Optimal. Leaf size=223 \[ \frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{128 d^{9/2} e^{9/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 223, 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, 393, 211} \begin {gather*} \frac {\left (35 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 )}{128 d^{9/2} e^{9/2}}-\frac {x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {x \left (7 a^2-\frac {18 a c d^2}{e^2}-\frac {25 c^2 d^4}{e^4}\right )}{48 d^2 \left (d+e x^2\right )^3}+\frac {x \left (35 a^2+\frac {6 a c d^2}{e^2}+\frac {163 c^2 d^4}{e^4}\right )}{192 d^3 \left (d+e x^2\right )^2}+\frac {x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
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 )^5} \, dx &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\int \frac {-7 a^2+\frac {c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}-\frac {8 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac {8 c^2 d^2 x^4}{e^2}-\frac {8 c^2 d x^6}{e}}{\left (d+e x^2\right )^4} \, dx}{8 d}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\int \frac {35 a^2+\frac {19 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}-\frac {96 c^2 d^3 x^2}{e^3}+\frac {48 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^3} \, dx}{48 d^2}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-3 \left (35 a^2-\frac {29 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right )-\frac {192 c^2 d^3 x^2}{e^3}}{\left (d+e x^2\right )^2} \, dx}{192 d^3}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \int \frac {1}{d+e x^2} \, dx}{128 d^4 e^4}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{128 d^{9/2} e^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 200, normalized size = 0.90 \begin {gather*} \frac {\frac {\sqrt {d} \sqrt {e} x \left (-6 a c d^2 e^2 \left (3 d^3+11 d^2 e x^2-11 d e^2 x^4-3 e^3 x^6\right )+a^2 e^4 \left (279 d^3+511 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )-c^2 d^4 \left (105 d^3+385 d^2 e x^2+511 d e^2 x^4+279 e^3 x^6\right )\right )}{\left (d+e x^2\right )^4}+3 \left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{384 d^{9/2} e^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 212, normalized size = 0.95
method | result | size |
default | \(\frac {\frac {\left (35 a^{2} e^{4}+6 a c \,d^{2} e^{2}-93 c^{2} d^{4}\right ) x^{7}}{128 d^{4} e}+\frac {\left (385 a^{2} e^{4}+66 a c \,d^{2} e^{2}-511 c^{2} d^{4}\right ) x^{5}}{384 d^{3} e^{2}}+\frac {\left (511 a^{2} e^{4}-66 a c \,d^{2} e^{2}-385 c^{2} d^{4}\right ) x^{3}}{384 d^{2} e^{3}}+\frac {\left (93 a^{2} e^{4}-6 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) x}{128 d \,e^{4}}}{\left (e \,x^{2}+d \right )^{4}}+\frac {\left (35 a^{2} e^{4}+6 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 d^{4} e^{4} \sqrt {d e}}\) | \(212\) |
