Optimal. Leaf size=166 \[ -\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {2 b e n x}{3 d^3 \sqrt {d+e x^2}}+\frac {8 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {b n}{d^2 x \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {271, 192, 191, 2350, 12, 1265, 385, 217, 206} \[ -\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {2 b e n x}{3 d^3 \sqrt {d+e x^2}}-\frac {b n}{d^2 x \sqrt {d+e x^2}}+\frac {8 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 206
Rule 217
Rule 271
Rule 385
Rule 1265
Rule 2350
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^{5/2}} \, dx &=-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-(b n) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{3 d^3 x^2 \left (d+e x^2\right )^{3/2}} \, dx\\ &=-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{3 d^3}\\ &=-\frac {b n}{d^2 x \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {6 d^2 e+8 d e^2 x^2}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d^4}\\ &=-\frac {b n}{d^2 x \sqrt {d+e x^2}}-\frac {2 b e n x}{3 d^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(8 b e n) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^3}\\ &=-\frac {b n}{d^2 x \sqrt {d+e x^2}}-\frac {2 b e n x}{3 d^3 \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(8 b e n) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^3}\\ &=-\frac {b n}{d^2 x \sqrt {d+e x^2}}-\frac {2 b e n x}{3 d^3 \sqrt {d+e x^2}}+\frac {8 b \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {a+b \log \left (c x^n\right )}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 144, normalized size = 0.87 \[ \frac {-3 a d^2-12 a d e x^2-8 a e^2 x^4-b \left (3 d^2+12 d e x^2+8 e^2 x^4\right ) \log \left (c x^n\right )-3 b d^2 n-5 b d e n x^2+8 b \sqrt {e} n x \left (d+e x^2\right )^{3/2} \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )-2 b e^2 n x^4}{3 d^3 x \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 399, normalized size = 2.40 \[ \left [\frac {4 \, {\left (b e^{2} n x^{5} + 2 \, b d e n x^{3} + b d^{2} n x\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, {\left (b e^{2} n + 4 \, a e^{2}\right )} x^{4} + 3 \, b d^{2} n + 3 \, a d^{2} + {\left (5 \, b d e n + 12 \, a d e\right )} x^{2} + {\left (8 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \relax (c) + {\left (8 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 3 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{3} e^{2} x^{5} + 2 \, d^{4} e x^{3} + d^{5} x\right )}}, -\frac {8 \, {\left (b e^{2} n x^{5} + 2 \, b d e n x^{3} + b d^{2} n x\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, {\left (b e^{2} n + 4 \, a e^{2}\right )} x^{4} + 3 \, b d^{2} n + 3 \, a d^{2} + {\left (5 \, b d e n + 12 \, a d e\right )} x^{2} + {\left (8 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \relax (c) + {\left (8 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 3 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{3} e^{2} x^{5} + 2 \, d^{4} e x^{3} + d^{5} x\right )}}\right ] \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a {\left (\frac {8 \, e x}{\sqrt {e x^{2} + d} d^{3}} + \frac {4 \, e x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2}} + \frac {3}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} d x}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{{\left (e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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