Optimal. Leaf size=137 \[ \frac {b c-a d}{5 a^2 x^5}-\frac {a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{9/2}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{7 a x^7} \]
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Rubi [A] time = 0.13, antiderivative size = 137, 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} \[ \frac {a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{a^4 x}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{9/2}}-\frac {a^2 e-a b d+b^2 c}{3 a^3 x^3}+\frac {b c-a d}{5 a^2 x^5}-\frac {c}{7 a x^7} \]
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^8 \left (a+b x^2\right )} \, dx &=\int \left (\frac {c}{a x^8}+\frac {-b c+a d}{a^2 x^6}+\frac {b^2 c-a b d+a^2 e}{a^3 x^4}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\left (b \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^4}\\ &=-\frac {c}{7 a x^7}+\frac {b c-a d}{5 a^2 x^5}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{a^4 x}+\frac {\sqrt {b} \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^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 139, normalized size = 1.01 \[ \frac {b c-a d}{5 a^2 x^5}+\frac {a^2 (-e)+a b d-b^2 c}{3 a^3 x^3}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{9/2}}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac {c}{7 a x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 292, normalized size = 2.13 \[ \left [-\frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 210 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} + 70 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} + 30 \, a^{3} c - 42 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{210 \, a^{4} x^{7}}, \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 151, normalized size = 1.10 \[ \frac {{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, b^{3} c x^{6} - 105 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 105 \, a^{2} b x^{6} e - 35 \, a b^{2} c x^{4} + 35 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 21 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 190, normalized size = 1.39 \[ -\frac {b f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {b^{4} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}-\frac {f}{a x}+\frac {b e}{a^{2} x}-\frac {b^{2} d}{a^{3} x}+\frac {b^{3} c}{a^{4} x}-\frac {e}{3 a \,x^{3}}+\frac {b d}{3 a^{2} x^{3}}-\frac {b^{2} c}{3 a^{3} x^{3}}-\frac {d}{5 a \,x^{5}}+\frac {b c}{5 a^{2} x^{5}}-\frac {c}{7 a \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 134, normalized size = 0.98 \[ \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 35 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{4} - 15 \, a^{3} c + 21 \, {\left (a^{2} b c - a^{3} d\right )} x^{2}}{105 \, a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 127, normalized size = 0.93 \[ \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{9/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^4}+\frac {x^2\,\left (a\,d-b\,c\right )}{5\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,a^3}}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 21.65, size = 301, normalized size = 2.20 \[ \frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b}{a^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b f - a^{2} b^{2} e + a b^{3} d - b^{4} c} + x \right )}}{2} + \frac {- 15 a^{3} c + x^{6} \left (- 105 a^{3} f + 105 a^{2} b e - 105 a b^{2} d + 105 b^{3} c\right ) + x^{4} \left (- 35 a^{3} e + 35 a^{2} b d - 35 a b^{2} c\right ) + x^{2} \left (- 21 a^{3} d + 21 a^{2} b c\right )}{105 a^{4} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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