Optimal. Leaf size=136 \[ -\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac {c d^2-e (2 b d-3 a e)}{d^4 x}-\frac {b d-2 a e}{3 d^3 x^3}-\frac {a}{5 d^2 x^5} \]
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Rubi [A] time = 0.25, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1259, 1802, 205} \[ -\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}-\frac {c d^2-e (2 b d-3 a e)}{d^4 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (5 b d-7 a e)\right )}{2 d^{9/2}}-\frac {b d-2 a e}{3 d^3 x^3}-\frac {a}{5 d^2 x^5} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1259
Rule 1802
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^2} \, dx &=-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\int \frac {-2 a d^3 e^2-2 d^2 e^2 (b d-a e) x^2-2 d e^2 \left (c d^2-b d e+a e^2\right ) x^4+e^3 \left (c d^2-b d e+a e^2\right ) x^6}{x^6 \left (d+e x^2\right )} \, dx}{2 d^4 e^2}\\ &=-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\int \left (-\frac {2 a d^2 e^2}{x^6}-\frac {2 d e^2 (b d-2 a e)}{x^4}+\frac {2 e^2 \left (-c d^2+e (2 b d-3 a e)\right )}{x^2}+\frac {e^3 \left (3 c d^2-e (5 b d-7 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^4 e^2}\\ &=-\frac {a}{5 d^2 x^5}-\frac {b d-2 a e}{3 d^3 x^3}-\frac {c d^2-e (2 b d-3 a e)}{d^4 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\left (e \left (3 c d^2-e (5 b d-7 a e)\right )\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^4}\\ &=-\frac {a}{5 d^2 x^5}-\frac {b d-2 a e}{3 d^3 x^3}-\frac {c d^2-e (2 b d-3 a e)}{d^4 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{2 d^4 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (3 c d^2-e (5 b d-7 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 135, normalized size = 0.99 \[ -\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (7 a e^2-5 b d e+3 c d^2\right )}{2 d^{9/2}}-\frac {e x \left (a e^2-b d e+c d^2\right )}{2 d^4 \left (d+e x^2\right )}+\frac {-3 a e^2+2 b d e-c d^2}{d^4 x}+\frac {2 a e-b d}{3 d^3 x^3}-\frac {a}{5 d^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 360, normalized size = 2.65 \[ \left [-\frac {30 \, {\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 20 \, {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 12 \, a d^{3} + 4 \, {\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} - 15 \, {\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} + {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} - 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right )}{60 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}, -\frac {15 \, {\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 10 \, {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 6 \, a d^{3} + 2 \, {\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2} + 15 \, {\left ({\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{7} + {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{5}\right )} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right )}{30 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 131, normalized size = 0.96 \[ -\frac {{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {9}{2}}} - \frac {c d^{2} x e - b d x e^{2} + a x e^{3}}{2 \, {\left (x^{2} e + d\right )} d^{4}} - \frac {15 \, c d^{2} x^{4} - 30 \, b d x^{4} e + 45 \, a x^{4} e^{2} + 5 \, b d^{2} x^{2} - 10 \, a d x^{2} e + 3 \, a d^{2}}{15 \, d^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 183, normalized size = 1.35 \[ -\frac {a \,e^{3} x}{2 \left (e \,x^{2}+d \right ) d^{4}}-\frac {7 a \,e^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{4}}+\frac {b \,e^{2} x}{2 \left (e \,x^{2}+d \right ) d^{3}}+\frac {5 b \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{3}}-\frac {c e x}{2 \left (e \,x^{2}+d \right ) d^{2}}-\frac {3 c e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{2}}-\frac {3 a \,e^{2}}{d^{4} x}+\frac {2 b e}{d^{3} x}-\frac {c}{d^{2} x}+\frac {2 a e}{3 d^{3} x^{3}}-\frac {b}{3 d^{2} x^{3}}-\frac {a}{5 d^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 139, normalized size = 1.02 \[ -\frac {15 \, {\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} x^{6} + 10 \, {\left (3 \, c d^{3} - 5 \, b d^{2} e + 7 \, a d e^{2}\right )} x^{4} + 6 \, a d^{3} + 2 \, {\left (5 \, b d^{3} - 7 \, a d^{2} e\right )} x^{2}}{30 \, {\left (d^{4} e x^{7} + d^{5} x^{5}\right )}} - \frac {{\left (3 \, c d^{2} e - 5 \, b d e^{2} + 7 \, a e^{3}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 128, normalized size = 0.94 \[ -\frac {\frac {a}{5\,d}-\frac {x^2\,\left (7\,a\,e-5\,b\,d\right )}{15\,d^2}+\frac {x^4\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{3\,d^3}+\frac {e\,x^6\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{2\,d^4}}{e\,x^7+d\,x^5}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,c\,d^2-5\,b\,d\,e+7\,a\,e^2\right )}{2\,d^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.13, size = 284, normalized size = 2.09 \[ \frac {\sqrt {- \frac {e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log {\left (- \frac {d^{5} \sqrt {- \frac {e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} - \frac {\sqrt {- \frac {e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right ) \log {\left (\frac {d^{5} \sqrt {- \frac {e}{d^{9}}} \left (7 a e^{2} - 5 b d e + 3 c d^{2}\right )}{7 a e^{3} - 5 b d e^{2} + 3 c d^{2} e} + x \right )}}{4} + \frac {- 6 a d^{3} + x^{6} \left (- 105 a e^{3} + 75 b d e^{2} - 45 c d^{2} e\right ) + x^{4} \left (- 70 a d e^{2} + 50 b d^{2} e - 30 c d^{3}\right ) + x^{2} \left (14 a d^{2} e - 10 b d^{3}\right )}{30 d^{5} x^{5} + 30 d^{4} e x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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