Optimal. Leaf size=155 \[ -\frac {3 c^2 d x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (3 a^2-\frac {13 c^2 d^4}{e^4}-\frac {10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1172, 1828,
1167, 211} \begin {gather*} \frac {\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}}+\frac {x \left (3 a^2-\frac {10 a c d^2}{e^2}-\frac {13 c^2 d^4}{e^4}\right )}{8 d^2 \left (d+e x^2\right )}+\frac {x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^3}{3 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1167
Rule 1172
Rule 1828
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx &=\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\int \frac {-3 a^2+\frac {c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}-\frac {4 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac {4 c^2 d^2 x^4}{e^2}-\frac {4 c^2 d x^6}{e}}{\left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (3 a^2-\frac {13 c^2 d^4}{e^4}-\frac {10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac {\int \frac {3 a^2+\frac {11 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}-\frac {16 c^2 d^3 x^2}{e^3}+\frac {8 c^2 d^2 x^4}{e^2}}{d+e x^2} \, dx}{8 d^2}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (3 a^2-\frac {13 c^2 d^4}{e^4}-\frac {10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac {\int \left (-\frac {24 c^2 d^3}{e^4}+\frac {8 c^2 d^2 x^2}{e^3}+\frac {35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (3 a^2-\frac {13 c^2 d^4}{e^4}-\frac {10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 e^4}\\ &=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (3 a^2-\frac {13 c^2 d^4}{e^4}-\frac {10 a c d^2}{e^2}\right ) x}{8 d^2 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 154, normalized size = 0.99 \begin {gather*} \frac {x \left (3 a^2 e^4 \left (5 d+3 e x^2\right )-6 a c d^2 e^2 \left (3 d+5 e x^2\right )-c^2 d^2 \left (105 d^3+175 d^2 e x^2+56 d e^2 x^4-8 e^3 x^6\right )\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 152, normalized size = 0.98
method | result | size |
default | \(-\frac {c^{2} \left (-\frac {1}{3} e \,x^{3}+3 d x \right )}{e^{4}}+\frac {\frac {\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{8 d}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 d^{2} \sqrt {d e}}}{e^{4}}\) | \(152\) |
risch | \(\frac {c^{2} x^{3}}{3 e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\frac {e \left (3 a^{2} e^{4}-10 a c \,d^{2} e^{2}-13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{8 d}}{e^{4} \left (e \,x^{2}+d \right )^{2}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a^{2}}{16 \sqrt {-d e}\, d^{2}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a c}{8 e^{2} \sqrt {-d e}}-\frac {35 d^{2} \ln \left (e x +\sqrt {-d e}\right ) c^{2}}{16 e^{4} \sqrt {-d e}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a^{2}}{16 \sqrt {-d e}\, d^{2}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a c}{8 e^{2} \sqrt {-d e}}+\frac {35 d^{2} \ln \left (-e x +\sqrt {-d e}\right ) c^{2}}{16 e^{4} \sqrt {-d e}}\) | \(263\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 152, normalized size = 0.98 \begin {gather*} \frac {1}{3} \, {\left (c^{2} x^{3} e - 9 \, c^{2} d x\right )} e^{\left (-4\right )} + \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{8 \, d^{\frac {5}{2}}} - \frac {{\left (13 \, c^{2} d^{4} e + 10 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{3} + {\left (11 \, c^{2} d^{5} + 6 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}\right )} x}{8 \, {\left (d^{2} x^{4} e^{6} + 2 \, d^{3} x^{2} e^{5} + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 509, normalized size = 3.28 \begin {gather*} \left [-\frac {350 \, c^{2} d^{5} x^{3} e^{2} + 210 \, c^{2} d^{6} x e - 18 \, a^{2} d x^{3} e^{6} - 30 \, a^{2} d^{2} x e^{5} + 3 \, {\left (70 \, c^{2} d^{5} x^{2} e + 35 \, c^{2} d^{6} + 12 \, a c d^{3} x^{2} e^{3} + 3 \, a^{2} x^{4} e^{6} + 6 \, a^{2} d x^{2} e^{5} + 3 \, {\left (2 \, a c d^{2} x^{4} + a^{2} d^{2}\right )} e^{4} + {\left (35 \, c^{2} d^{4} x^{4} + 6 \, a c d^{4}\right )} e^{2}\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) - 4 \, {\left (4 \, c^{2} d^{3} x^{7} - 15 \, a c d^{3} x^{3}\right )} e^{4} + 4 \, {\left (28 \, c^{2} d^{4} x^{5} + 9 \, a c d^{4} x\right )} e^{3}}{48 \, {\left (d^{3} x^{4} e^{7} + 2 \, d^{4} x^{2} e^{6} + d^{5} e^{5}\right )}}, -\frac {175 \, c^{2} d^{5} x^{3} e^{2} + 105 \, c^{2} d^{6} x e - 9 \, a^{2} d x^{3} e^{6} - 15 \, a^{2} d^{2} x e^{5} - 3 \, {\left (70 \, c^{2} d^{5} x^{2} e + 35 \, c^{2} d^{6} + 12 \, a c d^{3} x^{2} e^{3} + 3 \, a^{2} x^{4} e^{6} + 6 \, a^{2} d x^{2} e^{5} + 3 \, {\left (2 \, a c d^{2} x^{4} + a^{2} d^{2}\right )} e^{4} + {\left (35 \, c^{2} d^{4} x^{4} + 6 \, a c d^{4}\right )} e^{2}\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} - 2 \, {\left (4 \, c^{2} d^{3} x^{7} - 15 \, a c d^{3} x^{3}\right )} e^{4} + 2 \, {\left (28 \, c^{2} d^{4} x^{5} + 9 \, a c d^{4} x\right )} e^{3}}{24 \, {\left (d^{3} x^{4} e^{7} + 2 \, d^{4} x^{2} e^{6} + d^{5} e^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.83, size = 257, normalized size = 1.66 \begin {gather*} - \frac {3 c^{2} d x}{e^{4}} + \frac {c^{2} x^{3}}{3 e^{3}} - \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (- d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a^{2} e^{5} - 10 a c d^{2} e^{3} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 6 a c d^{3} e^{2} - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.77, size = 145, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, {\left (c^{2} x^{3} e^{6} - 9 \, c^{2} d x e^{5}\right )} e^{\left (-9\right )} + \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{8 \, d^{\frac {5}{2}}} - \frac {{\left (13 \, c^{2} d^{4} x^{3} e + 11 \, c^{2} d^{5} x + 10 \, a c d^{2} x^{3} e^{3} + 6 \, a c d^{3} x e^{2} - 3 \, a^{2} x^{3} e^{5} - 5 \, a^{2} d x e^{4}\right )} e^{\left (-4\right )}}{8 \, {\left (x^{2} e + d\right )}^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.41, size = 164, normalized size = 1.06 \begin {gather*} \frac {c^2\,x^3}{3\,e^3}-\frac {\frac {x^3\,\left (-3\,a^2\,e^5+10\,a\,c\,d^2\,e^3+13\,c^2\,d^4\,e\right )}{8\,d^2}+\frac {x\,\left (-5\,a^2\,e^4+6\,a\,c\,d^2\,e^2+11\,c^2\,d^4\right )}{8\,d}}{d^2\,e^4+2\,d\,e^5\,x^2+e^6\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{8\,d^{5/2}\,e^{9/2}}-\frac {3\,c^2\,d\,x}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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