Optimal. Leaf size=173 \[ \frac {d x \left (17 c d^2-e (13 b d-9 a e)\right )}{8 e^5 \left (d+e x^2\right )}+\frac {x \left (6 c d^2-e (3 b d-a e)\right )}{e^5}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{4 e^5 \left (d+e x^2\right )^2}-\frac {x^3 (3 c d-b e)}{3 e^4}+\frac {c x^5}{5 e^3} \]
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Rubi [A] time = 0.32, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1257, 1814, 1810, 205} \[ \frac {d x \left (17 c d^2-e (13 b d-9 a e)\right )}{8 e^5 \left (d+e x^2\right )}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{4 e^5 \left (d+e x^2\right )^2}+\frac {x \left (6 c d^2-e (3 b d-a e)\right )}{e^5}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}-\frac {x^3 (3 c d-b e)}{3 e^4}+\frac {c x^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1257
Rule 1810
Rule 1814
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}-\frac {\int \frac {-d^2 \left (c d^2-b d e+a e^2\right )+4 d e \left (c d^2-b d e+a e^2\right ) x^2-4 e^2 \left (c d^2-b d e+a e^2\right ) x^4+4 e^3 (c d-b e) x^6-4 c e^4 x^8}{\left (d+e x^2\right )^2} \, dx}{4 e^5}\\ &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\int \frac {-d^2 \left (15 c d^2-e (11 b d-7 a e)\right )+8 d e \left (3 c d^2-e (2 b d-a e)\right ) x^2-8 d e^2 (2 c d-b e) x^4+8 c d e^3 x^6}{d+e x^2} \, dx}{8 d e^5}\\ &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\int \left (8 d \left (6 c d^2-e (3 b d-a e)\right )-8 d e (3 c d-b e) x^2+8 c d e^2 x^4+\frac {-63 c d^4+35 b d^3 e-15 a d^2 e^2}{d+e x^2}\right ) \, dx}{8 d e^5}\\ &=\frac {\left (6 c d^2-e (3 b d-a e)\right ) x}{e^5}-\frac {(3 c d-b e) x^3}{3 e^4}+\frac {c x^5}{5 e^3}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\left (-63 c d^4+35 b d^3 e-15 a d^2 e^2\right ) \int \frac {1}{d+e x^2} \, dx}{8 d e^5}\\ &=\frac {\left (6 c d^2-e (3 b d-a e)\right ) x}{e^5}-\frac {(3 c d-b e) x^3}{3 e^4}+\frac {c x^5}{5 e^3}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}-\frac {\sqrt {d} \left (63 c d^2-5 e (7 b d-3 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 e^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 170, normalized size = 0.98 \[ \frac {x \left (d e (9 a e-13 b d)+17 c d^3\right )}{8 e^5 \left (d+e x^2\right )}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 e (3 a e-7 b d)+63 c d^2\right )}{8 e^{11/2}}+\frac {x \left (e (a e-3 b d)+6 c d^2\right )}{e^5}-\frac {x \left (d^2 e (a e-b d)+c d^4\right )}{4 e^5 \left (d+e x^2\right )^2}+\frac {x^3 (b e-3 c d)}{3 e^4}+\frac {c x^5}{5 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 504, normalized size = 2.91 \[ \left [\frac {48 \, c e^{4} x^{9} - 16 \, {\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 16 \, {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 50 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} + 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} + {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 30 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{240 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}, \frac {24 \, c e^{4} x^{9} - 8 \, {\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 8 \, {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 25 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} - 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} + {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{120 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 160, normalized size = 0.92 \[ -\frac {{\left (63 \, c d^{3} - 35 \, b d^{2} e + 15 \, a