Optimal. Leaf size=104 \[ \frac{1}{3} a^2 x^3 \left (a e^2+3 c d^2\right )+a^3 d^2 x+\frac{1}{7} c^2 x^7 \left (3 a e^2+c d^2\right )+\frac{3}{5} a c x^5 \left (a e^2+c d^2\right )+\frac{d e \left (a+c x^2\right )^4}{4 c}+\frac{1}{9} c^3 e^2 x^9 \]
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Rubi [A] time = 0.0630388, antiderivative size = 104, 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} \[ \frac{1}{3} a^2 x^3 \left (a e^2+3 c d^2\right )+a^3 d^2 x+\frac{1}{7} c^2 x^7 \left (3 a e^2+c d^2\right )+\frac{3}{5} a c x^5 \left (a e^2+c d^2\right )+\frac{d e \left (a+c x^2\right )^4}{4 c}+\frac{1}{9} c^3 e^2 x^9 \]
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 )^3 \, dx &=\frac{d e \left (a+c x^2\right )^4}{4 c}+\int \left (a+c x^2\right )^3 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac{d e \left (a+c x^2\right )^4}{4 c}+\int \left (a^3 d^2+a^2 \left (3 c d^2+a e^2\right ) x^2+3 a c \left (c d^2+a e^2\right ) x^4+c^2 \left (c d^2+3 a e^2\right ) x^6+c^3 e^2 x^8\right ) \, dx\\ &=a^3 d^2 x+\frac{1}{3} a^2 \left (3 c d^2+a e^2\right ) x^3+\frac{3}{5} a c \left (c d^2+a e^2\right ) x^5+\frac{1}{7} c^2 \left (c d^2+3 a e^2\right ) x^7+\frac{1}{9} c^3 e^2 x^9+\frac{d e \left (a+c x^2\right )^4}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0231168, size = 116, normalized size = 1.12 \[ \frac{1}{10} a^2 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+a^3 \left (d^2 x+d e x^2+\frac{e^2 x^3}{3}\right )+\frac{1}{35} a c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac{1}{252} c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 129, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{de{c}^{3}{x}^{8}}{4}}+{\frac{ \left ( 3\,{e}^{2}a{c}^{2}+{d}^{2}{c}^{3} \right ){x}^{7}}{7}}+a{c}^{2}de{x}^{6}+{\frac{ \left ( 3\,{e}^{2}{a}^{2}c+3\,{d}^{2}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{3\,de{a}^{2}c{x}^{4}}{2}}+{\frac{ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ){x}^{3}}{3}}+de{a}^{3}{x}^{2}+{a}^{3}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14436, size = 170, normalized size = 1.63 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{4} \, c^{3} d e x^{8} + a c^{2} d e x^{6} + \frac{3}{2} \, a^{2} c d e x^{4} + \frac{1}{7} \,{\left (c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{7} + a^{3} d e x^{2} + a^{3} d^{2} x + \frac{3}{5} \,{\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} c d^{2} + a^{3} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60838, size = 282, normalized size = 2.71 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{3}{7} x^{7} e^{2} c^{2} a + x^{6} e d c^{2} a + \frac{3}{5} x^{5} d^{2} c^{2} a + \frac{3}{5} x^{5} e^{2} c a^{2} + \frac{3}{2} x^{4} e d c a^{2} + x^{3} d^{2} c a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09911, size = 139, normalized size = 1.34 \begin{align*} a^{3} d^{2} x + a^{3} d e x^{2} + \frac{3 a^{2} c d e x^{4}}{2} + a c^{2} d e x^{6} + \frac{c^{3} d e x^{8}}{4} + \frac{c^{3} e^{2} x^{9}}{9} + x^{7} \left (\frac{3 a c^{2} e^{2}}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac{3 a^{2} c e^{2}}{5} + \frac{3 a c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + a^{2} c d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28823, size = 174, normalized size = 1.67 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{7} \, a c^{2} x^{7} e^{2} + a c^{2} d x^{6} e + \frac{3}{5} \, a c^{2} d^{2} x^{5} + \frac{3}{5} \, a^{2} c x^{5} e^{2} + \frac{3}{2} \, a^{2} c d x^{4} e + a^{2} c d^{2} x^{3} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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