Optimal. Leaf size=69 \[ \frac {1}{3} x^3 (a f+b d)+a d x+\frac {1}{2} a e x^2+\frac {1}{5} x^5 (b f+c d)+\frac {1}{4} b e x^4+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \]
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Rubi [A] time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1657} \begin {gather*} \frac {1}{3} x^3 (a f+b d)+a d x+\frac {1}{2} a e x^2+\frac {1}{5} x^5 (b f+c d)+\frac {1}{4} b e x^4+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin {align*} \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx\\ &=a d x+\frac {1}{2} a e x^2+\frac {1}{3} (b d+a f) x^3+\frac {1}{4} b e x^4+\frac {1}{5} (c d+b f) x^5+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 (a f+b d)+a d x+\frac {1}{2} a e x^2+\frac {1}{5} x^5 (b f+c d)+\frac {1}{4} b e x^4+\frac {1}{6} c e x^6+\frac {1}{7} c f 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+f 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 = 0.70, size = 61, normalized size = 0.88 \begin {gather*} \frac {1}{7} x^{7} f c + \frac {1}{6} x^{6} e c + \frac {1}{5} x^{5} d c + \frac {1}{5} x^{5} f b + \frac {1}{4} x^{4} e b + \frac {1}{3} x^{3} d b + \frac {1}{3} x^{3} f a + \frac {1}{2} x^{2} e a + x d a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 64, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, c f x^{7} + \frac {1}{6} \, c x^{6} e + \frac {1}{5} \, c d x^{5} + \frac {1}{5} \, b f x^{5} + \frac {1}{4} \, b x^{4} e + \frac {1}{3} \, b d x^{3} + \frac {1}{3} \, a f x^{3} + \frac {1}{2} \, a x^{2} 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 = 58, normalized size = 0.84 \begin {gather*} \frac {c f \,x^{7}}{7}+\frac {c e \,x^{6}}{6}+\frac {b e \,x^{4}}{4}+\frac {\left (b f +c d \right ) x^{5}}{5}+\frac {a e \,x^{2}}{2}+a d x +\frac {\left (a f +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 = 1.21, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{7} \, c f x^{7} + \frac {1}{6} \, c e x^{6} + \frac {1}{4} \, b e x^{4} + \frac {1}{5} \, {\left (c d + b f\right )} x^{5} + \frac {1}{2} \, a e x^{2} + \frac {1}{3} \, {\left (b d + a f\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 = 0.03, size = 59, normalized size = 0.86 \begin {gather*} \frac {c\,f\,x^7}{7}+\frac {c\,e\,x^6}{6}+\left (\frac {c\,d}{5}+\frac {b\,f}{5}\right )\,x^5+\frac {b\,e\,x^4}{4}+\left (\frac {b\,d}{3}+\frac {a\,f}{3}\right )\,x^3+\frac {a\,e\,x^2}{2}+a\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 65, normalized size = 0.94 \begin {gather*} a d x + \frac {a e x^{2}}{2} + \frac {b e x^{4}}{4} + \frac {c e x^{6}}{6} + \frac {c f x^{7}}{7} + x^{5} \left (\frac {b f}{5} + \frac {c d}{5}\right ) + x^{3} \left (\frac {a f}{3} + \frac {b d}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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