Optimal. Leaf size=124 \[ \frac {x \left (9 c d^2-e (5 b d-a e)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac {x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac {c x}{e^3} \]
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Rubi [A] time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, 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 = {1257, 1157, 388, 205} \[ \frac {x \left (9 c d^2-e (5 b d-a e)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac {x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac {c x}{e^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 1157
Rule 1257
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-c d^2+b d e-a e^2+4 e (c d-b e) x^2-4 c e^2 x^4}{\left (d+e x^2\right )^2} \, dx}{4 e^3}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac {\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 c d^2+e (3 b d+a e)+8 c d e x^2}{d+e x^2} \, dx}{8 d e^3}\\ &=\frac {c x}{e^3}-\frac {\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac {\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}-\frac {\left (15 c d^2-e (3 b d+a e)\right ) \int \frac {1}{d+e x^2} \, dx}{8 d e^3}\\ &=\frac {c x}{e^3}-\frac {\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac {\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}-\frac {\left (15 c d^2-e (3 b d+a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 122, normalized size = 0.98 \[ \frac {x \left (a e^2-5 b d e+9 c d^2\right )}{8 d e^3 \left (d+e x^2\right )}-\frac {x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-a e^2-3 b d e+15 c d^2\right )}{8 d^{3/2} e^{7/2}}+\frac {c x}{e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 421, normalized size = 3.40 \[ \left [\frac {16 \, c d^{2} e^{3} x^{5} + 2 \, {\left (25 \, c d^{3} e^{2} - 5 \, b d^{2} e^{3} + a d e^{4}\right )} x^{3} + {\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} + {\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \, {\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (15 \, c d^{4} e - 3 \, b d^{3} e^{2} - a d^{2} e^{3}\right )} x}{16 \, {\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )}}, \frac {8 \, c d^{2} e^{3} x^{5} + {\left (25 \, c d^{3} e^{2} - 5 \, b d^{2} e^{3} + a d e^{4}\right )} x^{3} - {\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} + {\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \, {\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (15 \, c d^{4} e - 3 \, b d^{3} e^{2} - a d^{2} e^{3}\right )} x}{8 \, {\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 107, normalized size = 0.86 \[ c x e^{\left (-3\right )} - \frac {{\left (15 \, c d^{2} - 3 \, b d e - a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {7}{2}\right )}}{8 \, d^{\frac {3}{2}}} + \frac {{\left (9 \, c d^{2} x^{3} e - 5 \, b d x^{3} e^{2} + 7 \, c d^{3} x + a x^{3} e^{3} - 3 \, b d^{2} x e - a d x e^{2}\right )} e^{\left (-3\right )}}{8 \, {\left (x^{2} e + d\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 179, normalized size = 1.44 \[ \frac {a \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} d}-\frac {5 b \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e}+\frac {9 c d \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{2}}-\frac {a x}{8 \left (e \,x^{2}+d \right )^{2} e}-\frac {3 b d x}{8 \left (e \,x^{2}+d \right )^{2} e^{2}}+\frac {7 c \,d^{2} x}{8 \left (e \,x^{2}+d \right )^{2} e^{3}}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, d e}+\frac {3 b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{2}}-\frac {15 c d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{3}}+\frac {c x}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 126, normalized size = 1.02 \[ \frac {{\left (9 \, c d^{2} e - 5 \, b d e^{2} + a e^{3}\right )} x^{3} + {\left (7 \, c d^{3} - 3 \, b d^{2} e - a d e^{2}\right )} x}{8 \, {\left (d e^{5} x^{4} + 2 \, d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}} + \frac {c x}{e^{3}} - \frac {{\left (15 \, c d^{2} - 3 \, b d e - a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 118, normalized size = 0.95 \[ \frac {c\,x}{e^3}-\frac {x\,\left (-\frac {7\,c\,d^2}{8}+\frac {3\,b\,d\,e}{8}+\frac {a\,e^2}{8}\right )-\frac {x^3\,\left (9\,c\,d^2\,e-5\,b\,d\,e^2+a\,e^3\right )}{8\,d}}{d^2\,e^3+2\,d\,e^4\,x^2+e^5\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (-15\,c\,d^2+3\,b\,d\,e+a\,e^2\right )}{8\,d^{3/2}\,e^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.62, size = 201, normalized size = 1.62 \[ \frac {c x}{e^{3}} - \frac {\sqrt {- \frac {1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log {\left (- d^{2} e^{3} \sqrt {- \frac {1}{d^{3} e^{7}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log {\left (d^{2} e^{3} \sqrt {- \frac {1}{d^{3} e^{7}}} + x \right )}}{16} + \frac {x^{3} \left (a e^{3} - 5 b d e^{2} + 9 c d^{2} e\right ) + x \left (- a d e^{2} - 3 b d^{2} e + 7 c d^{3}\right )}{8 d^{3} e^{3} + 16 d^{2} e^{4} x^{2} + 8 d e^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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