Optimal. Leaf size=77 \[ -\frac {(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac {(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac {b B (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ -\frac {(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac {(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac {b B (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^3 \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^3}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^4}{e^2}+\frac {b B (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {(b d-a e) (B d-A e) (d+e x)^4}{4 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^5}{5 e^3}+\frac {b B (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 130, normalized size = 1.69 \[ \frac {1}{2} d^2 x^2 (3 a A e+a B d+A b d)+\frac {1}{5} e^2 x^5 (a B e+A b e+3 b B d)+\frac {1}{4} e x^4 (a e (A e+3 B d)+3 b d (A e+B d))+\frac {1}{3} d x^3 (3 a e (A e+B d)+b d (3 A e+B d))+a A d^3 x+\frac {1}{6} b B e^3 x^6 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 163, normalized size = 2.12 \[ \frac {1}{6} x^{6} e^{3} b B + \frac {3}{5} x^{5} e^{2} d b B + \frac {1}{5} x^{5} e^{3} a B + \frac {1}{5} x^{5} e^{3} b A + \frac {3}{4} x^{4} e d^{2} b B + \frac {3}{4} x^{4} e^{2} d a B + \frac {3}{4} x^{4} e^{2} d b A + \frac {1}{4} x^{4} e^{3} a A + \frac {1}{3} x^{3} d^{3} b B + x^{3} e d^{2} a B + x^{3} e d^{2} b A + x^{3} e^{2} d a A + \frac {1}{2} x^{2} d^{3} a B + \frac {1}{2} x^{2} d^{3} b A + \frac {3}{2} x^{2} e d^{2} a A + x d^{3} a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 159, normalized size = 2.06 \[ \frac {1}{6} \, B b x^{6} e^{3} + \frac {3}{5} \, B b d x^{5} e^{2} + \frac {3}{4} \, B b d^{2} x^{4} e + \frac {1}{3} \, B b d^{3} x^{3} + \frac {1}{5} \, B a x^{5} e^{3} + \frac {1}{5} \, A b x^{5} e^{3} + \frac {3}{4} \, B a d x^{4} e^{2} + \frac {3}{4} \, A b d x^{4} e^{2} + B a d^{2} x^{3} e + A b d^{2} x^{3} e + \frac {1}{2} \, B a d^{3} x^{2} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac {3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 135, normalized size = 1.75 \[ \frac {B b \,e^{3} x^{6}}{6}+A a \,d^{3} x +\frac {\left (3 B b d \,e^{2}+\left (A b +B a \right ) e^{3}\right ) x^{5}}{5}+\frac {\left (A a \,e^{3}+3 B b \,d^{2} e +3 \left (A b +B a \right ) d \,e^{2}\right ) x^{4}}{4}+\frac {\left (3 A a d \,e^{2}+B b \,d^{3}+3 \left (A b +B a \right ) d^{2} e \right ) x^{3}}{3}+\frac {\left (3 A a \,d^{2} e +\left (A b +B a \right ) d^{3}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 134, normalized size = 1.74 \[ \frac {1}{6} \, B b e^{3} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B b d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B b d^{2} e + A a e^{3} + 3 \, {\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{3} + 3 \, A a d e^{2} + 3 \, {\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a d^{2} e + {\left (B a + A b\right )} d^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 141, normalized size = 1.83 \[ x^2\,\left (\frac {A\,b\,d^3}{2}+\frac {B\,a\,d^3}{2}+\frac {3\,A\,a\,d^2\,e}{2}\right )+x^5\,\left (\frac {A\,b\,e^3}{5}+\frac {B\,a\,e^3}{5}+\frac {3\,B\,b\,d\,e^2}{5}\right )+x^3\,\left (\frac {B\,b\,d^3}{3}+A\,a\,d\,e^2+A\,b\,d^2\,e+B\,a\,d^2\,e\right )+x^4\,\left (\frac {A\,a\,e^3}{4}+\frac {3\,A\,b\,d\,e^2}{4}+\frac {3\,B\,a\,d\,e^2}{4}+\frac {3\,B\,b\,d^2\,e}{4}\right )+A\,a\,d^3\,x+\frac {B\,b\,e^3\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 168, normalized size = 2.18 \[ A a d^{3} x + \frac {B b e^{3} x^{6}}{6} + x^{5} \left (\frac {A b e^{3}}{5} + \frac {B a e^{3}}{5} + \frac {3 B b d e^{2}}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A b d e^{2}}{4} + \frac {3 B a d e^{2}}{4} + \frac {3 B b d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + B a d^{2} e + \frac {B b d^{3}}{3}\right ) + x^{2} \left (\frac {3 A a d^{2} e}{2} + \frac {A b d^{3}}{2} + \frac {B a d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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