Optimal. Leaf size=117 \[ \frac{2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5} \]
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Rubi [A] time = 0.131277, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^4}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^5}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^6}{e^4}-\frac{4 c^2 d (d+e x)^7}{e^4}+\frac{c^2 (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac{2 c d \left (c d^2+a e^2\right ) (d+e x)^6}{3 e^5}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^7}{7 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}\\ \end{align*}
Mathematica [A] time = 0.0188699, size = 167, normalized size = 1.43 \[ \frac{1}{5} x^5 \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+2 a^2 d^3 e x^2+a^2 d^4 x+\frac{2}{7} c e^2 x^7 \left (a e^2+3 c d^2\right )+\frac{2}{3} c d e x^6 \left (2 a e^2+c d^2\right )+a d e x^4 \left (a e^2+2 c d^2\right )+\frac{2}{3} a d^2 x^3 \left (3 a e^2+c d^2\right )+\frac{1}{2} c^2 d e^3 x^8+\frac{1}{9} c^2 e^4 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 169, normalized size = 1.4 \begin{align*}{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{d{e}^{3}{c}^{2}{x}^{8}}{2}}+{\frac{ \left ( 2\,{e}^{4}ac+6\,{d}^{2}{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 8\,d{e}^{3}ac+4\,{d}^{3}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{e}^{4}+12\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{2}+8\,{d}^{3}eac \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{2}+2\,{d}^{4}ac \right ){x}^{3}}{3}}+2\,{d}^{3}e{a}^{2}{x}^{2}+{d}^{4}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05288, size = 220, normalized size = 1.88 \begin{align*} \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{2} \, c^{2} d e^{3} x^{8} + 2 \, a^{2} d^{3} e x^{2} + \frac{2}{7} \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac{2}{3} \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 12 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{5} +{\left (2 \, a c d^{3} e + a^{2} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (a c d^{4} + 3 \, a^{2} d^{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 = 1.60657, size = 375, normalized size = 3.21 \begin{align*} \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{2}{7} x^{7} e^{4} c a + \frac{2}{3} x^{6} e d^{3} c^{2} + \frac{4}{3} x^{6} e^{3} d c a + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{12}{5} x^{5} e^{2} d^{2} c a + \frac{1}{5} x^{5} e^{4} a^{2} + 2 x^{4} e d^{3} c a + x^{4} e^{3} d a^{2} + \frac{2}{3} x^{3} d^{4} c a + 2 x^{3} e^{2} d^{2} a^{2} + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.138325, size = 182, normalized size = 1.56 \begin{align*} a^{2} d^{4} x + 2 a^{2} d^{3} e x^{2} + \frac{c^{2} d e^{3} x^{8}}{2} + \frac{c^{2} e^{4} x^{9}}{9} + x^{7} \left (\frac{2 a c e^{4}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{4 a c d e^{3}}{3} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{12 a c d^{2} e^{2}}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 2 a c d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{2 a c d^{4}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33699, size = 224, normalized size = 1.91 \begin{align*} \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{2}{7} \, a c x^{7} e^{4} + \frac{4}{3} \, a c d x^{6} e^{3} + \frac{12}{5} \, a c d^{2} x^{5} e^{2} + 2 \, a c d^{3} x^{4} e + \frac{2}{3} \, a c d^{4} x^{3} + \frac{1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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