Optimal. Leaf size=148 \[ \frac{3}{4} a^2 c x^4 (A e+B d)+\frac{1}{3} a^2 x^3 (a B e+3 A c d)+\frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{1}{7} c^2 x^7 (3 a B e+A c d)+\frac{1}{2} a c^2 x^6 (A e+B d)+\frac{3}{5} a c x^5 (a B e+A c d)+\frac{1}{8} c^3 x^8 (A e+B d)+\frac{1}{9} B c^3 e x^9 \]
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Rubi [A] time = 0.210507, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ \frac{3}{4} a^2 c x^4 (A e+B d)+\frac{1}{3} a^2 x^3 (a B e+3 A c d)+\frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{1}{7} c^2 x^7 (3 a B e+A c d)+\frac{1}{2} a c^2 x^6 (A e+B d)+\frac{3}{5} a c x^5 (a B e+A c d)+\frac{1}{8} c^3 x^8 (A e+B d)+\frac{1}{9} B c^3 e x^9 \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A d+a^3 (B d+A e) x+a^2 (3 A c d+a B e) x^2+3 a^2 c (B d+A e) x^3+3 a c (A c d+a B e) x^4+3 a c^2 (B d+A e) x^5+c^2 (A c d+3 a B e) x^6+c^3 (B d+A e) x^7+B c^3 e x^8\right ) \, dx\\ &=a^3 A d x+\frac{1}{2} a^3 (B d+A e) x^2+\frac{1}{3} a^2 (3 A c d+a B e) x^3+\frac{3}{4} a^2 c (B d+A e) x^4+\frac{3}{5} a c (A c d+a B e) x^5+\frac{1}{2} a c^2 (B d+A e) x^6+\frac{1}{7} c^2 (A c d+3 a B e) x^7+\frac{1}{8} c^3 (B d+A e) x^8+\frac{1}{9} B c^3 e x^9\\ \end{align*}
Mathematica [A] time = 0.0522302, size = 135, normalized size = 0.91 \[ \frac{1}{20} a^2 c x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac{1}{6} a^3 x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac{1}{70} a c^2 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac{1}{504} c^3 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 143, normalized size = 1. \begin{align*}{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{{c}^{3} \left ( Ae+Bd \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\,Bea{c}^{2} \right ){x}^{7}}{7}}+{\frac{a{c}^{2} \left ( Ae+Bd \right ){x}^{6}}{2}}+{\frac{ \left ( 3\,Ada{c}^{2}+3\,Be{a}^{2}c \right ){x}^{5}}{5}}+{\frac{3\,{a}^{2}c \left ( Ae+Bd \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,Ad{a}^{2}c+Be{a}^{3} \right ){x}^{3}}{3}}+{\frac{{a}^{3} \left ( Ae+Bd \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 = 1.0688, size = 208, normalized size = 1.41 \begin{align*} \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{8} \,{\left (B c^{3} d + A c^{3} e\right )} x^{8} + \frac{1}{7} \,{\left (A c^{3} d + 3 \, B a c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (B a c^{2} d + A a c^{2} e\right )} x^{6} + A a^{3} d x + \frac{3}{5} \,{\left (A a c^{2} d + B a^{2} c e\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} c d + A a^{2} c e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} c d + B a^{3} e\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 = 1.59752, size = 396, normalized size = 2.68 \begin{align*} \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \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} e c a^{2} B + \frac{3}{5} x^{5} d c^{2} a A + \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} 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 = 0.152043, size = 182, normalized size = 1.23 \begin{align*} A a^{3} d x + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{B a c^{2} d}{2}\right ) + x^{5} \left (\frac{3 A a c^{2} d}{5} + \frac{3 B a^{2} c e}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 B a^{2} c d}{4}\right ) + x^{3} \left (A a^{2} c d + \frac{B a^{3} e}{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 = 1.18841, size = 234, normalized size = 1.58 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{1}{8} \, A c^{3} x^{8} e + \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} \, A a c^{2} x^{6} e + \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{3}{4} \, A a^{2} c x^{4} e + 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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