n \left (d+e x^2\right )^{5/2} \log (x)}{5 e}+\sqrt {d+e x^2} \left (\frac {1}{25} e x^4 \left (5 a-b n+5 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+\frac {d^2 \left (15 a-23 b n+15 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{75 e}+\frac {1}{75} d x^2 \left (30 a-11 b n+30 b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\frac {b d^{5/2} n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 100, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} b e^{\left (-1\right )} \log \left (c x^{n}\right ) + \frac {1}{5} \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} a e^{\left (-1\right )} + \frac {1}{75} \, {\left (15 \, d^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {\sqrt {d} e^{\left (-\frac {1}{2}\right )}}{{\left | x \right |}}\right ) - 3 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} - 5 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} d - 15 \, \sqrt {x^{2} e + d} d^{2}\right )} b n e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 299, normalized size = 2.39 \begin {gather*} \left [\frac {1}{150} \, {\left (15 \, b d^{\frac {5}{2}} n \log \left (-\frac {x^{2} e + 2 \, \sqrt {x^{2} e + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (b n - 5 \, a\right )} x^{4} e^{2} + 23 \, b d^{2} n + {\left (11 \, b d n - 30 \, a d\right )} x^{2} e - 15 \, a d^{2} - 15 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}, -\frac {1}{75} \, {\left (15 \, b \sqrt {-d} d^{2} n \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) + {\left (3 \, {\left (b n - 5 \, a\right )} x^{4} e^{2} + 23 \, b d^{2} n + {\left (11 \, b d n - 30 \, a d\right )} x^{2} e - 15 \, a d^{2} - 15 \, {\left (b x^{4} e^{2} + 2 \, b d x^{2} e + b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\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 x \left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{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 x\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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