Optimal. Leaf size=168 \[ -\frac {d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac {(2 c d-b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {d^{3/2} \left (9 c d^2-e (7 b d-5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{11/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 168, 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 = {1271, 1824,
211} \begin {gather*} \frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^{11/2}}-\frac {d x \left (4 c d^2-e (3 b d-2 a e)\right )}{e^5}+\frac {x^3 \left (3 c d^2-e (2 b d-a e)\right )}{3 e^4}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}-\frac {x^5 (2 c d-b e)}{5 e^3}+\frac {c x^7}{7 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1271
Rule 1824
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}-\frac {\int \frac {-d^2 \left (c d^2-b d e+a e^2\right )+2 d e \left (c d^2-b d e+a e^2\right ) x^2-2 e^2 \left (c d^2-b d e+a e^2\right ) x^4+2 e^3 (c d-b e) x^6-2 c e^4 x^8}{d+e x^2} \, dx}{2 e^5}\\ &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}-\frac {\int \left (2 d \left (4 c d^2-e (3 b d-2 a e)\right )-2 e \left (3 c d^2-e (2 b d-a e)\right ) x^2+2 e^2 (2 c d-b e) x^4-2 c e^3 x^6+\frac {-9 c d^4+7 b d^3 e-5 a d^2 e^2}{d+e x^2}\right ) \, dx}{2 e^5}\\ &=-\frac {d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac {(2 c d-b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {\left (d^2 \left (9 c d^2-e (7 b d-5 a e)\right )\right ) \int \frac {1}{d+e x^2} \, dx}{2 e^5}\\ &=-\frac {d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac {(2 c d-b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {d^{3/2} \left (9 c d^2-e (7 b d-5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 165, normalized size = 0.98 \begin {gather*} -\frac {d \left (4 c d^2-3 b d e+2 a e^2\right ) x}{e^5}+\frac {\left (3 c d^2-2 b d e+a e^2\right ) x^3}{3 e^4}+\frac {(-2 c d+b e) x^5}{5 e^3}+\frac {c x^7}{7 e^2}-\frac {\left (c d^4-b d^3 e+a d^2 e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac {d^{3/2} \left (9 c d^2-7 b d e+5 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 e^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 159, normalized size = 0.95
method | result | size |
default | \(-\frac {-\frac {1}{7} c \,x^{7} e^{3}-\frac {1}{5} b \,e^{3} x^{5}+\frac {2}{5} c d \,x^{5} e^{2}-\frac {1}{3} a \,e^{3} x^{3}+\frac {2}{3} b d \,e^{2} x^{3}-c \,d^{2} e \,x^{3}+2 d \,e^{2} a x -3 d^{2} e b x +4 c \,d^{3} x}{e^{5}}+\frac {d^{2} \left (\frac {\left (-\frac {1}{2} a \,e^{2}+\frac {1}{2} d e b -\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (5 a \,e^{2}-7 d e b +9 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{e^{5}}\) | \(159\) |
risch | \(\frac {c \,x^{7}}{7 e^{2}}+\frac {b \,x^{5}}{5 e^{2}}-\frac {2 c d \,x^{5}}{5 e^{3}}+\frac {a \,x^{3}}{3 e^{2}}-\frac {2 b d \,x^{3}}{3 e^{3}}+\frac {c \,d^{2} x^{3}}{e^{4}}-\frac {2 d a x}{e^{3}}+\frac {3 d^{2} b x}{e^{4}}-\frac {4 c \,d^{3} x}{e^{5}}+\frac {\left (-\frac {1}{2} d^{2} e^{2} a +\frac {1}{2} d^{3} e b -\frac {1}{2} d^{4} c \right ) x}{e^{5} \left (e \,x^{2}+d \right )}+\frac {5 \sqrt {-d e}\, d \ln \left (-\sqrt {-d e}\, x +d \right ) a}{4 e^{4}}-\frac {7 \sqrt {-d e}\, d^{2} \ln \left (-\sqrt {-d e}\, x +d \right ) b}{4 e^{5}}+\frac {9 \sqrt {-d e}\, d^{3} \ln \left (-\sqrt {-d e}\, x +d \right ) c}{4 e^{6}}-\frac {5 \sqrt {-d e}\, d \ln \left (\sqrt {-d e}\, x +d \right ) a}{4 e^{4}}+\frac {7 \sqrt {-d e}\, d^{2} \ln \left (\sqrt {-d e}\, x +d \right ) b}{4 e^{5}}-\frac {9 \sqrt {-d e}\, d^{3} \ln \left (\sqrt {-d e}\, x +d \right ) c}{4 e^{6}}\) | \(280\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 154, normalized size = 0.92 \begin {gather*} \frac {{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {11}{2}\right )}}{2 \, \sqrt {d}} + \frac {1}{105} \, {\left (15 \, c x^{7} e^{3} - 21 \, {\left (2 \, c d e^{2} - b e^{3}\right )} x^{5} + 35 \, {\left (3 \, c d^{2} e - 2 \, b