Optimal. Leaf size=184 \[ \frac {c^2 x}{e^4}+\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac {\left (5 a^2-\frac {19 c^2 d^4}{e^4}-\frac {14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac {\left (5 a^2+\frac {29 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac {\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{9/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1172, 1828,
1171, 396, 211} \begin {gather*} -\frac {\left (-5 a^2 e^4-2 a c d^2 e^2+35 c^2 d^4\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{9/2}}+\frac {x \left (5 a^2-\frac {14 a c d^2}{e^2}-\frac {19 c^2 d^4}{e^4}\right )}{24 d^2 \left (d+e x^2\right )^2}+\frac {x \left (5 a^2+\frac {2 a c d^2}{e^2}+\frac {29 c^2 d^4}{e^4}\right )}{16 d^3 \left (d+e x^2\right )}+\frac {x \left (a e^2+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac {c^2 x}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rule 1171
Rule 1172
Rule 1828
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^4} \, dx &=\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac {\int \frac {-5 a^2+\frac {c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}-\frac {6 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac {6 c^2 d^2 x^4}{e^2}-\frac {6 c^2 d x^6}{e}}{\left (d+e x^2\right )^3} \, dx}{6 d}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac {\left (5 a^2-\frac {19 c^2 d^4}{e^4}-\frac {14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac {\int \frac {3 \left (5 a^2+\frac {5 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right )-\frac {48 c^2 d^3 x^2}{e^3}+\frac {24 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^2} \, dx}{24 d^2}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac {\left (5 a^2-\frac {19 c^2 d^4}{e^4}-\frac {14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac {\left (5 a^2+\frac {29 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac {\int \frac {-3 \left (5 a^2-\frac {19 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right )-\frac {48 c^2 d^3 x^2}{e^3}}{d+e x^2} \, dx}{48 d^3}\\ &=\frac {c^2 x}{e^4}+\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac {\left (5 a^2-\frac {19 c^2 d^4}{e^4}-\frac {14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac {\left (5 a^2+\frac {29 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac {\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \int \frac {1}{d+e x^2} \, dx}{16 d^3 e^4}\\ &=\frac {c^2 x}{e^4}+\frac {\left (c d^2+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}+\frac {\left (5 a^2-\frac {19 c^2 d^4}{e^4}-\frac {14 a c d^2}{e^2}\right ) x}{24 d^2 \left (d+e x^2\right )^2}+\frac {\left (5 a^2+\frac {29 c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}\right ) x}{16 d^3 \left (d+e x^2\right )}-\frac {\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 174, normalized size = 0.95 \begin {gather*} \frac {x \left (-2 a c d^2 e^2 \left (3 d^2+8 d e x^2-3 e^2 x^4\right )+a^2 e^4 \left (33 d^2+40 d e x^2+15 e^2 x^4\right )+c^2 d^3 \left (105 d^3+280 d^2 e x^2+231 d e^2 x^4+48 e^3 x^6\right )\right )}{48 d^3 e^4 \left (d+e x^2\right )^3}-\frac {\left (35 c^2 d^4-2 a c d^2 e^2-5 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 179, normalized size = 0.97
