Optimal. Leaf size=212 \[ \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^4}-\frac {b d^2 n}{3 e^4 \sqrt {d+e x^2}}+\frac {8 b d n \sqrt {d+e x^2}}{3 e^4}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^4} \]
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Rubi [A] time = 0.32, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {266, 43, 2350, 12, 1799, 1619, 63, 208} \[ \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac {b d^2 n}{3 e^4 \sqrt {d+e x^2}}-\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^4}+\frac {8 b d n \sqrt {d+e x^2}}{3 e^4}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 208
Rule 266
Rule 1619
Rule 1799
Rule 2350
Rubi steps
\begin {align*} \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-(b n) \int \frac {-16 d^3-24 d^2 e x^2-6 d e^2 x^4+e^3 x^6}{3 e^4 x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac {(b n) \int \frac {-16 d^3-24 d^2 e x^2-6 d e^2 x^4+e^3 x^6}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^4}\\ &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac {(b n) \operatorname {Subst}\left (\int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^4}\\ &=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}-\frac {(b n) \operatorname {Subst}\left (\int \left (-\frac {d^2 e}{(d+e x)^{3/2}}-\frac {7 d e}{\sqrt {d+e x}}-\frac {16 d^2}{x \sqrt {d+e x}}+\frac {e^2 x}{\sqrt {d+e x}}\right ) \, dx,x,x^2\right )}{6 e^4}\\ &=-\frac {b d^2 n}{3 e^4 \sqrt {d+e x^2}}+\frac {7 b d n \sqrt {d+e x^2}}{3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {\left (8 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^4}-\frac {(b n) \operatorname {Subst}\left (\int \frac {x}{\sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac {b d^2 n}{3 e^4 \sqrt {d+e x^2}}+\frac {7 b d n \sqrt {d+e x^2}}{3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {\left (16 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^5}-\frac {(b n) \operatorname {Subst}\left (\int \left (-\frac {d}{e \sqrt {d+e x}}+\frac {\sqrt {d+e x}}{e}\right ) \, dx,x,x^2\right )}{6 e^2}\\ &=-\frac {b d^2 n}{3 e^4 \sqrt {d+e x^2}}+\frac {8 b d n \sqrt {d+e x^2}}{3 e^4}-\frac {b n \left (d+e x^2\right )^{3/2}}{9 e^4}-\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \left (d+e x^2\right )^{3/2}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d+e x^2}}-\frac {3 d \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^4}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 240, normalized size = 1.13 \[ \frac {-48 a d^3-72 a d^2 e x^2-18 a d e^2 x^4+3 a e^3 x^6-3 b \left (16 d^3+24 d^2 e x^2+6 d e^2 x^4-e^3 x^6\right ) \log \left (c x^n\right )-48 b d^{3/2} e n x^2 \sqrt {d+e x^2} \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )+48 b d^{3/2} n \log (x) \left (d+e x^2\right )^{3/2}-48 b d^{5/2} n \sqrt {d+e x^2} \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )+20 b d^3 n+42 b d^2 e n x^2+21 b d e^2 n x^4-b e^3 n x^6}{9 e^4 \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 504, normalized size = 2.38 \[ \left [\frac {24 \, {\left (b d e^{2} n x^{4} + 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left ({\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} - 20 \, b d^{3} n - 3 \, {\left (7 \, b d e^{2} n - 6 \, a d e^{2}\right )} x^{4} + 48 \, a d^{3} - 6 \, {\left (7 \, b d^{2} e n - 12 \, a d^{2} e\right )} x^{2} - 3 \, {\left (b e^{3} x^{6} - 6 \, b d e^{2} x^{4} - 24 \, b d^{2} e x^{2} - 16 \, b d^{3}\right )} \log \relax (c) - 3 \, {\left (b e^{3} n x^{6} - 6 \, b d e^{2} n x^{4} - 24 \, b d^{2} e n x^{2} - 16 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{9 \, {\left (e^{6} x^{4} + 2 \, d e^{5} x^{2} + d^{2} e^{4}\right )}}, \frac {48 \, {\left (b d e^{2} n x^{4} + 2 \, b d^{2} e n x^{2} + b d^{3} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) - {\left ({\left (b e^{3} n - 3 \, a e^{3}\right )} x^{6} - 20 \, b d^{3} n - 3 \, {\left (7 \, b d e^{2} n - 6 \, a d e^{2}\right )} x^{4} + 48 \, a d^{3} - 6 \, {\left (7 \, b d^{2} e n - 12 \, a d^{2} e\right )} x^{2} - 3 \, {\left (b e^{3} x^{6} - 6 \, b d e^{2} x^{4} - 24 \, b d^{2} e x^{2} - 16 \, b d^{3}\right )} \log \relax (c) - 3 \, {\left (b e^{3} n x^{6} - 6 \, b d e^{2} n x^{4} - 24 \, b d^{2} e n x^{2} - 16 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{9 \, {\left (e^{6} x^{4} + 2 \, d e^{5} x^{2} + d^{2} e^{4}\right )}}\right ] \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{7}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 246, normalized size = 1.16 \[ \frac {1}{9} \, b n {\left (\frac {24 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{e^{4}} - \frac {3 \, d^{2}}{\sqrt {e x^{2} + d} e^{4}} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} - 24 \, \sqrt {e x^{2} + d} d}{e^{4}}\right )} + \frac {1}{3} \, {\left (\frac {x^{6}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} - \frac {6 \, d x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} - \frac {24 \, d^{2} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}} - \frac {16 \, d^{3}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{4}}\right )} b \log \left (c x^{n}\right ) + \frac {1}{3} \, {\left (\frac {x^{6}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} - \frac {6 \, d x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} - \frac {24 \, d^{2} x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}} - \frac {16 \, d^{3}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^7\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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