Optimal. Leaf size=106 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (c d^2-e (3 b d-5 a e)\right )}{2 d^{7/2} \sqrt {e}}+\frac {x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}-\frac {b d-2 a e}{d^3 x}-\frac {a}{3 d^2 x^3} \]
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Rubi [A] time = 0.14, antiderivative size = 106, 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, 1261, 205} \[ \frac {x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (c d^2-e (3 b d-5 a e)\right )}{2 d^{7/2} \sqrt {e}}-\frac {b d-2 a e}{d^3 x}-\frac {a}{3 d^2 x^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1259
Rule 1261
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^4 \left (d+e x^2\right )^2} \, dx &=\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac {\int \frac {2 a d^2 e^2+2 d e^2 (b d-a e) x^2+e^2 \left (c d^2-b d e+a e^2\right ) x^4}{x^4 \left (d+e x^2\right )} \, dx}{2 d^3 e^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac {\int \left (\frac {2 a d e^2}{x^4}-\frac {2 e^2 (-b d+2 a e)}{x^2}+\frac {e^2 \left (c d^2-e (3 b d-5 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^3 e^2}\\ &=-\frac {a}{3 d^2 x^3}-\frac {b d-2 a e}{d^3 x}+\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac {\left (c d^2-e (3 b d-5 a e)\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^3}\\ &=-\frac {a}{3 d^2 x^3}-\frac {b d-2 a e}{d^3 x}+\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac {\left (c d^2-e (3 b d-5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{7/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 105, normalized size = 0.99 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 a e^2-3 b d e+c d^2\right )}{2 d^{7/2} \sqrt {e}}+\frac {x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}+\frac {2 a e-b d}{d^3 x}-\frac {a}{3 d^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 316, normalized size = 2.98 \[ \left [-\frac {4 \, a d^{3} e - 6 \, {\left (c d^{3} e - 3 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{4} + 4 \, {\left (3 \, b d^{3} e - 5 \, a d^{2} e^{2}\right )} x^{2} + 3 \, {\left ({\left (c d^{2} e - 3 \, b d e^{2} + 5 \, a e^{3}\right )} x^{5} + {\left (c d^{3} - 3 \, b d^{2} e + 5 \, a d e^{2}\right )} x^{3}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{12 \, {\left (d^{4} e^{2} x^{5} + d^{5} e x^{3}\right )}}, -\frac {2 \, a d^{3} e - 3 \, {\left (c d^{3} e - 3 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{4} + 2 \, {\left (3 \, b d^{3} e - 5 \, a d^{2} e^{2}\right )} x^{2} - 3 \, {\left ({\left (c d^{2} e - 3 \, b d e^{2} + 5 \, a e^{3}\right )} x^{5} + {\left (c d^{3} - 3 \, b d^{2} e + 5 \, a d e^{2}\right )} x^{3}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right )}{6 \, {\left (d^{4} e^{2} x^{5} + d^{5} e x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 94, normalized size = 0.89 \[ \frac {{\left (c d^{2} - 3 \, b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {7}{2}}} + \frac {c d^{2} x - b d x e + a x e^{2}}{2 \, {\left (x^{2} e + d\right )} d^{3}} - \frac {3 \, b d x^{2} - 6 \, a x^{2} e + a d}{3 \, d^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 1.38 \[ \frac {a \,e^{2} x}{2 \left (e \,x^{2}+d \right ) d^{3}}+\frac {5 a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{3}}-\frac {b e x}{2 \left (e \,x^{2}+d \right ) d^{2}}-\frac {3 b e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d^{2}}+\frac {c x}{2 \left (e \,x^{2}+d \right ) d}+\frac {c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}\, d}+\frac {2 a e}{d^{3} x}-\frac {b}{d^{2} x}-\frac {a}{3 d^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 103, normalized size = 0.97 \[ \frac {3 \, {\left (c d^{2} - 3 \, b d e + 5 \, a e^{2}\right )} x^{4} - 2 \, a d^{2} - 2 \, {\left (3 \, b d^{2} - 5 \, a d e\right )} x^{2}}{6 \, {\left (d^{3} e x^{5} + d^{4} x^{3}\right )}} + \frac {{\left (c d^{2} - 3 \, b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 98, normalized size = 0.92 \[ \frac {\frac {x^2\,\left (5\,a\,e-3\,b\,d\right )}{3\,d^2}-\frac {a}{3\,d}+\frac {x^4\,\left (c\,d^2-3\,b\,d\,e+5\,a\,e^2\right )}{2\,d^3}}{e\,x^5+d\,x^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (c\,d^2-3\,b\,d\,e+5\,a\,e^2\right )}{2\,d^{7/2}\,\sqrt {e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 167, normalized size = 1.58 \[ - \frac {\sqrt {- \frac {1}{d^{7} e}} \left (5 a e^{2} - 3 b d e + c d^{2}\right ) \log {\left (- d^{4} \sqrt {- \frac {1}{d^{7} e}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{7} e}} \left (5 a e^{2} - 3 b d e + c d^{2}\right ) \log {\left (d^{4} \sqrt {- \frac {1}{d^{7} e}} + x \right )}}{4} + \frac {- 2 a d^{2} + x^{4} \left (15 a e^{2} - 9 b d e + 3 c d^{2}\right ) + x^{2} \left (10 a d e - 6 b d^{2}\right )}{6 d^{4} x^{3} + 6 d^{3} e x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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