method | result | size |
default | \(\frac {c^{2} x}{e^{4}}+\frac {\frac {\frac {e^{2} \left (5 a^{2} e^{4}+2 a c \,d^{2} e^{2}+29 c^{2} d^{4}\right ) x^{5}}{16 d^{3}}+\frac {e \left (5 a^{2} e^{4}-2 a c \,d^{2} e^{2}+17 c^{2} d^{4}\right ) x^{3}}{6 d^{2}}+\frac {\left (11 a^{2} e^{4}-2 a c \,d^{2} e^{2}+19 c^{2} d^{4}\right ) x}{16 d}}{\left (e \,x^{2}+d \right )^{3}}+\frac {\left (5 a^{2} e^{4}+2 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{16 d^{3} \sqrt {d e}}}{e^{4}}\) | \(179\) |
risch | \(\frac {c^{2} x}{e^{4}}+\frac {\frac {e^{2} \left (5 a^{2} e^{4}+2 a c \,d^{2} e^{2}+29 c^{2} d^{4}\right ) x^{5}}{16 d^{3}}+\frac {e \left (5 a^{2} e^{4}-2 a c \,d^{2} e^{2}+17 c^{2} d^{4}\right ) x^{3}}{6 d^{2}}+\frac {\left (11 a^{2} e^{4}-2 a c \,d^{2} e^{2}+19 c^{2} d^{4}\right ) x}{16 d}}{e^{4} \left (e \,x^{2}+d \right )^{3}}-\frac {5 \ln \left (e x +\sqrt {-d e}\right ) a^{2}}{32 \sqrt {-d e}\, d^{3}}-\frac {\ln \left (e x +\sqrt {-d e}\right ) a c}{16 e^{2} \sqrt {-d e}\, d}+\frac {35 d \ln \left (e x +\sqrt {-d e}\right ) c^{2}}{32 e^{4} \sqrt {-d e}}+\frac {5 \ln \left (-e x +\sqrt {-d e}\right ) a^{2}}{32 \sqrt {-d e}\, d^{3}}+\frac {\ln \left (-e x +\sqrt {-d e}\right ) a c}{16 e^{2} \sqrt {-d e}\, d}-\frac {35 d \ln \left (-e x +\sqrt {-d e}\right ) c^{2}}{32 e^{4} \sqrt {-d e}}\) | \(290\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 185, normalized size = 1.01 \begin {gather*} c^{2} x e^{\left (-4\right )} + \frac {3 \, {\left (29 \, c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + 5 \, a^{2} e^{6}\right )} x^{5} + 8 \, {\left (17 \, c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + 5 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (19 \, c^{2} d^{6} - 2 \, a c d^{4} e^{2} + 11 \, a^{2} d^{2} e^{4}\right )} x}{48 \, {\left (d^{3} x^{6} e^{7} + 3 \, d^{4} x^{4} e^{6} + 3 \, d^{5} x^{2} e^{5} + d^{6} e^{4}\right )}} - \frac {{\left (35 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{16 \, d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 661, normalized size = 3.59 \begin {gather*} \left [\frac {560 \, c^{2} d^{6} x^{3} e^{2} + 210 \, c^{2} d^{7} x e + 30 \, a^{2} d x^{5} e^{7} + 80 \, a^{2} d^{2} x^{3} e^{6} + 3 \, {\left (105 \, c^{2} d^{6} x^{2} e + 35 \, c^{2} d^{7} - 5 \, a^{2} x^{6} e^{7} - 15 \, a^{2} d x^{4} e^{6} - {\left (2 \, a c d^{2} x^{6} + 15 \, a^{2} d^{2} x^{2}\right )} e^{5} - {\left (6 \, a c d^{3} x^{4} + 5 \, a^{2} d^{3}\right )} e^{4} + {\left (35 \, c^{2} d^{4} x^{6} - 6 \, a c d^{4} x^{2}\right )} e^{3} + {\left (105 \, c^{2} d^{5} x^{4} - 2 \, a c d^{5}\right )} e^{2}\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) + 6 \, {\left (2 \, a c d^{3} x^{5} + 11 \, a^{2} d^{3} x\right )} e^{5} + 32 \, {\left (3 \, c^{2} d^{4} x^{7} - a c d^{4} x^{3}\right )} e^{4} + 6 \, {\left (77 \, c^{2} d^{5} x^{5} - 2 \, a c d^{5} x\right )} e^{3}}{96 \, {\left (d^{4} x^{6} e^{8} + 3 \, d^{5} x^{4} e^{7} + 3 \, d^{6} x^{2} e^{6} + d^{7} e^{5}\right )}}, \frac {280 \, c^{2} d^{6} x^{3} e^{2} + 105 \, c^{2} d^{7} x e + 15 \, a^{2} d x^{5} e^{7} + 40 \, a^{2} d^{2} x^{3} e^{6} - 3 \, {\left (105 \, c^{2} d^{6} x^{2} e + 35 \, c^{2} d^{7} - 5 \, a^{2} x^{6} e^{7} - 15 \, a^{2} d x^{4} e^{6} - {\left (2 \, a c d^{2} x^{6} + 15 \, a^{2} d^{2} x^{2}\right )} e^{5} - {\left (6 \, a c d^{3} x^{4} + 5 \, a^{2} d^{3}\right )} e^{4} + {\left (35 \, c^{2} d^{4} x^{6} - 6 \, a c d^{4} x^{2}\right )} e^{3} + {\left (105 \, c^{2} d^{5} x^{4} - 2 \, a c d^{5}\right )} e^{2}\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} + 3 \, {\left (2 \, a c d^{3} x^{5} + 11 \, a^{2} d^{3} x\right )} e^{5} + 16 \, {\left (3 \, c^{2} d^{4} x^{7} - a c d^{4} x^{3}\right )} e^{4} + 3 \, {\left (77 \, c^{2} d^{5} x^{5} - 2 \, a c d^{5} x\right )} e^{3}}{48 \, {\left (d^{4} x^{6} e^{8} + 3 \, d^{5} x^{4} e^{7} + 3 \, d^{6} x^{2} e^{6} + d^{7} e^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.89, size = 292, normalized size = 1.59 \begin {gather*} \frac {c^{2} x}{e^{4}} - \frac {\sqrt {- \frac {1}{d^{7} e^{9}}} \cdot \left (5 a^{2} e^{4} + 2 a c d^{2} e^{2} - 35 c^{2} d^{4}\right ) \log {\left (- d^{4} e^{4} \sqrt {- \frac {1}{d^{7} e^{9}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{d^{7} e^{9}}} \cdot \left (5 a^{2} e^{4} + 2 a c d^{2} e^{2} - 35 c^{2} d^{4}\right ) \log {\left (d^{4} e^{4} \sqrt {- \frac {1}{d^{7} e^{9}}} + x \right )}}{32} + \frac {x^{5} \cdot \left (15 a^{2} e^{6} + 6 a c d^{2} e^{4} + 87 c^{2} d^{4} e^{2}\right ) + x^{3} \cdot \left (40 a^{2} d e^{5} - 16 a c d^{3} e^{3} + 136 c^{2} d^{5} e\right ) + x \left (33 a^{2} d^{2} e^{4} - 6 a c d^{4} e^{2} + 57 c^{2} d^{6}\right )}{48 d^{6} e^{4} + 144 d^{5} e^{5} x^{2} + 144 d^{4} e^{6} x^{4} + 48 d^{3} e^{7} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.62, size = 167, normalized size = 0.91 \begin {gather*} c^{2} x e^{\left (-4\right )} - \frac {{\left (35 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{16 \, d^{\frac {7}{2}}} + \frac {{\left (87 \, c^{2} d^{4} x^{5} e^{2} + 136 \, c^{2} d^{5} x^{3} e + 6 \, a c d^{2} x^{5} e^{4} + 57 \, c^{2} d^{6} x - 16 \, a c d^{3} x^{3} e^{3} + 15 \, a^{2} x^{5} e^{6} - 6 \, a c d^{4} x e^{2} + 40 \, a^{2} d x^{3} e^{5} + 33 \, a^{2} d^{2} x e^{4}\right )} e^{\left (-4\right )}}{48 \, {\left (x^{2} e + d\right )}^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.49, size = 199, normalized size = 1.08 \begin {gather*} \frac {\frac {x^3\,\left (5\,a^2\,e^5-2\,a\,c\,d^2\,e^3+17\,c^2\,d^4\,e\right )}{6\,d^2}+\frac {x\,\left (11\,a^2\,e^4-2\,a\,c\,d^2\,e^2+19\,c^2\,d^4\right )}{16\,d}+\frac {x^5\,\left (5\,a^2\,e^6+2\,a\,c\,d^2\,e^4+29\,c^2\,d^4\,e^2\right )}{16\,d^3}}{d^3\,e^4+3\,d^2\,e^5\,x^2+3\,d\,e^6\,x^4+e^7\,x^6}+\frac {c^2\,x}{e^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (5\,a^2\,e^4+2\,a\,c\,d^2\,e^2-35\,c^2\,d^4\right )}{16\,d^{7/2}\,e^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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