Optimal. Leaf size=108 \[ \frac {\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {c x^3 \left (2 a e^2+c d^2\right )}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e} \]
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Rubi [A] time = 0.08, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1154, 205} \begin {gather*} \frac {c x^3 \left (2 a e^2+c d^2\right )}{3 e^3}-\frac {c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac {\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1154
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx &=\int \left (-\frac {c d \left (c d^2+2 a e^2\right )}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^2}{e^3}-\frac {c^2 d x^4}{e^2}+\frac {c^2 x^6}{e}+\frac {c^2 d^4+2 a c d^2 e^2+a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2+a e^2\right )^2 \int \frac {1}{d+e x^2} \, dx}{e^4}\\ &=-\frac {c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac {c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac {c^2 d x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 97, normalized size = 0.90 \begin {gather*} \frac {\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}+\frac {c x \left (70 a e^2 \left (e x^2-3 d\right )+c \left (-105 d^3+35 d^2 e x^2-21 d e^2 x^4+15 e^3 x^6\right )\right )}{105 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^4\right )^2}{d+e x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.27, size = 268, normalized size = 2.48 \begin {gather*} \left [\frac {30 \, c^{2} d e^{4} x^{7} - 42 \, c^{2} d^{2} e^{3} x^{5} + 70 \, {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} - 105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 210 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac {15 \, c^{2} d e^{4} x^{7} - 21 \, c^{2} d^{2} e^{3} x^{5} + 35 \, {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} + 105 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 105 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3}\right )} x}{105 \, d e^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 105, normalized size = 0.97 \begin {gather*} \frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{\sqrt {d}} + \frac {1}{105} \, {\left (15 \, c^{2} x^{7} e^{6} - 21 \, c^{2} d x^{5} e^{5} + 35 \, c^{2} d^{2} x^{3} e^{4} - 105 \, c^{2} d^{3} x e^{3} + 70 \, a c x^{3} e^{6} - 210 \, a c d x e^{5}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 136, normalized size = 1.26 \begin {gather*} \frac {c^{2} x^{7}}{7 e}-\frac {c^{2} d \,x^{5}}{5 e^{2}}+\frac {2 a c \,x^{3}}{3 e}+\frac {c^{2} d^{2} x^{3}}{3 e^{3}}+\frac {a^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {2 a c \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{2}}+\frac {c^{2} d^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{4}}-\frac {2 a c d x}{e^{2}}-\frac {c^{2} d^{3} x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.45, size = 113, normalized size = 1.05 \begin {gather*} \frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {15 \, c^{2} e^{3} x^{7} - 21 \, c^{2} d e^{2} x^{5} + 35 \, {\left (c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{3} - 105 \, {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x}{105 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 141, normalized size = 1.31 \begin {gather*} x^3\,\left (\frac {c^2\,d^2}{3\,e^3}+\frac {2\,a\,c}{3\,e}\right )+\frac {c^2\,x^7}{7\,e}-\frac {c^2\,d\,x^5}{5\,e^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,{\left (c\,d^2+a\,e^2\right )}^2}{\sqrt {d}\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}\right )\,{\left (c\,d^2+a\,e^2\right )}^2}{\sqrt {d}\,e^{9/2}}-\frac {d\,x\,\left (\frac {c^2\,d^2}{e^3}+\frac {2\,a\,c}{e}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.50, size = 236, normalized size = 2.19 \begin {gather*} - \frac {c^{2} d x^{5}}{5 e^{2}} + \frac {c^{2} x^{7}}{7 e} + x^{3} \left (\frac {2 a c}{3 e} + \frac {c^{2} d^{2}}{3 e^{3}}\right ) + x \left (- \frac {2 a c d}{e^{2}} - \frac {c^{2} d^{3}}{e^{4}}\right ) - \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (- \frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log {\left (\frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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