Optimal. Leaf size=175 \[ \frac {b c-a d}{7 a^2 x^7}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{11/2}}-\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5 x}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {c}{9 a x^9} \]
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Rubi [A] time = 0.15, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1802, 205} \[ \frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5 x}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{11/2}}-\frac {a^2 e-a b d+b^2 c}{5 a^3 x^5}+\frac {b c-a d}{7 a^2 x^7}-\frac {c}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1802
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^{10}}+\frac {-b c+a d}{a^2 x^8}+\frac {b^2 c-a b d+a^2 e}{a^3 x^6}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^4}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^2}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {\left (b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{a^5}\\ &=-\frac {c}{9 a x^9}+\frac {b c-a d}{7 a^2 x^7}-\frac {b^2 c-a b d+a^2 e}{5 a^3 x^5}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^5 x}-\frac {b^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 174, normalized size = 0.99 \[ \frac {b c-a d}{7 a^2 x^7}+\frac {a^2 (-e)+a b d-b^2 c}{5 a^3 x^5}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{11/2}}+\frac {b \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^5 x}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{3 a^4 x^3}-\frac {c}{9 a x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 374, normalized size = 2.14 \[ \left [-\frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 630 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 210 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 70 \, a^{4} c + 126 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 90 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{630 \, a^{5} x^{9}}, -\frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{9} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 201, normalized size = 1.15 \[ -\frac {{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, b^{4} c x^{8} - 315 \, a b^{3} d x^{8} - 315 \, a^{3} b f x^{8} + 315 \, a^{2} b^{2} x^{8} e - 105 \, a b^{3} c x^{6} + 105 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 105 \, a^{3} b x^{6} e + 63 \, a^{2} b^{2} c x^{4} - 63 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 45 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{5} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 238, normalized size = 1.36 \[ \frac {b^{2} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {b^{4} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {b^{5} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{5}}+\frac {b f}{a^{2} x}-\frac {b^{2} e}{a^{3} x}+\frac {b^{3} d}{a^{4} x}-\frac {b^{4} c}{a^{5} x}-\frac {f}{3 a \,x^{3}}+\frac {b e}{3 a^{2} x^{3}}-\frac {b^{2} d}{3 a^{3} x^{3}}+\frac {b^{3} c}{3 a^{4} x^{3}}-\frac {e}{5 a \,x^{5}}+\frac {b d}{5 a^{2} x^{5}}-\frac {b^{2} c}{5 a^{3} x^{5}}-\frac {d}{7 a \,x^{7}}+\frac {b c}{7 a^{2} x^{7}}-\frac {c}{9 a \,x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 175, normalized size = 1.00 \[ -\frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {315 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{8} - 105 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} + 35 \, a^{4} c + 63 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{4} - 45 \, {\left (a^{3} b c - a^{4} d\right )} x^{2}}{315 \, a^{5} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 161, normalized size = 0.92 \[ -\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{7\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{5\,a^3}+\frac {b\,x^8\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^5}}{x^9}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 32.72, size = 354, normalized size = 2.02 \[ - \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b^{3}}{a^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{2} f - a^{2} b^{3} e + a b^{4} d - b^{5} c} + x \right )}}{2} + \frac {- 35 a^{4} c + x^{8} \left (315 a^{3} b f - 315 a^{2} b^{2} e + 315 a b^{3} d - 315 b^{4} c\right ) + x^{6} \left (- 105 a^{4} f + 105 a^{3} b e - 105 a^{2} b^{2} d + 105 a b^{3} c\right ) + x^{4} \left (- 63 a^{4} e + 63 a^{3} b d - 63 a^{2} b^{2} c\right ) + x^{2} \left (- 45 a^{4} d + 45 a^{3} b c\right )}{315 a^{5} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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