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 = 0.125778, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1671} \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{5} d x^5 \left (2 a c+b^2\right )+\frac{1}{6} e x^6 \left (2 a c+b^2\right )+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{2}{7} b c d x^7+\frac{1}{4} b c e x^8+\frac{1}{9} c^2 d x^9+\frac{1}{10} c^2 e x^{10} \]
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*}
Mathematica [A] time = 0.0506573, size = 97, normalized size = 0.87 \[ \frac{630 a^2 x (2 d+e x)+42 a \left (5 b x^3 (4 d+3 e x)+2 c x^5 (6 d+5 e x)\right )+42 b^2 x^5 (6 d+5 e x)+45 b c x^7 (8 d+7 e x)+14 c^2 x^9 (10 d+9 e x)}{1260} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 95, normalized size = 0.9 \begin{align*}{a}^{2}dx+{\frac{{a}^{2}e{x}^{2}}{2}}+{\frac{2\,abd{x}^{3}}{3}}+{\frac{abe{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ) d{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ) e{x}^{6}}{6}}+{\frac{2\,bcd{x}^{7}}{7}}+{\frac{bce{x}^{8}}{4}}+{\frac{{c}^{2}d{x}^{9}}{9}}+{\frac{{c}^{2}e{x}^{10}}{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951583, size = 127, normalized size = 1.13 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47995, size = 252, normalized size = 2.25 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08321, size = 116, normalized size = 1.04 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0943, size = 143, normalized size = 1.28 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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