d e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {11}{2}\right )}}{8 \, \sqrt {d}} + \frac {1}{15} \, {\left (3 \, c x^{5} e^{12} - 15 \, c d x^{3} e^{11} + 5 \, b x^{3} e^{12} + 90 \, c d^{2} x e^{10} - 45 \, b d x e^{11} + 15 \, a x e^{12}\right )} e^{\left (-15\right )} + \frac {{\left (17 \, c d^{3} x^{3} e - 13 \, b d^{2} x^{3} e^{2} + 15 \, c d^{4} x + 9 \, a d x^{3} e^{3} - 11 \, b d^{3} x e + 7 \, a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{8 \, {\left (x^{2} e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 239, normalized size = 1.38 \[ \frac {9 a d \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{2}}-\frac {13 b \,d^{2} x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{3}}+\frac {17 c \,d^{3} x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{4}}+\frac {c \,x^{5}}{5 e^{3}}+\frac {7 a \,d^{2} x}{8 \left (e \,x^{2}+d \right )^{2} e^{3}}-\frac {11 b \,d^{3} x}{8 \left (e \,x^{2}+d \right )^{2} e^{4}}+\frac {b \,x^{3}}{3 e^{3}}+\frac {15 c \,d^{4} x}{8 \left (e \,x^{2}+d \right )^{2} e^{5}}-\frac {c d \,x^{3}}{e^{4}}-\frac {15 a d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{3}}+\frac {35 b \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{4}}-\frac {63 c \,d^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{5}}+\frac {a x}{e^{3}}-\frac {3 b d x}{e^{4}}+\frac {6 c \,d^{2} x}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 175, normalized size = 1.01 \[ \frac {{\left (17 \, c d^{3} e - 13 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{3} + {\left (15 \, c d^{4} - 11 \, b d^{3} e + 7 \, a d^{2} e^{2}\right )} x}{8 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}} - \frac {{\left (63 \, c d^{3} - 35 \, b d^{2} e + 15 \, a d e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} e^{5}} + \frac {3 \, c e^{2} x^{5} - 5 \, {\left (3 \, c d e - b e^{2}\right )} x^{3} + 15 \, {\left (6 \, c d^{2} - 3 \, b d e + a e^{2}\right )} x}{15 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 223, normalized size = 1.29 \[ x^3\,\left (\frac {b}{3\,e^3}-\frac {c\,d}{e^4}\right )-x\,\left (\frac {3\,c\,d^2}{e^5}-\frac {a}{e^3}+\frac {3\,d\,\left (\frac {b}{e^3}-\frac {3\,c\,d}{e^4}\right )}{e}\right )+\frac {\left (\frac {17\,c\,d^3\,e}{8}-\frac {13\,b\,d^2\,e^2}{8}+\frac {9\,a\,d\,e^3}{8}\right )\,x^3+\left (\frac {15\,c\,d^4}{8}-\frac {11\,b\,d^3\,e}{8}+\frac {7\,a\,d^2\,e^2}{8}\right )\,x}{d^2\,e^5+2\,d\,e^6\,x^2+e^7\,x^4}+\frac {c\,x^5}{5\,e^3}-\frac {\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e}\,x\,\left (63\,c\,d^2-35\,b\,d\,e+15\,a\,e^2\right )}{63\,c\,d^3-35\,b\,d^2\,e+15\,a\,d\,e^2}\right )\,\left (63\,c\,d^2-35\,b\,d\,e+15\,a\,e^2\right )}{8\,e^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.58, size = 235, normalized size = 1.36 \[ \frac {c x^{5}}{5 e^{3}} + x^{3} \left (\frac {b}{3 e^{3}} - \frac {c d}{e^{4}}\right ) + x \left (\frac {a}{e^{3}} - \frac {3 b d}{e^{4}} + \frac {6 c d^{2}}{e^{5}}\right ) + \frac {\sqrt {- \frac {d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log {\left (- e^{5} \sqrt {- \frac {d}{e^{11}}} + x \right )}}{16} - \frac {\sqrt {- \frac {d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log {\left (e^{5} \sqrt {- \frac {d}{e^{11}}} + x \right )}}{16} + \frac {x^{3} \left (9 a d e^{3} - 13 b d^{2} e^{2} + 17 c d^{3} e\right ) + x \left (7 a d^{2} e^{2} - 11 b d^{3} e + 15 c d^{4}\right )}{8 d^{2} e^{5} + 16 d e^{6} x^{2} + 8 e^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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