3.756 \(\int e^{(a+b x)^3} x \, dx\)

Optimal. Leaf size=80 \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]

[Out]

(a*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(1/3)) - ((a + b*x)
^2*Gamma[2/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(2/3))

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Rubi [A]  time = 0.0795241, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[E^(a + b*x)^3*x,x]

[Out]

(a*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(1/3)) - ((a + b*x)
^2*Gamma[2/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(2/3))

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Rubi in Sympy [A]  time = 8.10108, size = 70, normalized size = 0.88 \[ \frac{a \left (a + b x\right ) \Gamma{\left (\frac{1}{3},- \left (a + b x\right )^{3} \right )}}{3 b^{2} \sqrt [3]{- \left (a + b x\right )^{3}}} - \frac{\left (a + b x\right )^{2} \Gamma{\left (\frac{2}{3},- \left (a + b x\right )^{3} \right )}}{3 b^{2} \left (- \left (a + b x\right )^{3}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp((b*x+a)**3)*x,x)

[Out]

a*(a + b*x)*Gamma(1/3, -(a + b*x)**3)/(3*b**2*(-(a + b*x)**3)**(1/3)) - (a + b*x
)**2*Gamma(2/3, -(a + b*x)**3)/(3*b**2*(-(a + b*x)**3)**(2/3))

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Mathematica [A]  time = 0.26031, size = 0, normalized size = 0. \[ \int e^{(a+b x)^3} x \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[E^(a + b*x)^3*x,x]

[Out]

Integrate[E^(a + b*x)^3*x, x]

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \[ \int{{\rm e}^{ \left ( bx+a \right ) ^{3}}}x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp((b*x+a)^3)*x,x)

[Out]

int(exp((b*x+a)^3)*x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int x e^{\left ({\left (b x + a\right )}^{3}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^((b*x + a)^3),x, algorithm="maxima")

[Out]

integrate(x*e^((b*x + a)^3), x)

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Fricas [A]  time = 0.256183, size = 120, normalized size = 1.5 \[ \frac{\left (-b^{3}\right )^{\frac{1}{3}} a \Gamma \left (\frac{1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) - b \Gamma \left (\frac{2}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, \left (-b^{3}\right )^{\frac{2}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^((b*x + a)^3),x, algorithm="fricas")

[Out]

1/3*((-b^3)^(1/3)*a*gamma(1/3, -b^3*x^3 - 3*a*b^2*x^2 - 3*a^2*b*x - a^3) - b*gam
ma(2/3, -b^3*x^3 - 3*a*b^2*x^2 - 3*a^2*b*x - a^3))/((-b^3)^(2/3)*b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp((b*x+a)**3)*x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int x e^{\left ({\left (b x + a\right )}^{3}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^((b*x + a)^3),x, algorithm="giac")

[Out]

integrate(x*e^((b*x + a)^3), x)