Optimal. Leaf size=169 \[ \frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+a^3 A d x+\frac{1}{2} a^2 c C e x^6+\frac{1}{4} a^3 C e x^4+\frac{1}{7} c^2 x^7 (3 a (B e+C d)+A c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{\left (a+c x^2\right )^4 (A e+B d)}{8 c}+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10} \]
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Rubi [A] time = 0.187031, antiderivative size = 169, 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} \[ \frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+a^3 A d x+\frac{1}{2} a^2 c C e x^6+\frac{1}{4} a^3 C e x^4+\frac{1}{7} c^2 x^7 (3 a (B e+C d)+A c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{\left (a+c x^2\right )^4 (A e+B d)}{8 c}+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10} \]
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 )^3 \left (A+B x+C x^2\right ) \, dx &=\frac{(B d+A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \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 )^4}{8 c}+\int \left (a^3 A d+a^2 (3 A c d+a C d+a B e) x^2+a^3 C e x^3+3 a c (A c d+a C d+a B e) x^4+3 a^2 c C e x^5+c^2 (A c d+3 a (C d+B e)) x^6+3 a c^2 C e x^7+c^3 (C d+B e) x^8+c^3 C e x^9\right ) \, dx\\ &=a^3 A d x+\frac{1}{3} a^2 (3 A c d+a C d+a B e) x^3+\frac{1}{4} a^3 C e x^4+\frac{3}{5} a c (A c d+a C d+a B e) x^5+\frac{1}{2} a^2 c C e x^6+\frac{1}{7} c^2 (A c d+3 a (C d+B e)) x^7+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 (C d+B e) x^9+\frac{1}{10} c^3 C e x^{10}+\frac{(B d+A e) \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.0751338, size = 196, normalized size = 1.16 \[ \frac{1}{4} a^2 x^4 (a C e+3 A c e+3 B c d)+\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+\frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{1}{8} c^2 x^8 (3 a C e+A c e+B c d)+\frac{1}{7} c^2 x^7 (3 a B e+3 a C d+A c d)+\frac{1}{2} a c x^6 (a C e+A c e+B c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 223, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}Ce{x}^{10}}{10}}+{\frac{ \left ({c}^{3}eB+{c}^{3}dC \right ){x}^{9}}{9}}+{\frac{ \left ({c}^{3}eA+{c}^{3}dB+3\,ea{c}^{2}C \right ){x}^{8}}{8}}+{\frac{ \left ({c}^{3}dA+3\,ea{c}^{2}B+3\,a{c}^{2}dC \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,ea{c}^{2}A+3\,a{c}^{2}dB+3\,e{a}^{2}cC \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,a{c}^{2}dA+3\,e{a}^{2}cB+3\,cd{a}^{2}C \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,e{a}^{2}cA+3\,cd{a}^{2}B+e{a}^{3}C \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,cd{a}^{2}A+e{a}^{3}B+d{a}^{3}C \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{3}A+d{a}^{3}B \right ){x}^{2}}{2}}+{a}^{3}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.321, size = 300, normalized size = 1.78 \begin{align*} \frac{1}{10} \, C c^{3} e x^{10} + \frac{1}{9} \,{\left (C c^{3} d + B c^{3} e\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, C a c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B a c^{2} e +{\left (3 \, C a c^{2} + A c^{3}\right )} d\right )} x^{7} + \frac{1}{2} \,{\left (B a c^{2} d +{\left (C a^{2} c + A a c^{2}\right )} e\right )} x^{6} + A a^{3} d x + \frac{3}{5} \,{\left (B a^{2} c e +{\left (C a^{2} c + A a c^{2}\right )} d\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d +{\left (C a^{3} + 3 \, A a^{2} c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} e +{\left (C a^{3} + 3 \, A a^{2} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d + A a^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45735, size = 603, normalized size = 3.57 \begin{align*} \frac{1}{10} x^{10} e c^{3} C + \frac{1}{9} x^{9} d c^{3} C + \frac{1}{9} x^{9} e c^{3} B + \frac{3}{8} x^{8} e c^{2} a C + \frac{1}{8} x^{8} d c^{3} B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} a C + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \frac{1}{2} x^{6} e c a^{2} C + \frac{1}{2} x^{6} d c^{2} a B + \frac{1}{2} x^{6} e c^{2} a A + \frac{3}{5} x^{5} d c a^{2} C + \frac{3}{5} x^{5} e c a^{2} B + \frac{3}{5} x^{5} d c^{2} a A + \frac{1}{4} x^{4} e a^{3} C + \frac{3}{4} x^{4} d c a^{2} B + \frac{3}{4} x^{4} e c a^{2} A + \frac{1}{3} x^{3} d a^{3} C + \frac{1}{3} x^{3} e a^{3} B + x^{3} d c a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101808, size = 265, normalized size = 1.57 \begin{align*} A a^{3} d x + \frac{C c^{3} e x^{10}}{10} + x^{9} \left (\frac{B c^{3} e}{9} + \frac{C c^{3} d}{9}\right ) + x^{8} \left (\frac{A c^{3} e}{8} + \frac{B c^{3} d}{8} + \frac{3 C a c^{2} e}{8}\right ) + x^{7} \left (\frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7} + \frac{3 C a c^{2} d}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{B a c^{2} d}{2} + \frac{C a^{2} c e}{2}\right ) + x^{5} \left (\frac{3 A a c^{2} d}{5} + \frac{3 B a^{2} c e}{5} + \frac{3 C a^{2} c d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 B a^{2} c d}{4} + \frac{C a^{3} e}{4}\right ) + x^{3} \left (A a^{2} c d + \frac{B a^{3} e}{3} + \frac{C a^{3} d}{3}\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{B a^{3} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14016, size = 352, normalized size = 2.08 \begin{align*} \frac{1}{10} \, C c^{3} x^{10} e + \frac{1}{9} \, C c^{3} d x^{9} + \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, C a c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, C a c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B a c^{2} x^{7} e + \frac{1}{2} \, B a c^{2} d x^{6} + \frac{1}{2} \, C a^{2} c x^{6} e + \frac{1}{2} \, A a c^{2} x^{6} e + \frac{3}{5} \, C a^{2} c d x^{5} + \frac{3}{5} \, A a c^{2} d x^{5} + \frac{3}{5} \, B a^{2} c x^{5} e + \frac{3}{4} \, B a^{2} c d x^{4} + \frac{1}{4} \, C a^{3} x^{4} e + \frac{3}{4} \, A a^{2} c x^{4} e + \frac{1}{3} \, C a^{3} d x^{3} + A a^{2} c d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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