Optimal. Leaf size=127 \[ -\frac {x \left (a e^2-b d e+c d^2\right )}{4 d^2 e \left (d+e x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{8 d^{7/2} e^{3/2}}+\frac {x \left (e (3 b d-7 a e)+c d^2\right )}{8 d^3 e \left (d+e x^2\right )}-\frac {a}{d^3 x} \]
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Rubi [A] time = 0.20, antiderivative size = 124, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1259, 456, 453, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{8 d^{7/2} e^{3/2}}+\frac {x \left (e (3 b d-7 a e)+c d^2\right )}{8 d^3 e \left (d+e x^2\right )}-\frac {x \left (\frac {c}{e}-\frac {b d-a e}{d^2}\right )}{4 \left (d+e x^2\right )^2}-\frac {a}{d^3 x} \]
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rule 456
Rule 1259
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^2 \left (d+e x^2\right )^3} \, dx &=-\frac {\left (\frac {c}{e}-\frac {b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}-\frac {\int \frac {-4 a d e^2-e \left (c d^2+3 e (b d-a e)\right ) x^2}{x^2 \left (d+e x^2\right )^2} \, dx}{4 d^2 e^2}\\ &=-\frac {\left (\frac {c}{e}-\frac {b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac {\int \frac {8 a e^2+e \left (c d+e \left (3 b-\frac {7 a e}{d}\right )\right ) x^2}{x^2 \left (d+e x^2\right )} \, dx}{8 d^2 e^2}\\ &=-\frac {a}{d^3 x}-\frac {\left (\frac {c}{e}-\frac {b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac {\left (c d^2+3 e (b d-5 a e)\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^3 e}\\ &=-\frac {a}{d^3 x}-\frac {\left (\frac {c}{e}-\frac {b d-a e}{d^2}\right ) x}{4 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (3 b d-7 a e)\right ) x}{8 d^3 e \left (d+e x^2\right )}+\frac {\left (c d^2+3 e (b d-5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2} e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 124, normalized size = 0.98 \[ \frac {\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 e (b d-5 a e)+c d^2\right )}{e^{3/2}}+\frac {\sqrt {d} \left (d x^2 \left (b e \left (5 d+3 e x^2\right )+c d \left (e x^2-d\right )\right )-a e \left (8 d^2+25 d e x^2+15 e^2 x^4\right )\right )}{e x \left (d+e x^2\right )^2}}{8 d^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 421, normalized size = 3.31 \[ \left [-\frac {16 \, a d^{3} e^{2} - 2 \, {\left (c d^{3} e^{2} + 3 \, b d^{2} e^{3} - 15 \, a d e^{4}\right )} x^{4} + 2 \, {\left (c d^{4} e - 5 \, b d^{3} e^{2} + 25 \, a d^{2} e^{3}\right )} x^{2} - {\left ({\left (c d^{2} e^{2} + 3 \, b d e^{3} - 15 \, a e^{4}\right )} x^{5} + 2 \, {\left (c d^{3} e + 3 \, b d^{2} e^{2} - 15 \, a d e^{3}\right )} x^{3} + {\left (c d^{4} + 3 \, b d^{3} e - 15 \, a d^{2} e^{2}\right )} x\right )} \sqrt {-d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{16 \, {\left (d^{4} e^{4} x^{5} + 2 \, d^{5} e^{3} x^{3} + d^{6} e^{2} x\right )}}, -\frac {8 \, a d^{3} e^{2} - {\left (c d^{3} e^{2} + 3 \, b d^{2} e^{3} - 15 \, a d e^{4}\right )} x^{4} + {\left (c d^{4} e - 5 \, b d^{3} e^{2} + 25 \, a d^{2} e^{3}\right )} x^{2} - {\left ({\left (c d^{2} e^{2} + 3 \, b d e^{3} - 15 \, a e^{4}\right )} x^{5} + 2 \, {\left (c d^{3} e + 3 \, b d^{2} e^{2} - 15 \, a d e^{3}\right )} x^{3} + {\left (c d^{4} + 3 \, b d^{3} e - 15 \, a d^{2} e^{2}\right )} x\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right )}{8 \, {\left (d^{4} e^{4} x^{5} + 2 \, d^{5} e^{3} x^{3} + d^{6} e^{2} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 110, normalized size = 0.87 \[ \frac {{\left (c d^{2} + 3 \, b d e - 15 \, a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {3}{2}\right )}}{8 \, d^{\frac {7}{2}}} - \frac {a}{d^{3} x} + \frac {{\left (c d^{2} x^{3} e + 3 \, b d x^{3} e^{2} - c d^{3} x - 7 \, a x^{3} e^{3} + 5 \, b d^{2} x e - 9 \, a d x e^{2}\right )} e^{\left (-1\right )}}{8 \, {\left (x^{2} e + d\right )}^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 182, normalized size = 1.43 \[ -\frac {7 a \,e^{2} x^{3}}{8 \left (e \,x^{2}+d \right )^{2} d^{3}}+\frac {3 b e \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} d^{2}}+\frac {c \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} d}-\frac {9 a e x}{8 \left (e \,x^{2}+d \right )^{2} d^{2}}+\frac {5 b x}{8 \left (e \,x^{2}+d \right )^{2} d}-\frac {c x}{8 \left (e \,x^{2}+d \right )^{2} e}-\frac {15 a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, d^{3}}+\frac {3 b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, d^{2}}+\frac {c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, d e}-\frac {a}{d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.65, size = 129, normalized size = 1.02 \[ \frac {{\left (c d^{2} e + 3 \, b d e^{2} - 15 \, a e^{3}\right )} x^{4} - 8 \, a d^{2} e - {\left (c d^{3} - 5 \, b d^{2} e + 25 \, a d e^{2}\right )} x^{2}}{8 \, {\left (d^{3} e^{3} x^{5} + 2 \, d^{4} e^{2} x^{3} + d^{5} e x\right )}} + \frac {{\left (c d^{2} + 3 \, b d e - 15 \, a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} d^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 118, normalized size = 0.93 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (c\,d^2+3\,b\,d\,e-15\,a\,e^2\right )}{8\,d^{7/2}\,e^{3/2}}-\frac {\frac {a}{d}-\frac {x^4\,\left (c\,d^2+3\,b\,d\,e-15\,a\,e^2\right )}{8\,d^3}+\frac {x^2\,\left (c\,d^2-5\,b\,d\,e+25\,a\,e^2\right )}{8\,d^2\,e}}{d^2\,x+2\,d\,e\,x^3+e^2\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 202, normalized size = 1.59 \[ \frac {\sqrt {- \frac {1}{d^{7} e^{3}}} \left (15 a e^{2} - 3 b d e - c d^{2}\right ) \log {\left (- d^{4} e \sqrt {- \frac {1}{d^{7} e^{3}}} + x \right )}}{16} - \frac {\sqrt {- \frac {1}{d^{7} e^{3}}} \left (15 a e^{2} - 3 b d e - c d^{2}\right ) \log {\left (d^{4} e \sqrt {- \frac {1}{d^{7} e^{3}}} + x \right )}}{16} + \frac {- 8 a d^{2} e + x^{4} \left (- 15 a e^{3} + 3 b d e^{2} + c d^{2} e\right ) + x^{2} \left (- 25 a d e^{2} + 5 b d^{2} e - c d^{3}\right )}{8 d^{5} e x + 16 d^{4} e^{2} x^{3} + 8 d^{3} e^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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