Optimal. Leaf size=154 \[ a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{2} a b e x^4+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{4} b c e x^8+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \]
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Rubi [A] time = 0.13, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1657} \begin {gather*} a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{2} a b e x^4+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{4} b c e x^8+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \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 )^2 \, dx &=\int \left (a^2 d+a^2 e x+a (2 b d+a f) x^2+2 a b e x^3+\left (b^2 d+2 a c d+2 a b f\right ) x^4+\left (b^2+2 a c\right ) e x^5+\left (2 b c d+b^2 f+2 a c f\right ) x^6+2 b c e x^7+c (c d+2 b f) x^8+c^2 e x^9+c^2 f x^{10}\right ) \, dx\\ &=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 154, normalized size = 1.00 \begin {gather*} a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{2} a b e x^4+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{4} b c e x^8+\frac {1}{10} c^2 e x^{10}+\frac {1}{11} c^2 f x^{11} \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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.75, size = 151, normalized size = 0.98 \begin {gather*} \frac {1}{11} x^{11} f c^{2} + \frac {1}{10} x^{10} e c^{2} + \frac {1}{9} x^{9} d c^{2} + \frac {2}{9} x^{9} f c b + \frac {1}{4} x^{8} e c b + \frac {2}{7} x^{7} d c b + \frac {1}{7} x^{7} f b^{2} + \frac {2}{7} x^{7} f c a + \frac {1}{6} x^{6} e b^{2} + \frac {1}{3} x^{6} e c a + \frac {1}{5} x^{5} d b^{2} + \frac {2}{5} x^{5} d c a + \frac {2}{5} x^{5} f b a + \frac {1}{2} x^{4} e b a + \frac {2}{3} x^{3} d b a + \frac {1}{3} x^{3} f a^{2} + \frac {1}{2} x^{2} e a^{2} + x d a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 157, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} x^{10} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c f x^{9} + \frac {1}{4} \, b c x^{8} e + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} f x^{7} + \frac {2}{7} \, a c f x^{7} + \frac {1}{6} \, b^{2} x^{6} e + \frac {1}{3} \, a c x^{6} e + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b f x^{5} + \frac {1}{2} \, a b x^{4} e + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 139, normalized size = 0.90 \begin {gather*} \frac {c^{2} f \,x^{11}}{11}+\frac {c^{2} e \,x^{10}}{10}+\frac {b c e \,x^{8}}{4}+\frac {\left (2 f b c +c^{2} d \right ) x^{9}}{9}+\frac {a b e \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) e \,x^{6}}{6}+\frac {\left (2 b c d +\left (2 a c +b^{2}\right ) f \right ) x^{7}}{7}+\frac {a^{2} e \,x^{2}}{2}+\frac {\left (2 a b f +\left (2 a c +b^{2}\right ) d \right ) x^{5}}{5}+a^{2} d x +\frac {\left (f \,a^{2}+2 a 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 = 1.03, size = 138, normalized size = 0.90 \begin {gather*} \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{4} \, b c e x^{8} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 138, normalized size = 0.90 \begin {gather*} x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,e\,x^{10}}{10}+\frac {c^2\,f\,x^{11}}{11}+\frac {e\,x^6\,\left (b^2+2\,a\,c\right )}{6}+a^2\,d\,x+\frac {a\,b\,e\,x^4}{2}+\frac {b\,c\,e\,x^8}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 165, normalized size = 1.07 \begin {gather*} a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {a b e x^{4}}{2} + \frac {b c e x^{8}}{4} + \frac {c^{2} e x^{10}}{10} + \frac {c^{2} f x^{11}}{11} + x^{9} \left (\frac {2 b c f}{9} + \frac {c^{2} d}{9}\right ) + x^{7} \left (\frac {2 a c f}{7} + \frac {b^{2} f}{7} + \frac {2 b c d}{7}\right ) + x^{6} \left (\frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \left (\frac {2 a b f}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b d}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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