Optimal. Leaf size=225 \[ \frac {1}{4} A b^3 d^2 x^4+\frac {1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac {1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac {1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac {1}{10} B c^3 e^2 x^{10} \]
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Rubi [A] time = 0.30, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \[ \frac {1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac {1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac {1}{4} A b^3 d^2 x^4+\frac {1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac {1}{10} B c^3 e^2 x^{10} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d^2 x^3+b^2 d (b B d+3 A c d+2 A b e) x^4+b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^5+\left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^6+c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^7+c^2 e (2 B c d+3 b B e+A c e) x^8+B c^3 e^2 x^9\right ) \, dx\\ &=\frac {1}{4} A b^3 d^2 x^4+\frac {1}{5} b^2 d (b B d+3 A c d+2 A b e) x^5+\frac {1}{6} b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^6+\frac {1}{7} \left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^7+\frac {1}{8} c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^8+\frac {1}{9} c^2 e (2 B c d+3 b B e+A c e) x^9+\frac {1}{10} B c^3 e^2 x^{10}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 225, normalized size = 1.00 \[ \frac {1}{4} A b^3 d^2 x^4+\frac {1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac {1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac {1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac {1}{10} B c^3 e^2 x^{10} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 292, normalized size = 1.30 \[ \frac {1}{10} x^{10} e^{2} c^{3} B + \frac {2}{9} x^{9} e d c^{3} B + \frac {1}{3} x^{9} e^{2} c^{2} b B + \frac {1}{9} x^{9} e^{2} c^{3} A + \frac {1}{8} x^{8} d^{2} c^{3} B + \frac {3}{4} x^{8} e d c^{2} b B + \frac {3}{8} x^{8} e^{2} c b^{2} B + \frac {1}{4} x^{8} e d c^{3} A + \frac {3}{8} x^{8} e^{2} c^{2} b A + \frac {3}{7} x^{7} d^{2} c^{2} b B + \frac {6}{7} x^{7} e d c b^{2} B + \frac {1}{7} x^{7} e^{2} b^{3} B + \frac {1}{7} x^{7} d^{2} c^{3} A + \frac {6}{7} x^{7} e d c^{2} b A + \frac {3}{7} x^{7} e^{2} c b^{2} A + \frac {1}{2} x^{6} d^{2} c b^{2} B + \frac {1}{3} x^{6} e d b^{3} B + \frac {1}{2} x^{6} d^{2} c^{2} b A + x^{6} e d c b^{2} A + \frac {1}{6} x^{6} e^{2} b^{3} A + \frac {1}{5} x^{5} d^{2} b^{3} B + \frac {3}{5} x^{5} d^{2} c b^{2} A + \frac {2}{5} x^{5} e d b^{3} A + \frac {1}{4} x^{4} d^{2} b^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 292, normalized size = 1.30 \[ \frac {1}{10} \, B c^{3} x^{10} e^{2} + \frac {2}{9} \, B c^{3} d x^{9} e + \frac {1}{8} \, B c^{3} d^{2} x^{8} + \frac {1}{3} \, B b c^{2} x^{9} e^{2} + \frac {1}{9} \, A c^{3} x^{9} e^{2} + \frac {3}{4} \, B b c^{2} d x^{8} e + \frac {1}{4} \, A c^{3} d x^{8} e + \frac {3}{7} \, B b c^{2} d^{2} x^{7} + \frac {1}{7} \, A c^{3} d^{2} x^{7} + \frac {3}{8} \, B b^{2} c x^{8} e^{2} + \frac {3}{8} \, A b c^{2} x^{8} e^{2} + \frac {6}{7} \, B b^{2} c d x^{7} e + \frac {6}{7} \, A b c^{2} d x^{7} e + \frac {1}{2} \, B b^{2} c d^{2} x^{6} + \frac {1}{2} \, A b c^{2} d^{2} x^{6} + \frac {1}{7} \, B b^{3} x^{7} e^{2} + \frac {3}{7} \, A b^{2} c x^{7} e^{2} + \frac {1}{3} \, B b^{3} d x^{6} e + A b^{2} c d x^{6} e + \frac {1}{5} \, B b^{3} d^{2} x^{5} + \frac {3}{5} \, A b^{2} c d^{2} x^{5} + \frac {1}{6} \, A b^{3} x^{6} e^{2} + \frac {2}{5} \, A b^{3} d x^{5} e + \frac {1}{4} \, A b^{3} d^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 240, normalized size = 1.07 \[ \frac {B \,c^{3} e^{2} x^{10}}{10}+\frac {A \,b^{3} d^{2} x^{4}}{4}+\frac {\left (3 B b \,c^{2} e^{2}+\left (A \,e^{2}+2 B d e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (3 B \,b^{2} c \,e^{2}+3 \left (A \,e^{2}+2 B d e \right ) b \,c^{2}+\left (2 A d e +B \,d^{2}\right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{2}+B \,b^{3} e^{2}+3 \left (A \,e^{2}+2 B d e \right ) b^{2} c +3 \left (2 A d e +B \,d^{2}\right ) b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2} d^{2}+\left (A \,e^{2}+2 B d e \right ) b^{3}+3 \left (2 A d e +B \,d^{2}\right ) b^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,b^{2} c \,d^{2}+\left (2 A d e +B \,d^{2}\right ) b^{3}\right ) x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 242, normalized size = 1.08 \[ \frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {1}{9} \, {\left (2 \, B c^{3} d e + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (A b^{3} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, A b^{3} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2}\right )} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 235, normalized size = 1.04 \[ x^7\,\left (\frac {B\,b^3\,e^2}{7}+\frac {6\,B\,b^2\,c\,d\,e}{7}+\frac {3\,A\,b^2\,c\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^2}{7}+\frac {6\,A\,b\,c^2\,d\,e}{7}+\frac {A\,c^3\,d^2}{7}\right )+x^6\,\left (\frac {B\,b^3\,d\,e}{3}+\frac {A\,b^3\,e^2}{6}+\frac {B\,b^2\,c\,d^2}{2}+A\,b^2\,c\,d\,e+\frac {A\,b\,c^2\,d^2}{2}\right )+x^8\,\left (\frac {3\,B\,b^2\,c\,e^2}{8}+\frac {3\,B\,b\,c^2\,d\,e}{4}+\frac {3\,A\,b\,c^2\,e^2}{8}+\frac {B\,c^3\,d^2}{8}+\frac {A\,c^3\,d\,e}{4}\right )+\frac {b^2\,d\,x^5\,\left (2\,A\,b\,e+3\,A\,c\,d+B\,b\,d\right )}{5}+\frac {c^2\,e\,x^9\,\left (A\,c\,e+3\,B\,b\,e+2\,B\,c\,d\right )}{9}+\frac {A\,b^3\,d^2\,x^4}{4}+\frac {B\,c^3\,e^2\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 303, normalized size = 1.35 \[ \frac {A b^{3} d^{2} x^{4}}{4} + \frac {B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac {A c^{3} e^{2}}{9} + \frac {B b c^{2} e^{2}}{3} + \frac {2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac {3 A b c^{2} e^{2}}{8} + \frac {A c^{3} d e}{4} + \frac {3 B b^{2} c e^{2}}{8} + \frac {3 B b c^{2} d e}{4} + \frac {B c^{3} d^{2}}{8}\right ) + x^{7} \left (\frac {3 A b^{2} c e^{2}}{7} + \frac {6 A b c^{2} d e}{7} + \frac {A c^{3} d^{2}}{7} + \frac {B b^{3} e^{2}}{7} + \frac {6 B b^{2} c d e}{7} + \frac {3 B b c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac {A b^{3} e^{2}}{6} + A b^{2} c d e + \frac {A b c^{2} d^{2}}{2} + \frac {B b^{3} d e}{3} + \frac {B b^{2} c d^{2}}{2}\right ) + x^{5} \left (\frac {2 A b^{3} d e}{5} + \frac {3 A b^{2} c d^{2}}{5} + \frac {B b^{3} d^{2}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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