Optimal. Leaf size=132 \[ \frac{2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac{1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+a^4 d^2 x+\frac{1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac{2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^5}{5 c}+\frac{1}{11} c^4 e^2 x^{11} \]
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Rubi [A] time = 0.0951265, antiderivative size = 132, 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} \[ \frac{2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac{1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+a^4 d^2 x+\frac{1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac{2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^5}{5 c}+\frac{1}{11} c^4 e^2 x^{11} \]
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 )^4 \, dx &=\frac{d e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac{d e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^2+a^3 \left (4 c d^2+a e^2\right ) x^2+2 a^2 c \left (3 c d^2+2 a e^2\right ) x^4+2 a c^2 \left (2 c d^2+3 a e^2\right ) x^6+c^3 \left (c d^2+4 a e^2\right ) x^8+c^4 e^2 x^{10}\right ) \, dx\\ &=a^4 d^2 x+\frac{1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac{2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac{2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac{1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac{1}{11} c^4 e^2 x^{11}+\frac{d e \left (a+c x^2\right )^5}{5 c}\\ \end{align*}
Mathematica [A] time = 0.0300655, size = 148, normalized size = 1.12 \[ \frac{2}{35} a^2 c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac{2}{15} a^3 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+a^4 \left (d^2 x+d e x^2+\frac{e^2 x^3}{3}\right )+\frac{1}{63} a c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+\frac{1}{495} c^4 x^9 \left (55 d^2+99 d e x+45 e^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 170, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}{e}^{2}{x}^{11}}{11}}+{\frac{de{c}^{4}{x}^{10}}{5}}+{\frac{ \left ( 4\,{e}^{2}a{c}^{3}+{d}^{2}{c}^{4} \right ){x}^{9}}{9}}+dea{c}^{3}{x}^{8}+{\frac{ \left ( 6\,{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{2}a{c}^{3} \right ){x}^{7}}{7}}+2\,de{a}^{2}{c}^{2}{x}^{6}+{\frac{ \left ( 4\,{e}^{2}{a}^{3}c+6\,{d}^{2}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+2\,de{a}^{3}c{x}^{4}+{\frac{ \left ({e}^{2}{a}^{4}+4\,{d}^{2}{a}^{3}c \right ){x}^{3}}{3}}+de{a}^{4}{x}^{2}+{a}^{4}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16173, size = 228, normalized size = 1.73 \begin{align*} \frac{1}{11} \, c^{4} e^{2} x^{11} + \frac{1}{5} \, c^{4} d e x^{10} + a c^{3} d e x^{8} + 2 \, a^{2} c^{2} d e x^{6} + 2 \, a^{3} c d e x^{4} + \frac{1}{9} \,{\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + a^{4} d e x^{2} + \frac{2}{7} \,{\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d^{2} x + \frac{2}{5} \,{\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (4 \, a^{3} c d^{2} + a^{4} e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59751, size = 375, normalized size = 2.84 \begin{align*} \frac{1}{11} x^{11} e^{2} c^{4} + \frac{1}{5} x^{10} e d c^{4} + \frac{1}{9} x^{9} d^{2} c^{4} + \frac{4}{9} x^{9} e^{2} c^{3} a + x^{8} e d c^{3} a + \frac{4}{7} x^{7} d^{2} c^{3} a + \frac{6}{7} x^{7} e^{2} c^{2} a^{2} + 2 x^{6} e d c^{2} a^{2} + \frac{6}{5} x^{5} d^{2} c^{2} a^{2} + \frac{4}{5} x^{5} e^{2} c a^{3} + 2 x^{4} e d c a^{3} + \frac{4}{3} x^{3} d^{2} c a^{3} + \frac{1}{3} x^{3} e^{2} a^{4} + x^{2} e d a^{4} + x d^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.107205, size = 187, normalized size = 1.42 \begin{align*} a^{4} d^{2} x + a^{4} d e x^{2} + 2 a^{3} c d e x^{4} + 2 a^{2} c^{2} d e x^{6} + a c^{3} d e x^{8} + \frac{c^{4} d e x^{10}}{5} + \frac{c^{4} e^{2} x^{11}}{11} + x^{9} \left (\frac{4 a c^{3} e^{2}}{9} + \frac{c^{4} d^{2}}{9}\right ) + x^{7} \left (\frac{6 a^{2} c^{2} e^{2}}{7} + \frac{4 a c^{3} d^{2}}{7}\right ) + x^{5} \left (\frac{4 a^{3} c e^{2}}{5} + \frac{6 a^{2} c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{4} e^{2}}{3} + \frac{4 a^{3} c d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3557, size = 231, normalized size = 1.75 \begin{align*} \frac{1}{11} \, c^{4} x^{11} e^{2} + \frac{1}{5} \, c^{4} d x^{10} e + \frac{1}{9} \, c^{4} d^{2} x^{9} + \frac{4}{9} \, a c^{3} x^{9} e^{2} + a c^{3} d x^{8} e + \frac{4}{7} \, a c^{3} d^{2} x^{7} + \frac{6}{7} \, a^{2} c^{2} x^{7} e^{2} + 2 \, a^{2} c^{2} d x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac{4}{5} \, a^{3} c x^{5} e^{2} + 2 \, a^{3} c d x^{4} e + \frac{4}{3} \, a^{3} c d^{2} x^{3} + \frac{1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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