Optimal. Leaf size=158 \[ \frac {5 b d n \sqrt {d+e x^2}}{3 e^3}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^3}-\frac {8 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {272, 45, 2392,
12, 1265, 911, 1167, 214} \begin {gather*} -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {8 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}+\frac {5 b d n \sqrt {d+e x^2}}{3 e^3}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 214
Rule 272
Rule 911
Rule 1167
Rule 1265
Rule 2392
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-(b n) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {d+e x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {(b n) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {d+e x^2}} \, dx}{3 e^3}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {(b n) \text {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {(b n) \text {Subst}\left (\int \frac {-3 d^2-6 d x^2+x^4}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {(b n) \text {Subst}\left (\int \left (-5 d e+e x^2-\frac {8 d^2}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4}\\ &=\frac {5 b d n \sqrt {d+e x^2}}{3 e^3}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {\left (8 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^4}\\ &=\frac {5 b d n \sqrt {d+e x^2}}{3 e^3}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^3}-\frac {8 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 160, normalized size = 1.01 \begin {gather*} \frac {-24 a d^2+14 b d^2 n-12 a d e x^2+13 b d e n x^2+3 a e^2 x^4-b e^2 n x^4+24 b d^{3/2} n \sqrt {d+e x^2} \log (x)-3 b \left (8 d^2+4 d e x^2-e^2 x^4\right ) \log \left (c x^n\right )-24 b d^{3/2} n \sqrt {d+e x^2} \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{9 e^3 \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 \[\int \frac {x^{5} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 191, normalized size = 1.21 \begin {gather*} \frac {1}{9} \, {\left (12 \, d^{\frac {3}{2}} e^{\left (-3\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}} - 15 \, \sqrt {x^{2} e + d} d\right )} e^{\left (-3\right )}\right )} b n + \frac {1}{3} \, {\left (\frac {x^{4} e^{\left (-1\right )}}{\sqrt {x^{2} e + d}} - \frac {4 \, d x^{2} e^{\left (-2\right )}}{\sqrt {x^{2} e + d}} - \frac {8 \, d^{2} e^{\left (-3\right )}}{\sqrt {x^{2} e + d}}\right )} b \log \left (c x^{n}\right ) + \frac {1}{3} \, {\left (\frac {x^{4} e^{\left (-1\right )}}{\sqrt {x^{2} e + d}} - \frac {4 \, d x^{2} e^{\left (-2\right )}}{\sqrt {x^{2} e + d}} - \frac {8 \, d^{2} e^{\left (-3\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 = 0.44, size = 351, normalized size = 2.22 \begin {gather*} \left [\frac {12 \, {\left (b d n x^{2} e + b d^{2} 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 - 3 \, a\right )} x^{4} e^{2} - 14 \, b d^{2} n - {\left (13 \, b d n - 12 \, a d\right )} x^{2} e + 24 \, a d^{2} - 3 \, {\left (b x^{4} e^{2} - 4 \, b d x^{2} e - 8 \, b d^{2}\right )} \log \left (c\right ) - 3 \, {\left (b n x^{4} e^{2} - 4 \, b d n x^{2} e - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{9 \, {\left (x^{2} e^{4} + d e^{3}\right )}}, \frac {24 \, {\left (b d n x^{2} e + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {x^{2} e + d}}\right ) - {\left ({\left (b n - 3 \, a\right )} x^{4} e^{2} - 14 \, b d^{2} n - {\left (13 \, b d n - 12 \, a d\right )} x^{2} e + 24 \, a d^{2} - 3 \, {\left (b x^{4} e^{2} - 4 \, b d x^{2} e - 8 \, b d^{2}\right )} \log \left (c\right ) - 3 \, {\left (b n x^{4} e^{2} - 4 \, b d n x^{2} e - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{9 \, {\left (x^{2} e^{4} + d e^{3}\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^{5} \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 \frac {x^5\,\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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