Optimal. Leaf size=148 \[ \frac {1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac {3}{4} a^2 c x^4 (A e+B d)+\frac {1}{3} a^2 x^3 (a B e+3 A c d)+\frac {1}{7} c^2 x^7 (3 a B e+A c d)+\frac {1}{2} a c^2 x^6 (A e+B d)+\frac {3}{5} a c x^5 (a B e+A c d)+\frac {1}{8} c^3 x^8 (A e+B d)+\frac {1}{9} B c^3 e x^9 \]
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Rubi [A] time = 0.21, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.05, size = 135, normalized size = 0.91 \[ \frac {1}{6} a^3 x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {1}{20} a^2 c x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac {1}{70} a c^2 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {1}{504} c^3 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 165, normalized size = 1.11 \[ \frac {1}{9} x^{9} e c^{3} B + \frac {1}{8} x^{8} d c^{3} B + \frac {1}{8} x^{8} e c^{3} A + \frac {3}{7} x^{7} e c^{2} a B + \frac {1}{7} x^{7} d c^{3} A + \frac {1}{2} x^{6} d c^{2} a B + \frac {1}{2} x^{6} e c^{2} a A + \frac {3}{5} x^{5} e c a^{2} B + \frac {3}{5} x^{5} d c^{2} a A + \frac {3}{4} x^{4} d c a^{2} B + \frac {3}{4} x^{4} e c a^{2} A + \frac {1}{3} x^{3} e a^{3} B + x^{3} d c a^{2} A + \frac {1}{2} x^{2} d a^{3} B + \frac {1}{2} x^{2} e a^{3} A + x d a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 173, normalized size = 1.17 \[ \frac {1}{9} \, B c^{3} x^{9} e + \frac {1}{8} \, B c^{3} d x^{8} + \frac {1}{8} \, A c^{3} x^{8} e + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B a c^{2} x^{7} e + \frac {1}{2} \, B a c^{2} d x^{6} + \frac {1}{2} \, A a c^{2} x^{6} e + \frac {3}{5} \, A a c^{2} d x^{5} + \frac {3}{5} \, B a^{2} c x^{5} e + \frac {3}{4} \, B a^{2} c d x^{4} + \frac {3}{4} \, A a^{2} c x^{4} e + A a^{2} c d x^{3} + \frac {1}{3} \, B a^{3} x^{3} e + \frac {1}{2} \, B a^{3} d x^{2} + \frac {1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 143, normalized size = 0.97 \[ \frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (A e +B d \right ) c^{3} x^{8}}{8}+\frac {\left (A e +B d \right ) a \,c^{2} x^{6}}{2}+\frac {3 \left (A e +B d \right ) a^{2} c \,x^{4}}{4}+\frac {\left (A d \,c^{3}+3 B e a \,c^{2}\right ) x^{7}}{7}+A \,a^{3} d x +\frac {\left (A e +B d \right ) a^{3} x^{2}}{2}+\frac {\left (3 A d a \,c^{2}+3 B e \,a^{2} c \right ) x^{5}}{5}+\frac {\left (3 A d \,a^{2} c +B e \,a^{3}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 154, normalized size = 1.04 \[ \frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + A c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d + 3 \, B a c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (B a c^{2} d + A a c^{2} e\right )} x^{6} + A a^{3} d x + \frac {3}{5} \, {\left (A a c^{2} d + B a^{2} c e\right )} x^{5} + \frac {3}{4} \, {\left (B a^{2} c d + A a^{2} c e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d + B a^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d + A a^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 140, normalized size = 0.95 \[ x^3\,\left (\frac {B\,e\,a^3}{3}+A\,c\,d\,a^2\right )+x^7\,\left (\frac {A\,d\,c^3}{7}+\frac {3\,B\,a\,e\,c^2}{7}\right )+x^5\,\left (\frac {3\,B\,e\,a^2\,c}{5}+\frac {3\,A\,d\,a\,c^2}{5}\right )+\frac {a^3\,x^2\,\left (A\,e+B\,d\right )}{2}+\frac {c^3\,x^8\,\left (A\,e+B\,d\right )}{8}+A\,a^3\,d\,x+\frac {B\,c^3\,e\,x^9}{9}+\frac {3\,a^2\,c\,x^4\,\left (A\,e+B\,d\right )}{4}+\frac {a\,c^2\,x^6\,\left (A\,e+B\,d\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 182, normalized size = 1.23 \[ A a^{3} d x + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \left (\frac {A c^{3} d}{7} + \frac {3 B a c^{2} e}{7}\right ) + x^{6} \left (\frac {A a c^{2} e}{2} + \frac {B a c^{2} d}{2}\right ) + x^{5} \left (\frac {3 A a c^{2} d}{5} + \frac {3 B a^{2} c e}{5}\right ) + x^{4} \left (\frac {3 A a^{2} c e}{4} + \frac {3 B a^{2} c d}{4}\right ) + x^{3} \left (A a^{2} c d + \frac {B a^{3} e}{3}\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {B a^{3} d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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