Optimal. Leaf size=56 \[ \frac {1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac {2}{3} a d e x^3+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1154} \begin {gather*} \frac {1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac {2}{3} a d e x^3+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 1154
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx &=\int \left (a d^2+2 a d e x^2+\left (c d^2+a e^2\right ) x^4+2 c d e x^6+c e^2 x^8\right ) \, dx\\ &=a d^2 x+\frac {2}{3} a d e x^3+\frac {1}{5} \left (c d^2+a e^2\right ) x^5+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^5 \left (a e^2+c d^2\right )+a d^2 x+\frac {2}{3} a d e x^3+\frac {2}{7} c d e x^7+\frac {1}{9} c e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d+e x^2\right )^2 \left (a+c x^4\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.57, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{9} x^{9} e^{2} c + \frac {2}{7} x^{7} e d c + \frac {1}{5} x^{5} d^{2} c + \frac {1}{5} x^{5} e^{2} a + \frac {2}{3} x^{3} e d a + x d^{2} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{9} \, c x^{9} e^{2} + \frac {2}{7} \, c d x^{7} e + \frac {1}{5} \, c d^{2} x^{5} + \frac {1}{5} \, a x^{5} e^{2} + \frac {2}{3} \, a d x^{3} e + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 0.88 \begin {gather*} \frac {c \,e^{2} x^{9}}{9}+\frac {2 c d e \,x^{7}}{7}+\frac {2 a d e \,x^{3}}{3}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) x^{5}}{5}+a \,d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 48, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, c e^{2} x^{9} + \frac {2}{7} \, c d e x^{7} + \frac {2}{3} \, a d e x^{3} + \frac {1}{5} \, {\left (c d^{2} + a e^{2}\right )} x^{5} + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 49, normalized size = 0.88 \begin {gather*} x^5\,\left (\frac {c\,d^2}{5}+\frac {a\,e^2}{5}\right )+\frac {c\,e^2\,x^9}{9}+a\,d^2\,x+\frac {2\,a\,d\,e\,x^3}{3}+\frac {2\,c\,d\,e\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 56, normalized size = 1.00 \begin {gather*} a d^{2} x + \frac {2 a d e x^{3}}{3} + \frac {2 c d e x^{7}}{7} + \frac {c e^{2} x^{9}}{9} + x^{5} \left (\frac {a e^{2}}{5} + \frac {c d^{2}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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