Optimal. Leaf size=80 \[ a^2 d^2 x+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^3}{3 c}+\frac{1}{7} c^2 e^2 x^7 \]
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Rubi [A] time = 0.0461326, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {696, 1810} \[ a^2 d^2 x+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^3}{3 c}+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Rule 696
Rule 1810
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx &=\frac{d e \left (a+c x^2\right )^3}{3 c}+\int \left (a+c x^2\right )^2 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac{d e \left (a+c x^2\right )^3}{3 c}+\int \left (a^2 d^2+a \left (2 c d^2+a e^2\right ) x^2+c \left (c d^2+2 a e^2\right ) x^4+c^2 e^2 x^6\right ) \, dx\\ &=a^2 d^2 x+\frac{1}{3} a \left (2 c d^2+a e^2\right ) x^3+\frac{1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac{1}{7} c^2 e^2 x^7+\frac{d e \left (a+c x^2\right )^3}{3 c}\\ \end{align*}
Mathematica [A] time = 0.011535, size = 91, normalized size = 1.14 \[ a^2 d^2 x+a^2 d e x^2+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+a c d e x^4+\frac{1}{3} c^2 d e x^6+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 88, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{de{c}^{2}{x}^{6}}{3}}+{\frac{ \left ( 2\,{e}^{2}ac+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+acde{x}^{4}+{\frac{ \left ({a}^{2}{e}^{2}+2\,{d}^{2}ac \right ){x}^{3}}{3}}+de{a}^{2}{x}^{2}+{a}^{2}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15998, size = 117, normalized size = 1.46 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac{1}{5} \,{\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{3} \,{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66084, size = 198, normalized size = 2.48 \begin{align*} \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{2}{5} x^{5} e^{2} c a + x^{4} e d c a + \frac{2}{3} x^{3} d^{2} c a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} e d a^{2} + x d^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122765, size = 95, normalized size = 1.19 \begin{align*} a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac{c^{2} d e x^{6}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{2 a c d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35347, size = 120, normalized size = 1.5 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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