Optimal. Leaf size=138 \[ -\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}+\frac {b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5} \]
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Rubi [A]
time = 0.08, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2373, 283, 223,
212} \begin {gather*} -\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d}-\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 283
Rule 2373
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2}}{x^6} \, dx}{5 d}\\ &=-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {(b e n) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{5 d}\\ &=-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {\left (b e^2 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{5 d}\\ &=-\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {\left (b e^3 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{5 d}\\ &=-\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}+\frac {\left (b e^3 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{5 d}\\ &=-\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}+\frac {b e^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 114, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {d+e x^2} \left (15 a \left (d+e x^2\right )^2+b n \left (3 d^2+11 d e x^2+23 e^2 x^4\right )\right )+15 b \left (d+e x^2\right )^{5/2} \log \left (c x^n\right )-15 b e^{5/2} n x^5 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{75 d x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{6}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 159, normalized size = 1.15 \begin {gather*} \frac {{\left (15 \, \operatorname {arsinh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {5}{2}} + \frac {10 \, {\left (x^{2} e + d\right )}^{\frac {3}{2}} x e^{3}}{d^{2}} + \frac {15 \, \sqrt {x^{2} e + d} x e^{3}}{d} - \frac {8 \, {\left (x^{2} e + d\right )}^{\frac {5}{2}} e^{2}}{d^{2} x} - \frac {2 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}} e}{d^{2} x^{3}} - \frac {3 \, {\left (x^{2} e + d\right )}^{\frac {7}{2}}}{d x^{5}}\right )} b n}{75 \, d} - \frac {{\left (x^{2} e + d\right )}^{\frac {5}{2}} b \log \left (c x^{n}\right )}{5 \, d x^{5}} - \frac {{\left (x^{2} e + d\right )}^{\frac {5}{2}} a}{5 \, d x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 156, normalized size = 1.13 \begin {gather*} \frac {15 \, b n x^{5} e^{\frac {5}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) - 2 \, {\left ({\left (23 \, b n + 15 \, a\right )} x^{4} e^{2} + 3 \, 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}}{150 \, d x^{5}} \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 {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, 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 {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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