Optimal. Leaf size=322 \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]
[Out]
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Rubi [A] time = 0.601621, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]
Antiderivative was successfully verified.
[In] Int[E^(e/(c + d*x)^2)*(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 66.527, size = 298, normalized size = 0.93 \[ - \frac{b^{3} e^{2} \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{2}} \right )}}{4 d^{4}} + \frac{b^{3} e \left (c + d x\right )^{2} e^{\frac{e}{\left (c + d x\right )^{2}}}}{4 d^{4}} + \frac{b^{3} \left (c + d x\right )^{4} e^{\frac{e}{\left (c + d x\right )^{2}}}}{4 d^{4}} - \frac{2 \sqrt{\pi } b^{2} e^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d^{4}} + \frac{2 b^{2} e \left (c + d x\right ) \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} + \frac{b^{2} \left (c + d x\right )^{3} \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} - \frac{3 b e \left (a d - b c\right )^{2} \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{2}} \right )}}{2 d^{4}} + \frac{3 b \left (c + d x\right )^{2} \left (a d - b c\right )^{2} e^{\frac{e}{\left (c + d x\right )^{2}}}}{2 d^{4}} - \frac{\sqrt{\pi } \sqrt{e} \left (a d - b c\right )^{3} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d^{4}} + \frac{\left (c + d x\right ) \left (a d - b c\right )^{3} e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(e/(d*x+c)**2)*(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.355645, size = 243, normalized size = 0.75 \[ \frac{4 \sqrt{\pi } \sqrt{e} (b c-a d) \left (a^2 d^2-2 a b c d+b^2 \left (c^2+2 e\right )\right ) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )-b e \left (6 a^2 d^2-12 a b c d+b^2 \left (6 c^2+e\right )\right ) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )+d x e^{\frac{e}{(c+d x)^2}} \left (4 a^3 d^3+6 a^2 b d^3 x+4 a b^2 d \left (d^2 x^2+2 e\right )+b^3 \left (-6 c e+d^3 x^3+d e x\right )\right )}{4 d^4}-\frac{c e^{\frac{e}{(c+d x)^2}} \left (-4 a^3 d^3+6 a^2 b c d^2-4 a b^2 d \left (c^2+2 e\right )+b^3 \left (c^3+7 c e\right )\right )}{4 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[E^(e/(c + d*x)^2)*(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.023, size = 560, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(e/(d*x+c)^2)*(b*x+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} +{\left (6 \, a^{2} b d^{3} + b^{3} d e\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{3} - 3 \, b^{3} c e + 4 \, a b^{2} d e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{3}} + \int \frac{{\left (3 \, b^{3} c^{4} e - 4 \, a b^{2} c^{3} d e -{\left (12 \, a b^{2} c d^{3} e - 6 \, a^{2} b d^{4} e -{\left (6 \, c^{2} d^{2} e + d^{2} e^{2}\right )} b^{3}\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{4} e - 2 \,{\left (3 \, c^{2} d^{2} e - 2 \, d^{2} e^{2}\right )} a b^{2} +{\left (4 \, c^{3} d e - 3 \, c d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \,{\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283888, size = 450, normalized size = 1.4 \[ \frac{4 \, \sqrt{\pi }{\left (2 \,{\left (b^{3} c - a b^{2} d\right )} e^{2} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) -{\left ({\left (b^{3} d e^{2} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e\right )}{\rm Ei}\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (b^{3} d^{5} x^{4} + 4 \, a b^{2} d^{5} x^{3} - b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 6 \, a^{2} b c^{2} d^{3} + 4 \, a^{3} c d^{4} +{\left (6 \, a^{2} b d^{5} + b^{3} d^{3} e\right )} x^{2} -{\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2}\right )} e + 2 \,{\left (2 \, a^{3} d^{5} -{\left (3 \, b^{3} c d^{2} - 4 \, a b^{2} d^{3}\right )} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}\right )} \sqrt{-\frac{e}{d^{2}}}}{4 \, d^{5} \sqrt{-\frac{e}{d^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right )^{3} e^{\frac{e}{c^{2} + 2 c d x + d^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(e/(d*x+c)**2)*(b*x+a)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{3} e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="giac")
[Out]