Optimal. Leaf size=42 \[ \frac {1}{3} x^3 (a e+b d)+a d x+\frac {1}{5} x^5 (b e+c d)+\frac {1}{7} c e x^7 \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1153} \begin {gather*} \frac {1}{3} x^3 (a e+b d)+a d x+\frac {1}{5} x^5 (b e+c d)+\frac {1}{7} c e x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d+(b d+a e) x^2+(c d+b e) x^4+c e x^6\right ) \, dx\\ &=a d x+\frac {1}{3} (b d+a e) x^3+\frac {1}{5} (c d+b e) x^5+\frac {1}{7} c e x^7\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 (a e+b d)+a d x+\frac {1}{5} x^5 (b e+c d)+\frac {1}{7} c e x^7 \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 ) \left (a+b x^2+c x^4\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.93, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{7} x^{7} e c + \frac {1}{5} x^{5} d c + \frac {1}{5} x^{5} e b + \frac {1}{3} x^{3} d b + \frac {1}{3} x^{3} e a + x d a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 43, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, c x^{7} e + \frac {1}{5} \, c d x^{5} + \frac {1}{5} \, b x^{5} e + \frac {1}{3} \, b d x^{3} + \frac {1}{3} \, a x^{3} e + a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.88 \begin {gather*} \frac {c e \,x^{7}}{7}+\frac {\left (b e +c d \right ) x^{5}}{5}+a d x +\frac {\left (a e +b d \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{7} \, c e x^{7} + \frac {1}{5} \, {\left (c d + b e\right )} x^{5} + \frac {1}{3} \, {\left (b d + a e\right )} x^{3} + a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.90 \begin {gather*} \frac {c\,e\,x^7}{7}+\left (\frac {b\,e}{5}+\frac {c\,d}{5}\right )\,x^5+\left (\frac {a\,e}{3}+\frac {b\,d}{3}\right )\,x^3+a\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 39, normalized size = 0.93 \begin {gather*} a d x + \frac {c e x^{7}}{7} + x^{5} \left (\frac {b e}{5} + \frac {c d}{5}\right ) + x^{3} \left (\frac {a e}{3} + \frac {b d}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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