Optimal. Leaf size=159 \[ \frac {e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5} \]
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Rubi [A] time = 0.25, antiderivative size = 159, 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 = {77} \[ \frac {e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^3 (A+B x) (d+e x)^3 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^3 (a+b x)^3}{b^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4}{b^4}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5}{b^4}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^6}{b^4}+\frac {B e^3 (a+b x)^7}{b^4}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^3 (a+b x)^4}{4 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^5}{5 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^6}{2 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^7}{7 b^5}+\frac {B e^3 (a+b x)^8}{8 b^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 297, normalized size = 1.87 \[ a^3 A d^3 x+a d x^3 \left (A \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B d (a e+b d)\right )+\frac {1}{2} b e x^6 \left (a^2 B e^2+a b e (A e+3 B d)+b^2 d (A e+B d)\right )+\frac {1}{2} a^2 d^2 x^2 (3 A (a e+b d)+a B d)+\frac {1}{5} x^5 \left (a^3 B e^3+3 a^2 b e^2 (A e+3 B d)+9 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac {1}{4} x^4 \left (3 a B d \left (a^2 e^2+3 a b d e+b^2 d^2\right )+A \left (a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{7} b^2 e^2 x^7 (3 a B e+A b e+3 b B d)+\frac {1}{8} b^3 B e^3 x^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 410, normalized size = 2.58 \[ \frac {1}{8} x^{8} e^{3} b^{3} B + \frac {3}{7} x^{7} e^{2} d b^{3} B + \frac {3}{7} x^{7} e^{3} b^{2} a B + \frac {1}{7} x^{7} e^{3} b^{3} A + \frac {1}{2} x^{6} e d^{2} b^{3} B + \frac {3}{2} x^{6} e^{2} d b^{2} a B + \frac {1}{2} x^{6} e^{3} b a^{2} B + \frac {1}{2} x^{6} e^{2} d b^{3} A + \frac {1}{2} x^{6} e^{3} b^{2} a A + \frac {1}{5} x^{5} d^{3} b^{3} B + \frac {9}{5} x^{5} e d^{2} b^{2} a B + \frac {9}{5} x^{5} e^{2} d b a^{2} B + \frac {1}{5} x^{5} e^{3} a^{3} B + \frac {3}{5} x^{5} e d^{2} b^{3} A + \frac {9}{5} x^{5} e^{2} d b^{2} a A + \frac {3}{5} x^{5} e^{3} b a^{2} A + \frac {3}{4} x^{4} d^{3} b^{2} a B + \frac {9}{4} x^{4} e d^{2} b a^{2} B + \frac {3}{4} x^{4} e^{2} d a^{3} B + \frac {1}{4} x^{4} d^{3} b^{3} A + \frac {9}{4} x^{4} e d^{2} b^{2} a A + \frac {9}{4} x^{4} e^{2} d b a^{2} A + \frac {1}{4} x^{4} e^{3} a^{3} A + x^{3} d^{3} b a^{2} B + x^{3} e d^{2} a^{3} B + x^{3} d^{3} b^{2} a A + 3 x^{3} e d^{2} b a^{2} A + x^{3} e^{2} d a^{3} A + \frac {1}{2} x^{2} d^{3} a^{3} B + \frac {3}{2} x^{2} d^{3} b a^{2} A + \frac {3}{2} x^{2} e d^{2} a^{3} A + x d^{3} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 402, normalized size = 2.53 \[ \frac {1}{8} \, B b^{3} x^{8} e^{3} + \frac {3}{7} \, B b^{3} d x^{7} e^{2} + \frac {1}{2} \, B b^{3} d^{2} x^{6} e + \frac {1}{5} \, B b^{3} d^{3} x^{5} + \frac {3}{7} \, B a b^{2} x^{7} e^{3} + \frac {1}{7} \, A b^{3} x^{7} e^{3} + \frac {3}{2} \, B a b^{2} d x^{6} e^{2} + \frac {1}{2} \, A b^{3} d x^{6} e^{2} + \frac {9}{5} \, B a b^{2} d^{2} x^{5} e + \frac {3}{5} \, A b^{3} d^{2} x^{5} e + \frac {3}{4} \, B a b^{2} d^{3} x^{4} + \frac {1}{4} \, A b^{3} d^{3} x^{4} + \frac {1}{2} \, B a^{2} b x^{6} e^{3} + \frac {1}{2} \, A a b^{2} x^{6} e^{3} + \frac {9}{5} \, B a^{2} b d x^{5} e^{2} + \frac {9}{5} \, A a b^{2} d x^{5} e^{2} + \frac {9}{4} \, B a^{2} b d^{2} x^{4} e + \frac {9}{4} \, A a b^{2} d^{2} x^{4} e + B a^{2} b d^{3} x^{3} + A a b^{2} d^{3} x^{3} + \frac {1}{5} \, B a^{3} x^{5} e^{3} + \frac {3}{5} \, A a^{2} b x^{5} e^{3} + \frac {3}{4} \, B a^{3} d x^{4} e^{2} + \frac {9}{4} \, A a^{2} b d x^{4} e^{2} + B a^{3} d^{2} x^{3} e + 3 \, A a^{2} b d^{2} x^{3} e + \frac {1}{2} \, B a^{3} d^{3} x^{2} + \frac {3}{2} \, A a^{2} b d^{3} x^{2} + \frac {1}{4} \, A a^{3} x^{4} e^{3} + A a^{3} d x^{3} e^{2} + \frac {3}{2} \, A a^{3} d^{2} x^{2} e + A a^{3} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 339, normalized size = 2.13 \[ \frac {B \,b^{3} e^{3} x^{8}}{8}+A \,a^{3} d^{3} x +\frac {\left (3 B \,b^{3} d \,e^{2}+\left (b^{3} A +3 a \,b^{2} B \right ) e^{3}\right ) x^{7}}{7}+\frac {\left (3 B \,b^{3} d^{2} e +3 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{3}\right ) x^{6}}{6}+\frac {\left (B \,b^{3} d^{3}+3 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e +3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{2}+\left (3 A \,a^{2} b +B \,a^{3}\right ) e^{3}\right ) x^{5}}{5}+\frac {\left (A \,a^{3} e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) d^{3}+3 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e +3 \left (3 A \,a^{2} b +B \,a^{3}\right ) d \,e^{2}\right ) x^{4}}{4}+\frac {\left (3 A \,a^{3} d \,e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3}+3 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{2} e \right ) x^{3}}{3}+\frac {\left (3 A \,a^{3} d^{2} e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d^{3}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 325, normalized size = 2.04 \[ \frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 334, normalized size = 2.10 \[ x^3\,\left (B\,a^3\,d^2\,e+A\,a^3\,d\,e^2+B\,a^2\,b\,d^3+3\,A\,a^2\,b\,d^2\,e+A\,a\,b^2\,d^3\right )+x^6\,\left (\frac {B\,a^2\,b\,e^3}{2}+\frac {3\,B\,a\,b^2\,d\,e^2}{2}+\frac {A\,a\,b^2\,e^3}{2}+\frac {B\,b^3\,d^2\,e}{2}+\frac {A\,b^3\,d\,e^2}{2}\right )+x^4\,\left (\frac {3\,B\,a^3\,d\,e^2}{4}+\frac {A\,a^3\,e^3}{4}+\frac {9\,B\,a^2\,b\,d^2\,e}{4}+\frac {9\,A\,a^2\,b\,d\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^3}{4}+\frac {9\,A\,a\,b^2\,d^2\,e}{4}+\frac {A\,b^3\,d^3}{4}\right )+x^5\,\left (\frac {B\,a^3\,e^3}{5}+\frac {9\,B\,a^2\,b\,d\,e^2}{5}+\frac {3\,A\,a^2\,b\,e^3}{5}+\frac {9\,B\,a\,b^2\,d^2\,e}{5}+\frac {9\,A\,a\,b^2\,d\,e^2}{5}+\frac {B\,b^3\,d^3}{5}+\frac {3\,A\,b^3\,d^2\,e}{5}\right )+\frac {a^2\,d^2\,x^2\,\left (3\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^2\,x^7\,\left (A\,b\,e+3\,B\,a\,e+3\,B\,b\,d\right )}{7}+A\,a^3\,d^3\,x+\frac {B\,b^3\,e^3\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 422, normalized size = 2.65 \[ A a^{3} d^{3} x + \frac {B b^{3} e^{3} x^{8}}{8} + x^{7} \left (\frac {A b^{3} e^{3}}{7} + \frac {3 B a b^{2} e^{3}}{7} + \frac {3 B b^{3} d e^{2}}{7}\right ) + x^{6} \left (\frac {A a b^{2} e^{3}}{2} + \frac {A b^{3} d e^{2}}{2} + \frac {B a^{2} b e^{3}}{2} + \frac {3 B a b^{2} d e^{2}}{2} + \frac {B b^{3} d^{2} e}{2}\right ) + x^{5} \left (\frac {3 A a^{2} b e^{3}}{5} + \frac {9 A a b^{2} d e^{2}}{5} + \frac {3 A b^{3} d^{2} e}{5} + \frac {B a^{3} e^{3}}{5} + \frac {9 B a^{2} b d e^{2}}{5} + \frac {9 B a b^{2} d^{2} e}{5} + \frac {B b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac {A a^{3} e^{3}}{4} + \frac {9 A a^{2} b d e^{2}}{4} + \frac {9 A a b^{2} d^{2} e}{4} + \frac {A b^{3} d^{3}}{4} + \frac {3 B a^{3} d e^{2}}{4} + \frac {9 B a^{2} b d^{2} e}{4} + \frac {3 B a b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + 3 A a^{2} b d^{2} e + A a b^{2} d^{3} + B a^{3} d^{2} e + B a^{2} b d^{3}\right ) + x^{2} \left (\frac {3 A a^{3} d^{2} e}{2} + \frac {3 A a^{2} b d^{3}}{2} + \frac {B a^{3} d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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