Optimal. Leaf size=84 \[ -\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{3/2} e}+\frac {b n}{3 d e \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2338, 266, 51, 63, 208} \[ -\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{3/2} e}+\frac {b n}{3 d e \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 2338
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b n) \int \frac {1}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e}\\ &=\frac {b n}{3 d e \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d e}\\ &=\frac {b n}{3 d e \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 d e^2}\\ &=\frac {b n}{3 d e \sqrt {d+e x^2}}-\frac {b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{3/2} e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 97, normalized size = 1.15 \[ -\frac {\frac {a}{\left (d+e x^2\right )^{3/2}}+\frac {b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}}+\frac {b n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{d^{3/2}}-\frac {b n \log (x)}{d^{3/2}}-\frac {b n}{d \sqrt {d+e x^2}}}{3 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 267, normalized size = 3.18 \[ \left [\frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (b d e n x^{2} - b d^{2} n \log \relax (x) + b d^{2} n - b d^{2} \log \relax (c) - a d^{2}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (b d e n x^{2} - b d^{2} n \log \relax (x) + b d^{2} n - b d^{2} \log \relax (c) - a d^{2}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 75, normalized size = 0.89 \[ -\frac {b n {\left (\frac {\operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {1}{\sqrt {e x^{2} + d} d}\right )}}{3 \, e} - \frac {b \log \left (c x^{n}\right )}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e} - \frac {a}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e} \]
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 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 33.60, size = 245, normalized size = 2.92 \[ - \frac {a}{3 e \left (d + e x^{2}\right )^{\frac {3}{2}}} + \frac {2 b d^{3} n \sqrt {1 + \frac {e x^{2}}{d}}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} + \frac {b d^{3} n \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} - \frac {2 b d^{3} n \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} + \frac {b d^{2} n x^{2} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} + 6 d^{\frac {7}{2}} e x^{2}} - \frac {2 b d^{2} n x^{2} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} + 6 d^{\frac {7}{2}} e x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{3 e \left (d + e x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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