d e^{2} + a e^{3}\right )} x^{3} - 105 \, {\left (4 \, c d^{3} - 3 \, b d^{2} e + 2 \, a d e^{2}\right )} x\right )} e^{\left (-5\right )} - \frac {{\left (c d^{4} - b d^{3} e + a d^{2} e^{2}\right )} x}{2 \, {\left (x^{2} e^{6} + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 415, normalized size = 2.47 \begin {gather*} \left [-\frac {1890 \, c d^{4} x - 105 \, {\left (9 \, c d^{4} + 5 \, a d x^{2} e^{3} - {\left (7 \, b d^{2} x^{2} - 5 \, a d^{2}\right )} e^{2} + {\left (9 \, c d^{3} x^{2} - 7 \, b d^{3}\right )} e\right )} \sqrt {-d e^{\left (-1\right )}} \log \left (\frac {x^{2} e + 2 \, \sqrt {-d e^{\left (-1\right )}} x e - d}{x^{2} e + d}\right ) - 4 \, {\left (15 \, c x^{9} + 21 \, b x^{7} + 35 \, a x^{5}\right )} e^{4} + 4 \, {\left (27 \, c d x^{7} + 49 \, b d x^{5} + 175 \, a d x^{3}\right )} e^{3} - 14 \, {\left (18 \, c d^{2} x^{5} + 70 \, b d^{2} x^{3} - 75 \, a d^{2} x\right )} e^{2} + 210 \, {\left (6 \, c d^{3} x^{3} - 7 \, b d^{3} x\right )} e}{420 \, {\left (x^{2} e^{6} + d e^{5}\right )}}, -\frac {945 \, c d^{4} x - 105 \, {\left (9 \, c d^{4} + 5 \, a d x^{2} e^{3} - {\left (7 \, b d^{2} x^{2} - 5 \, a d^{2}\right )} e^{2} + {\left (9 \, c d^{3} x^{2} - 7 \, b d^{3}\right )} e\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )} - 2 \, {\left (15 \, c x^{9} + 21 \, b x^{7} + 35 \, a x^{5}\right )} e^{4} + 2 \, {\left (27 \, c d x^{7} + 49 \, b d x^{5} + 175 \, a d x^{3}\right )} e^{3} - 7 \, {\left (18 \, c d^{2} x^{5} + 70 \, b d^{2} x^{3} - 75 \, a d^{2} x\right )} e^{2} + 105 \, {\left (6 \, c d^{3} x^{3} - 7 \, b d^{3} x\right )} e}{210 \, {\left (x^{2} e^{6} + d e^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs.
\(2 (153) = 306\).
time = 0.67, size = 320, normalized size = 1.90 \begin {gather*} \frac {c x^{7}}{7 e^{2}} + x^{5} \left (\frac {b}{5 e^{2}} - \frac {2 c d}{5 e^{3}}\right ) + x^{3} \left (\frac {a}{3 e^{2}} - \frac {2 b d}{3 e^{3}} + \frac {c d^{2}}{e^{4}}\right ) + x \left (- \frac {2 a d}{e^{3}} + \frac {3 b d^{2}}{e^{4}} - \frac {4 c d^{3}}{e^{5}}\right ) + \frac {x \left (- a d^{2} e^{2} + b d^{3} e - c d^{4}\right )}{2 d e^{5} + 2 e^{6} x^{2}} - \frac {\sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log {\left (- \frac {e^{5} \sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log {\left (\frac {e^{5} \sqrt {- \frac {d^{3}}{e^{11}}} \cdot \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.39, size = 160, normalized size = 0.95 \begin {gather*} \frac {{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {11}{2}\right )}}{2 \, \sqrt {d}} + \frac {1}{105} \, {\left (15 \, c x^{7} e^{12} - 42 \, c d x^{5} e^{11} + 21 \, b x^{5} e^{12} + 105 \, c d^{2} x^{3} e^{10} - 70 \, b d x^{3} e^{11} - 420 \, c d^{3} x e^{9} + 35 \, a x^{3} e^{12} + 315 \, b d^{2} x e^{10} - 210 \, a d x e^{11}\right )} e^{\left (-14\right )} - \frac {{\left (c d^{4} x - b d^{3} x e + a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{2 \, {\left (x^{2} e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 251, normalized size = 1.49 \begin {gather*} x^5\,\left (\frac {b}{5\,e^2}-\frac {2\,c\,d}{5\,e^3}\right )-x^3\,\left (\frac {c\,d^2}{3\,e^4}-\frac {a}{3\,e^2}+\frac {2\,d\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{3\,e}\right )+x\,\left (\frac {2\,d\,\left (\frac {c\,d^2}{e^4}-\frac {a}{e^2}+\frac {2\,d\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{e}\right )}{e}-\frac {d^2\,\left (\frac {b}{e^2}-\frac {2\,c\,d}{e^3}\right )}{e^2}\right )-\frac {x\,\left (\frac {c\,d^4}{2}-\frac {b\,d^3\,e}{2}+\frac {a\,d^2\,e^2}{2}\right )}{e^6\,x^2+d\,e^5}+\frac {c\,x^7}{7\,e^2}+\frac {d^{3/2}\,\mathrm {atan}\left (\frac {d^{3/2}\,\sqrt {e}\,x\,\left (9\,c\,d^2-7\,b\,d\,e+5\,a\,e^2\right )}{9\,c\,d^4-7\,b\,d^3\,e+5\,a\,d^2\,e^2}\right )\,\left (9\,c\,d^2-7\,b\,d\,e+5\,a\,e^2\right )}{2\,e^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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