Optimal. Leaf size=128 \[ a^2 A d x+\frac{1}{4} a^2 C e x^4+\frac{1}{5} c x^5 (2 a (B e+C d)+A c d)+\frac{1}{3} a x^3 (a B e+a C d+2 A c d)+\frac{\left (a+c x^2\right )^3 (A e+B d)}{6 c}+\frac{1}{3} a c C e x^6+\frac{1}{7} c^2 x^7 (B e+C d)+\frac{1}{8} c^2 C e x^8 \]
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Rubi [A] time = 0.15868, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1582, 1810} \[ a^2 A d x+\frac{1}{4} a^2 C e x^4+\frac{1}{5} c x^5 (2 a (B e+C d)+A c d)+\frac{1}{3} a x^3 (a B e+a C d+2 A c d)+\frac{\left (a+c x^2\right )^3 (A e+B d)}{6 c}+\frac{1}{3} a c C e x^6+\frac{1}{7} c^2 x^7 (B e+C d)+\frac{1}{8} c^2 C e x^8 \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin{align*} \int (d+e x) \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx &=\frac{(B d+A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a+c x^2\right )^2 \left (-(B d+A e) x+(d+e x) \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac{(B d+A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a^2 A d+a (2 A c d+a C d+a B e) x^2+a^2 C e x^3+c (A c d+2 a (C d+B e)) x^4+2 a c C e x^5+c^2 (C d+B e) x^6+c^2 C e x^7\right ) \, dx\\ &=a^2 A d x+\frac{1}{3} a (2 A c d+a C d+a B e) x^3+\frac{1}{4} a^2 C e x^4+\frac{1}{5} c (A c d+2 a (C d+B e)) x^5+\frac{1}{3} a c C e x^6+\frac{1}{7} c^2 (C d+B e) x^7+\frac{1}{8} c^2 C e x^8+\frac{(B d+A e) \left (a+c x^2\right )^3}{6 c}\\ \end{align*}
Mathematica [A] time = 0.050692, size = 144, normalized size = 1.12 \[ \frac{1}{2} a^2 x^2 (A e+B d)+a^2 A d x+\frac{1}{6} c x^6 (2 a C e+A c e+B c d)+\frac{1}{5} c x^5 (2 a B e+2 a C d+A c d)+\frac{1}{4} a x^4 (a C e+2 A c e+2 B c d)+\frac{1}{3} a x^3 (a B e+a C d+2 A c d)+\frac{1}{7} c^2 x^7 (B e+C d)+\frac{1}{8} c^2 C e x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 151, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}Ce{x}^{8}}{8}}+{\frac{ \left ({c}^{2}eB+{c}^{2}dC \right ){x}^{7}}{7}}+{\frac{ \left ({c}^{2}eA+{c}^{2}dB+2\,aceC \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{2}dA+2\,aceB+2\,acdC \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,aceA+2\,acdB+{a}^{2}eC \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,acdA+{a}^{2}eB+{a}^{2}dC \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}eA+{a}^{2}dB \right ){x}^{2}}{2}}+{a}^{2}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98917, size = 208, normalized size = 1.62 \begin{align*} \frac{1}{8} \, C c^{2} e x^{8} + \frac{1}{7} \,{\left (C c^{2} d + B c^{2} e\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d +{\left (2 \, C a c + A c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (2 \, B a c e +{\left (2 \, C a c + A c^{2}\right )} d\right )} x^{5} + A a^{2} d x + \frac{1}{4} \,{\left (2 \, B a c d +{\left (C a^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (B a^{2} e +{\left (C a^{2} + 2 \, A a c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d + A a^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57426, size = 428, normalized size = 3.34 \begin{align*} \frac{1}{8} x^{8} e c^{2} C + \frac{1}{7} x^{7} d c^{2} C + \frac{1}{7} x^{7} e c^{2} B + \frac{1}{3} x^{6} e c a C + \frac{1}{6} x^{6} d c^{2} B + \frac{1}{6} x^{6} e c^{2} A + \frac{2}{5} x^{5} d c a C + \frac{2}{5} x^{5} e c a B + \frac{1}{5} x^{5} d c^{2} A + \frac{1}{4} x^{4} e a^{2} C + \frac{1}{2} x^{4} d c a B + \frac{1}{2} x^{4} e c a A + \frac{1}{3} x^{3} d a^{2} C + \frac{1}{3} x^{3} e a^{2} B + \frac{2}{3} x^{3} d c a A + \frac{1}{2} x^{2} d a^{2} B + \frac{1}{2} x^{2} e a^{2} A + x d a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090758, size = 180, normalized size = 1.41 \begin{align*} A a^{2} d x + \frac{C c^{2} e x^{8}}{8} + x^{7} \left (\frac{B c^{2} e}{7} + \frac{C c^{2} d}{7}\right ) + x^{6} \left (\frac{A c^{2} e}{6} + \frac{B c^{2} d}{6} + \frac{C a c e}{3}\right ) + x^{5} \left (\frac{A c^{2} d}{5} + \frac{2 B a c e}{5} + \frac{2 C a c d}{5}\right ) + x^{4} \left (\frac{A a c e}{2} + \frac{B a c d}{2} + \frac{C a^{2} e}{4}\right ) + x^{3} \left (\frac{2 A a c d}{3} + \frac{B a^{2} e}{3} + \frac{C a^{2} d}{3}\right ) + x^{2} \left (\frac{A a^{2} e}{2} + \frac{B a^{2} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16791, size = 244, normalized size = 1.91 \begin{align*} \frac{1}{8} \, C c^{2} x^{8} e + \frac{1}{7} \, C c^{2} d x^{7} + \frac{1}{7} \, B c^{2} x^{7} e + \frac{1}{6} \, B c^{2} d x^{6} + \frac{1}{3} \, C a c x^{6} e + \frac{1}{6} \, A c^{2} x^{6} e + \frac{2}{5} \, C a c d x^{5} + \frac{1}{5} \, A c^{2} d x^{5} + \frac{2}{5} \, B a c x^{5} e + \frac{1}{2} \, B a c d x^{4} + \frac{1}{4} \, C a^{2} x^{4} e + \frac{1}{2} \, A a c x^{4} e + \frac{1}{3} \, C a^{2} d x^{3} + \frac{2}{3} \, A a c d x^{3} + \frac{1}{3} \, B a^{2} x^{3} e + \frac{1}{2} \, B a^{2} d x^{2} + \frac{1}{2} \, A a^{2} x^{2} e + A a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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