Optimal. Leaf size=112 \[ a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{5} d x^5 \left (2 a c+b^2\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {2}{3} a b d x^3+\frac {1}{2} a b e x^4+\frac {2}{7} b c d x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c^2 d x^9+\frac {1}{10} c^2 e x^{10} \]
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Rubi [A] time = 0.13, antiderivative size = 112, 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 = {1671} \begin {gather*} a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{5} d x^5 \left (2 a c+b^2\right )+\frac {1}{6} e x^6 \left (2 a c+b^2\right )+\frac {2}{3} a b d x^3+\frac {1}{2} a b e x^4+\frac {2}{7} b c d x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c^2 d x^9+\frac {1}{10} c^2 e x^{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 1671
Rubi steps
\begin {align*} \int (d+e x) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d+a^2 e x+2 a b d x^2+2 a b e x^3+\left (b^2+2 a c\right ) d x^4+\left (b^2+2 a c\right ) e x^5+2 b c d x^6+2 b c e x^7+c^2 d x^8+c^2 e x^9\right ) \, dx\\ &=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {2}{3} a b d x^3+\frac {1}{2} a b e x^4+\frac {1}{5} \left (b^2+2 a c\right ) d x^5+\frac {1}{6} \left (b^2+2 a c\right ) e x^6+\frac {2}{7} b c d x^7+\frac {1}{4} b c e x^8+\frac {1}{9} c^2 d x^9+\frac {1}{10} c^2 e x^{10}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 97, normalized size = 0.87 \begin {gather*} \frac {630 a^2 x (2 d+e x)+42 a \left (5 b x^3 (4 d+3 e x)+2 c x^5 (6 d+5 e x)\right )+42 b^2 x^5 (6 d+5 e x)+45 b c x^7 (8 d+7 e x)+14 c^2 x^9 (10 d+9 e x)}{1260} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \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 = 0.83, size = 100, normalized size = 0.89 \begin {gather*} \frac {1}{10} x^{10} e c^{2} + \frac {1}{9} x^{9} d c^{2} + \frac {1}{4} x^{8} e c b + \frac {2}{7} x^{7} d c b + \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 {1}{2} x^{4} e b a + \frac {2}{3} x^{3} d b a + \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 = 0.39, size = 106, normalized size = 0.95 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {1}{4} \, b c x^{8} e + \frac {2}{7} \, b c d 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 {1}{2} \, a b x^{4} e + \frac {2}{3} \, a b d 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 = 95, normalized size = 0.85 \begin {gather*} \frac {c^{2} e \,x^{10}}{10}+\frac {c^{2} d \,x^{9}}{9}+\frac {b c e \,x^{8}}{4}+\frac {2 b c d \,x^{7}}{7}+\frac {a b e \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) e \,x^{6}}{6}+\frac {2 a b d \,x^{3}}{3}+\frac {\left (2 a c +b^{2}\right ) d \,x^{5}}{5}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 94, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, c^{2} e x^{10} + \frac {1}{9} \, c^{2} d x^{9} + \frac {1}{4} \, b c e x^{8} + \frac {2}{7} \, b c d x^{7} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} e x^{6} + \frac {1}{2} \, a b e x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} d x^{5} + \frac {2}{3} \, a b d x^{3} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 94, normalized size = 0.84 \begin {gather*} \frac {a^2\,e\,x^2}{2}+\frac {c^2\,d\,x^9}{9}+\frac {c^2\,e\,x^{10}}{10}+\frac {d\,x^5\,\left (b^2+2\,a\,c\right )}{5}+\frac {e\,x^6\,\left (b^2+2\,a\,c\right )}{6}+a^2\,d\,x+\frac {2\,a\,b\,d\,x^3}{3}+\frac {a\,b\,e\,x^4}{2}+\frac {2\,b\,c\,d\,x^7}{7}+\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 = 0.08, size = 116, normalized size = 1.04 \begin {gather*} a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {2 a b d x^{3}}{3} + \frac {a b e x^{4}}{2} + \frac {2 b c d x^{7}}{7} + \frac {b c e x^{8}}{4} + \frac {c^{2} d x^{9}}{9} + \frac {c^{2} e x^{10}}{10} + x^{6} \left (\frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \left (\frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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