3.393 \(\int e^{e (c+d x)^3} (a+b x) \, dx\)

Optimal. Leaf size=92 \[ \frac{(c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac{b (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.109915, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac{b (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[E^(e*(c + d*x)^3)*(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.2215, size = 85, normalized size = 0.92 \[ - \frac{b \left (c + d x\right )^{2} \Gamma{\left (\frac{2}{3},- e \left (c + d x\right )^{3} \right )}}{3 d^{2} \left (- e \left (c + d x\right )^{3}\right )^{\frac{2}{3}}} - \frac{\left (c + d x\right ) \left (a d - b c\right ) \Gamma{\left (\frac{1}{3},- e \left (c + d x\right )^{3} \right )}}{3 d^{2} \sqrt [3]{- e \left (c + d x\right )^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.583249, size = 0, normalized size = 0. \[ \int e^{e (c+d x)^3} (a+b x) \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[E^(e*(c + d*x)^3)*(a + b*x),x]

[Out]

Integrate[E^(e*(c + d*x)^3)*(a + b*x), x]

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{{\rm e}^{e \left ( dx+c \right ) ^{3}}} \left ( bx+a \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e*(d*x+c)^3)*(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)*e^((d*x + c)^3*e), x)

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Fricas [A]  time = 0.255587, size = 144, normalized size = 1.57 \[ -\frac{b d \Gamma \left (\frac{2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - \left (-d^{3} e\right )^{\frac{1}{3}}{\left (b c - a d\right )} \Gamma \left (\frac{1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, \left (-d^{3} e\right )^{\frac{2}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*d*gamma(2/3, -d^3*e*x^3 - 3*c*d^2*e*x^2 - 3*c^2*d*e*x - c^3*e) - (-d^3*e
)^(1/3)*(b*c - a*d)*gamma(1/3, -d^3*e*x^3 - 3*c*d^2*e*x^2 - 3*c^2*d*e*x - c^3*e)
)/((-d^3*e)^(2/3)*d)

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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(e*(d*x+c)**3)*(b*x+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)*e^((d*x + c)^3*e), x)