Optimal. Leaf size=230 \[ -\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7} \]
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Rubi [A] time = 0.20, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {271, 264, 2350, 12, 1265, 451, 277, 217, 206} \[ -\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}+\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 217
Rule 264
Rule 271
Rule 277
Rule 451
Rule 1265
Rule 2350
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}-(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-15 d^2+12 d e x^2-8 e^2 x^4\right )}{105 d^3 x^8} \, dx\\ &=-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}-\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-15 d^2+12 d e x^2-8 e^2 x^4\right )}{x^8} \, dx}{105 d^3}\\ &=-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {(b n) \int \frac {\left (d+e x^2\right )^{3/2} \left (-114 d^2 e+56 d e^2 x^2\right )}{x^6} \, dx}{735 d^4}\\ &=-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {\left (8 b e^2 n\right ) \int \frac {\left (d+e x^2\right )^{3/2}}{x^4} \, dx}{105 d^3}\\ &=-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {\left (8 b e^3 n\right ) \int \frac {\sqrt {d+e x^2}}{x^2} \, dx}{105 d^3}\\ &=-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {\left (8 b e^4 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{105 d^3}\\ &=-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}+\frac {\left (8 b e^4 n\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{105 d^3}\\ &=-\frac {8 b e^3 n \sqrt {d+e x^2}}{105 d^3 x}-\frac {8 b e^2 n \left (d+e x^2\right )^{3/2}}{315 d^3 x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{49 d^2 x^7}+\frac {38 b e n \left (d+e x^2\right )^{5/2}}{1225 d^3 x^5}+\frac {8 b e^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{105 d^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{7 d x^7}+\frac {4 e \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{35 d^2 x^5}-\frac {8 e^2 \left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{105 d^3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 180, normalized size = 0.78 \[ -\frac {\sqrt {d+e x^2} \left (105 a \left (15 d^3+3 d^2 e x^2-4 d e^2 x^4+8 e^3 x^6\right )+b n \left (225 d^3+108 d^2 e x^2-179 d e^2 x^4+778 e^3 x^6\right )\right )+105 b \sqrt {d+e x^2} \left (15 d^3+3 d^2 e x^2-4 d e^2 x^4+8 e^3 x^6\right ) \log \left (c x^n\right )-840 b e^{7/2} n x^7 \log \left (\sqrt {e} \sqrt {d+e x^2}+e x\right )}{11025 d^3 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 426, normalized size = 1.85 \[ \left [\frac {420 \, b e^{\frac {7}{2}} n x^{7} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - {\left (2 \, {\left (389 \, b e^{3} n + 420 \, a e^{3}\right )} x^{6} + 225 \, b d^{3} n - {\left (179 \, b d e^{2} n + 420 \, a d e^{2}\right )} x^{4} + 1575 \, a d^{3} + 9 \, {\left (12 \, b d^{2} e n + 35 \, a d^{2} e\right )} x^{2} + 105 \, {\left (8 \, b e^{3} x^{6} - 4 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + 15 \, b d^{3}\right )} \log \relax (c) + 105 \, {\left (8 \, b e^{3} n x^{6} - 4 \, b d e^{2} n x^{4} + 3 \, b d^{2} e n x^{2} + 15 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{11025 \, d^{3} x^{7}}, -\frac {840 \, b \sqrt {-e} e^{3} n x^{7} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (2 \, {\left (389 \, b e^{3} n + 420 \, a e^{3}\right )} x^{6} + 225 \, b d^{3} n - {\left (179 \, b d e^{2} n + 420 \, a d e^{2}\right )} x^{4} + 1575 \, a d^{3} + 9 \, {\left (12 \, b d^{2} e n + 35 \, a d^{2} e\right )} x^{2} + 105 \, {\left (8 \, b e^{3} x^{6} - 4 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + 15 \, b d^{3}\right )} \log \relax (c) + 105 \, {\left (8 \, b e^{3} n x^{6} - 4 \, b d e^{2} n x^{4} + 3 \, b d^{2} e n x^{2} + 15 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{11025 \, d^{3} x^{7}}\right ] \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e \,x^{2}+d}}{x^{8}}\, dx \]
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 \[ -\frac {1}{105} \, a {\left (\frac {8 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}{d^{3} x^{3}} - \frac {12 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d^{2} x^{5}} + \frac {15 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{7}}\right )} + b \int \frac {\sqrt {e x^{2} + d} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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