Optimal. Leaf size=231 \[ \frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {8 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{315 e^3}-\frac {8 b d^4 n \sqrt {d+e x^2}}{315 e^3}-\frac {8 b d^3 n \left (d+e x^2\right )^{3/2}}{945 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{5/2}}{1575 e^3}+\frac {11 b d n \left (d+e x^2\right )^{7/2}}{441 e^3}-\frac {b n \left (d+e x^2\right )^{9/2}}{81 e^3} \]
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Rubi [A] time = 0.28, antiderivative size = 231, 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 = {266, 43, 2350, 12, 1251, 897, 1261, 208} \[ \frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {8 b d^4 n \sqrt {d+e x^2}}{315 e^3}-\frac {8 b d^3 n \left (d+e x^2\right )^{3/2}}{945 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{5/2}}{1575 e^3}+\frac {8 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{315 e^3}+\frac {11 b d n \left (d+e x^2\right )^{7/2}}{441 e^3}-\frac {b n \left (d+e x^2\right )^{9/2}}{81 e^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 208
Rule 266
Rule 897
Rule 1251
Rule 1261
Rule 2350
Rubi steps
\begin {align*} \int x^5 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (8 d^2-20 d e x^2+35 e^2 x^4\right )}{315 e^3 x} \, dx\\ &=\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{5/2} \left (8 d^2-20 d e x^2+35 e^2 x^4\right )}{x} \, dx}{315 e^3}\\ &=\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(b n) \operatorname {Subst}\left (\int \frac {(d+e x)^{5/2} \left (8 d^2-20 d e x+35 e^2 x^2\right )}{x} \, dx,x,x^2\right )}{630 e^3}\\ &=\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(b n) \operatorname {Subst}\left (\int \frac {x^6 \left (63 d^2-90 d x^2+35 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{315 e^4}\\ &=\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {(b n) \operatorname {Subst}\left (\int \left (8 d^4 e+8 d^3 e x^2+8 d^2 e x^4-55 d e x^6+35 e x^8+\frac {8 d^5}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{315 e^4}\\ &=-\frac {8 b d^4 n \sqrt {d+e x^2}}{315 e^3}-\frac {8 b d^3 n \left (d+e x^2\right )^{3/2}}{945 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{5/2}}{1575 e^3}+\frac {11 b d n \left (d+e x^2\right )^{7/2}}{441 e^3}-\frac {b n \left (d+e x^2\right )^{9/2}}{81 e^3}+\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {\left (8 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{315 e^4}\\ &=-\frac {8 b d^4 n \sqrt {d+e x^2}}{315 e^3}-\frac {8 b d^3 n \left (d+e x^2\right )^{3/2}}{945 e^3}-\frac {8 b d^2 n \left (d+e x^2\right )^{5/2}}{1575 e^3}+\frac {11 b d n \left (d+e x^2\right )^{7/2}}{441 e^3}-\frac {b n \left (d+e x^2\right )^{9/2}}{81 e^3}+\frac {8 b d^{9/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{315 e^3}+\frac {d^2 \left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^3}+\frac {\left (d+e x^2\right )^{9/2} \left (a+b \log \left (c x^n\right )\right )}{9 e^3}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 256, normalized size = 1.11 \[ \frac {\sqrt {d+e x^2} \left (2 d^4 \left (1260 a+1260 b \left (\log \left (c x^n\right )-n \log (x)\right )-1307 b n\right )-d^3 e x^2 \left (1260 a+1260 b \left (\log \left (c x^n\right )-n \log (x)\right )-677 b n\right )+3 d^2 e^2 x^4 \left (315 a+315 b \left (\log \left (c x^n\right )-n \log (x)\right )-143 b n\right )+25 d e^3 x^6 \left (630 a+630 b \left (\log \left (c x^n\right )-n \log (x)\right )-97 b n\right )+1225 e^4 x^8 \left (9 a+9 b \log \left (c x^n\right )-9 b n \log (x)-b n\right )\right )+2520 b d^{9/2} n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )-2520 b d^{9/2} n \log (x)+315 b n \log (x) \left (d+e x^2\right )^{5/2} \left (8 d^2-20 d e x^2+35 e^2 x^4\right )}{99225 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 514, normalized size = 2.23 \[ \left [\frac {1260 \, b d^{\frac {9}{2}} n \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (1225 \, {\left (b e^{4} n - 9 \, a e^{4}\right )} x^{8} + 25 \, {\left (97 \, b d e^{3} n - 630 \, a d e^{3}\right )} x^{6} + 2614 \, b d^{4} n - 2520 \, a d^{4} + 3 \, {\left (143 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{4} - {\left (677 \, b d^{3} e n - 1260 \, a d^{3} e\right )} x^{2} - 315 \, {\left (35 \, b e^{4} x^{8} + 50 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} - 4 \, b d^{3} e x^{2} + 8 \, b d^{4}\right )} \log \relax (c) - 315 \, {\left (35 \, b e^{4} n x^{8} + 50 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} - 4 \, b d^{3} e n x^{2} + 8 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{99225 \, e^{3}}, -\frac {2520 \, b \sqrt {-d} d^{4} n \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (1225 \, {\left (b e^{4} n - 9 \, a e^{4}\right )} x^{8} + 25 \, {\left (97 \, b d e^{3} n - 630 \, a d e^{3}\right )} x^{6} + 2614 \, b d^{4} n - 2520 \, a d^{4} + 3 \, {\left (143 \, b d^{2} e^{2} n - 315 \, a d^{2} e^{2}\right )} x^{4} - {\left (677 \, b d^{3} e n - 1260 \, a d^{3} e\right )} x^{2} - 315 \, {\left (35 \, b e^{4} x^{8} + 50 \, b d e^{3} x^{6} + 3 \, b d^{2} e^{2} x^{4} - 4 \, b d^{3} e x^{2} + 8 \, b d^{4}\right )} \log \relax (c) - 315 \, {\left (35 \, b e^{4} n x^{8} + 50 \, b d e^{3} n x^{6} + 3 \, b d^{2} e^{2} n x^{4} - 4 \, b d^{3} e n x^{2} + 8 \, b d^{4} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{99225 \, e^{3}}\right ] \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 234, normalized size = 1.01 \[ -\frac {1}{99225} \, {\left (\frac {1260 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{e^{3}} + \frac {1225 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 2475 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 504 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} + 840 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} + 2520 \, \sqrt {e x^{2} + d} d^{4}}{e^{3}}\right )} b n + \frac {1}{315} \, {\left (\frac {35 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{e} - \frac {20 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d x^{2}}{e^{2}} + \frac {8 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} b \log \left (c x^{n}\right ) + \frac {1}{315} \, {\left (\frac {35 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{e} - \frac {20 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d x^{2}}{e^{2}} + \frac {8 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2}}{e^{3}}\right )} a \]
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 \[ \int x^5\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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