risch | \(\frac {\frac {\left (35 a^{2} e^{4}+6 a c \,d^{2} e^{2}-93 c^{2} d^{4}\right ) x^{7}}{128 d^{4} e}+\frac {\left (385 a^{2} e^{4}+66 a c \,d^{2} e^{2}-511 c^{2} d^{4}\right ) x^{5}}{384 d^{3} e^{2}}+\frac {\left (511 a^{2} e^{4}-66 a c \,d^{2} e^{2}-385 c^{2} d^{4}\right ) x^{3}}{384 d^{2} e^{3}}+\frac {\left (93 a^{2} e^{4}-6 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) x}{128 d \,e^{4}}}{\left (e \,x^{2}+d \right )^{4}}-\frac {35 \ln \left (e x +\sqrt {-d e}\right ) a^{2}}{256 \sqrt {-d e}\, d^{4}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a c}{128 \sqrt {-d e}\, e^{2} d^{2}}-\frac {35 \ln \left (e x +\sqrt {-d e}\right ) c^{2}}{256 \sqrt {-d e}\, e^{4}}+\frac {35 \ln \left (-e x +\sqrt {-d e}\right ) a^{2}}{256 \sqrt {-d e}\, d^{4}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a c}{128 \sqrt {-d e}\, e^{2} d^{2}}+\frac {35 \ln \left (-e x +\sqrt {-d e}\right ) c^{2}}{256 \sqrt {-d e}\, e^{4}}\) | \(320\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 221, normalized size = 0.99 \begin {gather*} -\frac {3 \, {\left (93 \, c^{2} d^{4} e^{3} - 6 \, a c d^{2} e^{5} - 35 \, a^{2} e^{7}\right )} x^{7} + {\left (511 \, c^{2} d^{5} e^{2} - 66 \, a c d^{3} e^{4} - 385 \, a^{2} d e^{6}\right )} x^{5} + {\left (385 \, c^{2} d^{6} e + 66 \, a c d^{4} e^{3} - 511 \, a^{2} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{7} + 6 \, a c d^{5} e^{2} - 93 \, a^{2} d^{3} e^{4}\right )} x}{384 \, {\left (d^{4} x^{8} e^{8} + 4 \, d^{5} x^{6} e^{7} + 6 \, d^{6} x^{4} e^{6} + 4 \, d^{7} x^{2} e^{5} + d^{8} e^{4}\right )}} + \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{128 \, d^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 804, normalized size = 3.61 \begin {gather*} \left [-\frac {770 \, c^{2} d^{7} x^{3} e^{2} + 210 \, c^{2} d^{8} x e - 210 \, a^{2} d x^{7} e^{8} - 770 \, a^{2} d^{2} x^{5} e^{7} + 3 \, {\left (140 \, c^{2} d^{7} x^{2} e + 35 \, a^{2} x^{8} e^{8} + 35 \, c^{2} d^{8} + 140 \, a^{2} d x^{6} e^{7} + 6 \, {\left (a c d^{2} x^{8} + 35 \, a^{2} d^{2} x^{4}\right )} e^{6} + 4 \, {\left (6 \, a c d^{3} x^{6} + 35 \, a^{2} d^{3} x^{2}\right )} e^{5} + {\left (35 \, c^{2} d^{4} x^{8} + 36 \, a c d^{4} x^{4} + 35 \, a^{2} d^{4}\right )} e^{4} + 4 \, {\left (35 \, c^{2} d^{5} x^{6} + 6 \, a c d^{5} x^{2}\right )} e^{3} + 6 \, {\left (35 \, c^{2} d^{6} x^{4} + a c d^{6}\right )} e^{2}\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) - 2 \, {\left (18 \, a c d^{3} x^{7} + 511 \, a^{2} d^{3} x^{3}\right )} e^{6} - 6 \, {\left (22 \, a c d^{4} x^{5} + 93 \, a^{2} d^{4} x\right )} e^{5} + 6 \, {\left (93 \, c^{2} d^{5} x^{7} + 22 \, a c d^{5} x^{3}\right )} e^{4} + 2 \, {\left (511 \, c^{2} d^{6} x^{5} + 18 \, a c d^{6} x\right )} e^{3}}{768 \, {\left (d^{5} x^{8} e^{9} + 4 \, d^{6} x^{6} e^{8} + 6 \, d^{7} x^{4} e^{7} + 4 \, d^{8} x^{2} e^{6} + d^{9} e^{5}\right )}}, -\frac {385 \, c^{2} d^{7} x^{3} e^{2} + 105 \, c^{2} d^{8} x e - 105 \, a^{2} d x^{7} e^{8} - 385 \, a^{2} d^{2} x^{5} e^{7} - 3 \, {\left (140 \, c^{2} d^{7} x^{2} e + 35 \, a^{2} x^{8} e^{8} + 35 \, c^{2} d^{8} + 140 \, a^{2} d x^{6} e^{7} + 6 \, {\left (a c d^{2} x^{8} + 35 \, a^{2} d^{2} x^{4}\right )} e^{6} + 4 \, {\left (6 \, a c d^{3} x^{6} + 35 \, a^{2} d^{3} x^{2}\right )} e^{5} + {\left (35 \, c^{2} d^{4} x^{8} + 36 \, a c d^{4} x^{4} + 35 \, a^{2} d^{4}\right )} e^{4} + 4 \, {\left (35 \, c^{2} d^{5} x^{6} + 6 \, a c d^{5} x^{2}\right )} e^{3} + 6 \, {\left (35 \, c^{2} d^{6} x^{4} + a c d^{6}\right )} e^{2}\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} - {\left (18 \, a c d^{3} x^{7} + 511 \, a^{2} d^{3} x^{3}\right )} e^{6} - 3 \, {\left (22 \, a c d^{4} x^{5} + 93 \, a^{2} d^{4} x\right )} e^{5} + 3 \, {\left (93 \, c^{2} d^{5} x^{7} + 22 \, a c d^{5} x^{3}\right )} e^{4} + {\left (511 \, c^{2} d^{6} x^{5} + 18 \, a c d^{6} x\right )} e^{3}}{384 \, {\left (d^{5} x^{8} e^{9} + 4 \, d^{6} x^{6} e^{8} + 6 \, d^{7} x^{4} e^{7} + 4 \, d^{8} x^{2} e^{6} + d^{9} e^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.66, size = 335, normalized size = 1.50 \begin {gather*} - \frac {\sqrt {- \frac {1}{d^{9} e^{9}}} \cdot \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (- d^{5} e^{4} \sqrt {- \frac {1}{d^{9} e^{9}}} + x \right )}}{256} + \frac {\sqrt {- \frac {1}{d^{9} e^{9}}} \cdot \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (d^{5} e^{4} \sqrt {- \frac {1}{d^{9} e^{9}}} + x \right )}}{256} + \frac {x^{7} \cdot \left (105 a^{2} e^{7} + 18 a c d^{2} e^{5} - 279 c^{2} d^{4} e^{3}\right ) + x^{5} \cdot \left (385 a^{2} d e^{6} + 66 a c d^{3} e^{4} - 511 c^{2} d^{5} e^{2}\right ) + x^{3} \cdot \left (511 a^{2} d^{2} e^{5} - 66 a c d^{4} e^{3} - 385 c^{2} d^{6} e\right ) + x \left (279 a^{2} d^{3} e^{4} - 18 a c d^{5} e^{2} - 105 c^{2} d^{7}\right )}{384 d^{8} e^{4} + 1536 d^{7} e^{5} x^{2} + 2304 d^{6} e^{6} x^{4} + 1536 d^{5} e^{7} x^{6} + 384 d^{4} e^{8} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.29, size = 198, normalized size = 0.89 \begin {gather*} \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{128 \, d^{\frac {9}{2}}} - \frac {{\left (279 \, c^{2} d^{4} x^{7} e^{3} + 511 \, c^{2} d^{5} x^{5} e^{2} - 18 \, a c d^{2} x^{7} e^{5} + 385 \, c^{2} d^{6} x^{3} e - 66 \, a c d^{3} x^{5} e^{4} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} + 66 \, a c d^{4} x^{3} e^{3} - 385 \, a^{2} d x^{5} e^{6} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \, {\left (x^{2} e + d\right )}^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.49, size = 240, normalized size = 1.08 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (35\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{128\,d^{9/2}\,e^{9/2}}-\frac {\frac {x\,\left (-93\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{128\,d\,e^4}-\frac {x^7\,\left (35\,a^2\,e^4+6\,a\,c\,d^2\,e^2-93\,c^2\,d^4\right )}{128\,d^4\,e}+\frac {x^3\,\left (-511\,a^2\,e^4+66\,a\,c\,d^2\,e^2+385\,c^2\,d^4\right )}{384\,d^2\,e^3}-\frac {x^5\,\left (385\,a^2\,e^4+66\,a\,c\,d^2\,e^2-511\,c^2\,d^4\right )}{384\,d^3\,e^2}}{d^4+4\,d^3\,e\,x^2+6\,d^2\,e^2\,x^4+4\,d\,e^3\,x^6+e^4\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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