Optimal. Leaf size=184 \[ -\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x}}{b^2} \]
[Out]
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Rubi [A] time = 0.396527, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{e^{-a-b x} (a+b x)^4}{b^2}+\frac{a e^{-a-b x} (a+b x)^3}{b^2}-\frac{4 e^{-a-b x} (a+b x)^3}{b^2}+\frac{3 a e^{-a-b x} (a+b x)^2}{b^2}-\frac{12 e^{-a-b x} (a+b x)^2}{b^2}+\frac{6 a e^{-a-b x} (a+b x)}{b^2}-\frac{24 e^{-a-b x} (a+b x)}{b^2}+\frac{6 a e^{-a-b x}}{b^2}-\frac{24 e^{-a-b x}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[E^(-a - b*x)*x*(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 31.5669, size = 168, normalized size = 0.91 \[ \frac{a \left (a + b x\right )^{3} e^{- a - b x}}{b^{2}} + \frac{3 a \left (a + b x\right )^{2} e^{- a - b x}}{b^{2}} + \frac{6 a \left (a + b x\right ) e^{- a - b x}}{b^{2}} + \frac{6 a e^{- a - b x}}{b^{2}} - \frac{\left (a + b x\right )^{4} e^{- a - b x}}{b^{2}} - \frac{4 \left (a + b x\right )^{3} e^{- a - b x}}{b^{2}} - \frac{12 \left (a + b x\right )^{2} e^{- a - b x}}{b^{2}} - \frac{24 \left (a + b x\right ) e^{- a - b x}}{b^{2}} - \frac{24 e^{- a - b x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(-b*x-a)*x*(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0312783, size = 96, normalized size = 0.52 \[ \frac{e^{-a-b x} \left (-a^3 (b x+1)-3 a^2 \left (b^2 x^2+2 b x+2\right )-3 a \left (b^3 x^3+3 b^2 x^2+6 b x+6\right )-b^4 x^4-4 b^3 x^3-12 b^2 x^2-24 b x-24\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[E^(-a - b*x)*x*(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.007, size = 102, normalized size = 0.6 \[ -{\frac{ \left ({b}^{4}{x}^{4}+3\,{b}^{3}{x}^{3}a+3\,{a}^{2}{b}^{2}{x}^{2}+4\,{b}^{3}{x}^{3}+{a}^{3}bx+9\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+12\,{b}^{2}{x}^{2}+{a}^{3}+18\,abx+6\,{a}^{2}+24\,bx+18\,a+24 \right ){{\rm e}^{-bx-a}}}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(-b*x-a)*x*(b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.799639, size = 178, normalized size = 0.97 \[ -\frac{{\left (b x + 1\right )} a^{3} e^{\left (-b x - a\right )}}{b^{2}} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} e^{\left (-b x - a\right )}}{b^{2}} - \frac{3 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a e^{\left (-b x - a\right )}}{b^{2}} - \frac{{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*x*e^(-b*x - a),x, algorithm="maxima")
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Fricas [A] time = 0.248376, size = 105, normalized size = 0.57 \[ -\frac{{\left (b^{4} x^{4} +{\left (3 \, a + 4\right )} b^{3} x^{3} + 3 \,{\left (a^{2} + 3 \, a + 4\right )} b^{2} x^{2} + a^{3} +{\left (a^{3} + 6 \, a^{2} + 18 \, a + 24\right )} b x + 6 \, a^{2} + 18 \, a + 24\right )} e^{\left (-b x - a\right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*x*e^(-b*x - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.39774, size = 148, normalized size = 0.8 \[ \begin{cases} \frac{\left (- a^{3} b x - a^{3} - 3 a^{2} b^{2} x^{2} - 6 a^{2} b x - 6 a^{2} - 3 a b^{3} x^{3} - 9 a b^{2} x^{2} - 18 a b x - 18 a - b^{4} x^{4} - 4 b^{3} x^{3} - 12 b^{2} x^{2} - 24 b x - 24\right ) e^{- a - b x}}{b^{2}} & \text{for}\: b^{2} \neq 0 \\\frac{a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{5}}{5} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(-b*x-a)*x*(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.24142, size = 166, normalized size = 0.9 \[ -\frac{{\left (b^{7} x^{4} + 3 \, a b^{6} x^{3} + 3 \, a^{2} b^{5} x^{2} + 4 \, b^{6} x^{3} + a^{3} b^{4} x + 9 \, a b^{5} x^{2} + 6 \, a^{2} b^{4} x + 12 \, b^{5} x^{2} + a^{3} b^{3} + 18 \, a b^{4} x + 6 \, a^{2} b^{3} + 24 \, b^{4} x + 18 \, a b^{3} + 24 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*x*e^(-b*x - a),x, algorithm="giac")
[Out]