Optimal. Leaf size=163 \[ -\frac {b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac {3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac {(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5} \]
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Rubi [A] time = 0.29, antiderivative size = 163, 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} \[ -\frac {b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac {3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac {(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5} \]
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)^4 \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e) (d+e x)^4}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^5}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^6}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^7}{e^4}+\frac {b^3 B (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^3 (B d-A e) (d+e x)^5}{5 e^5}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^6}{6 e^5}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^7}{7 e^5}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^8}{8 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 397, normalized size = 2.44 \[ a^3 A d^4 x+\frac {1}{3} a d^2 x^3 \left (3 A \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+a B d (4 a e+3 b d)\right )+\frac {1}{7} b e^2 x^7 \left (3 a^2 B e^2+3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+\frac {1}{2} a^2 d^3 x^2 (4 a A e+a B d+3 A b d)+\frac {1}{6} e x^6 \left (a^3 B e^3+3 a^2 b e^2 (A e+4 B d)+6 a b^2 d e (2 A e+3 B d)+2 b^3 d^2 (3 A e+2 B d)\right )+\frac {1}{5} x^5 \left (a^3 e^3 (A e+4 B d)+6 a^2 b d e^2 (2 A e+3 B d)+6 a b^2 d^2 e (3 A e+2 B d)+b^3 d^3 (4 A e+B d)\right )+\frac {1}{4} d x^4 \left (3 a B d \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+A \left (4 a^3 e^3+18 a^2 b d e^2+12 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{8} b^2 e^3 x^8 (3 a B e+A b e+4 b B d)+\frac {1}{9} b^3 B e^4 x^9 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 534, normalized size = 3.28 \[ \frac {1}{9} x^{9} e^{4} b^{3} B + \frac {1}{2} x^{8} e^{3} d b^{3} B + \frac {3}{8} x^{8} e^{4} b^{2} a B + \frac {1}{8} x^{8} e^{4} b^{3} A + \frac {6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac {12}{7} x^{7} e^{3} d b^{2} a B + \frac {3}{7} x^{7} e^{4} b a^{2} B + \frac {4}{7} x^{7} e^{3} d b^{3} A + \frac {3}{7} x^{7} e^{4} b^{2} a A + \frac {2}{3} x^{6} e d^{3} b^{3} B + 3 x^{6} e^{2} d^{2} b^{2} a B + 2 x^{6} e^{3} d b a^{2} B + \frac {1}{6} x^{6} e^{4} a^{3} B + x^{6} e^{2} d^{2} b^{3} A + 2 x^{6} e^{3} d b^{2} a A + \frac {1}{2} x^{6} e^{4} b a^{2} A + \frac {1}{5} x^{5} d^{4} b^{3} B + \frac {12}{5} x^{5} e d^{3} b^{2} a B + \frac {18}{5} x^{5} e^{2} d^{2} b a^{2} B + \frac {4}{5} x^{5} e^{3} d a^{3} B + \frac {4}{5} x^{5} e d^{3} b^{3} A + \frac {18}{5} x^{5} e^{2} d^{2} b^{2} a A + \frac {12}{5} x^{5} e^{3} d b a^{2} A + \frac {1}{5} x^{5} e^{4} a^{3} A + \frac {3}{4} x^{4} d^{4} b^{2} a B + 3 x^{4} e d^{3} b a^{2} B + \frac {3}{2} x^{4} e^{2} d^{2} a^{3} B + \frac {1}{4} x^{4} d^{4} b^{3} A + 3 x^{4} e d^{3} b^{2} a A + \frac {9}{2} x^{4} e^{2} d^{2} b a^{2} A + x^{4} e^{3} d a^{3} A + x^{3} d^{4} b a^{2} B + \frac {4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} b^{2} a A + 4 x^{3} e d^{3} b a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac {1}{2} x^{2} d^{4} a^{3} B + \frac {3}{2} x^{2} d^{4} b a^{2} A + 2 x^{2} e d^{3} a^{3} A + x d^{4} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 518, normalized size = 3.18 \[ \frac {1}{9} \, B b^{3} x^{9} e^{4} + \frac {1}{2} \, B b^{3} d x^{8} e^{3} + \frac {6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac {2}{3} \, B b^{3} d^{3} x^{6} e + \frac {1}{5} \, B b^{3} d^{4} x^{5} + \frac {3}{8} \, B a b^{2} x^{8} e^{4} + \frac {1}{8} \, A b^{3} x^{8} e^{4} + \frac {12}{7} \, B a b^{2} d x^{7} e^{3} + \frac {4}{7} \, A b^{3} d x^{7} e^{3} + 3 \, B a b^{2} d^{2} x^{6} e^{2} + A b^{3} d^{2} x^{6} e^{2} + \frac {12}{5} \, B a b^{2} d^{3} x^{5} e + \frac {4}{5} \, A b^{3} d^{3} x^{5} e + \frac {3}{4} \, B a b^{2} d^{4} x^{4} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + \frac {3}{7} \, B a^{2} b x^{7} e^{4} + \frac {3}{7} \, A a b^{2} x^{7} e^{4} + 2 \, B a^{2} b d x^{6} e^{3} + 2 \, A a b^{2} d x^{6} e^{3} + \frac {18}{5} \, B a^{2} b d^{2} x^{5} e^{2} + \frac {18}{5} \, A a b^{2} d^{2} x^{5} e^{2} + 3 \, B a^{2} b d^{3} x^{4} e + 3 \, A a b^{2} d^{3} x^{4} e + B a^{2} b d^{4} x^{3} + A a b^{2} d^{4} x^{3} + \frac {1}{6} \, B a^{3} x^{6} e^{4} + \frac {1}{2} \, A a^{2} b x^{6} e^{4} + \frac {4}{5} \, B a^{3} d x^{5} e^{3} + \frac {12}{5} \, A a^{2} b d x^{5} e^{3} + \frac {3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac {9}{2} \, A a^{2} b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{3} d^{3} x^{3} e + 4 \, A a^{2} b d^{3} x^{3} e + \frac {1}{2} \, B a^{3} d^{4} x^{2} + \frac {3}{2} \, A a^{2} b d^{4} x^{2} + \frac {1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 434, normalized size = 2.66 \[ \frac {B \,b^{3} e^{4} x^{9}}{9}+A \,a^{3} d^{4} x +\frac {\left (4 B \,b^{3} d \,e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) e^{4}\right ) x^{8}}{8}+\frac {\left (6 B \,b^{3} d^{2} e^{2}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{3}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{4}\right ) x^{7}}{7}+\frac {\left (4 B \,b^{3} d^{3} e +6 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e^{2}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{3}+\left (3 A \,a^{2} b +B \,a^{3}\right ) e^{4}\right ) x^{6}}{6}+\frac {\left (A \,a^{3} e^{4}+B \,b^{3} d^{4}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d^{3} e +6 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e^{2}+4 \left (3 A \,a^{2} b +B \,a^{3}\right ) d \,e^{3}\right ) x^{5}}{5}+\frac {\left (4 A \,a^{3} d \,e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) d^{4}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3} e +6 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{2} e^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{3} d^{2} e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{4}+4 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{3} e \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} d^{3} e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d^{4}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 425, normalized size = 2.61 \[ \frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 439, normalized size = 2.69 \[ x^3\,\left (\frac {4\,B\,a^3\,d^3\,e}{3}+2\,A\,a^3\,d^2\,e^2+B\,a^2\,b\,d^4+4\,A\,a^2\,b\,d^3\,e+A\,a\,b^2\,d^4\right )+x^7\,\left (\frac {3\,B\,a^2\,b\,e^4}{7}+\frac {12\,B\,a\,b^2\,d\,e^3}{7}+\frac {3\,A\,a\,b^2\,e^4}{7}+\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}\right )+x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {18\,B\,a^2\,b\,d^2\,e^2}{5}+\frac {12\,A\,a^2\,b\,d\,e^3}{5}+\frac {12\,B\,a\,b^2\,d^3\,e}{5}+\frac {18\,A\,a\,b^2\,d^2\,e^2}{5}+\frac {B\,b^3\,d^4}{5}+\frac {4\,A\,b^3\,d^3\,e}{5}\right )+x^4\,\left (\frac {3\,B\,a^3\,d^2\,e^2}{2}+A\,a^3\,d\,e^3+3\,B\,a^2\,b\,d^3\,e+\frac {9\,A\,a^2\,b\,d^2\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^4}{4}+3\,A\,a\,b^2\,d^3\,e+\frac {A\,b^3\,d^4}{4}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+2\,B\,a^2\,b\,d\,e^3+\frac {A\,a^2\,b\,e^4}{2}+3\,B\,a\,b^2\,d^2\,e^2+2\,A\,a\,b^2\,d\,e^3+\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2\right )+\frac {a^2\,d^3\,x^2\,\left (4\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^3\,x^8\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{8}+A\,a^3\,d^4\,x+\frac {B\,b^3\,e^4\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 546, normalized size = 3.35 \[ A a^{3} d^{4} x + \frac {B b^{3} e^{4} x^{9}}{9} + x^{8} \left (\frac {A b^{3} e^{4}}{8} + \frac {3 B a b^{2} e^{4}}{8} + \frac {B b^{3} d e^{3}}{2}\right ) + x^{7} \left (\frac {3 A a b^{2} e^{4}}{7} + \frac {4 A b^{3} d e^{3}}{7} + \frac {3 B a^{2} b e^{4}}{7} + \frac {12 B a b^{2} d e^{3}}{7} + \frac {6 B b^{3} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{2} b e^{4}}{2} + 2 A a b^{2} d e^{3} + A b^{3} d^{2} e^{2} + \frac {B a^{3} e^{4}}{6} + 2 B a^{2} b d e^{3} + 3 B a b^{2} d^{2} e^{2} + \frac {2 B b^{3} d^{3} e}{3}\right ) + x^{5} \left (\frac {A a^{3} e^{4}}{5} + \frac {12 A a^{2} b d e^{3}}{5} + \frac {18 A a b^{2} d^{2} e^{2}}{5} + \frac {4 A b^{3} d^{3} e}{5} + \frac {4 B a^{3} d e^{3}}{5} + \frac {18 B a^{2} b d^{2} e^{2}}{5} + \frac {12 B a b^{2} d^{3} e}{5} + \frac {B b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + \frac {9 A a^{2} b d^{2} e^{2}}{2} + 3 A a b^{2} d^{3} e + \frac {A b^{3} d^{4}}{4} + \frac {3 B a^{3} d^{2} e^{2}}{2} + 3 B a^{2} b d^{3} e + \frac {3 B a b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + 4 A a^{2} b d^{3} e + A a b^{2} d^{4} + \frac {4 B a^{3} d^{3} e}{3} + B a^{2} b d^{4}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac {3 A a^{2} b d^{4}}{2} + \frac {B a^{3} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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