3.392 \(\int e^{e (c+d x)^3} (a+b x)^2 \, dx\)

Optimal. Leaf size=126 \[ \frac{2 b (c+d x)^2 (b c-a d) \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}}-\frac{(c+d x) (b c-a d)^2 \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac{b^2 e^{e (c+d x)^3}}{3 d^3 e} \]

[Out]

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

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Rubi [A]  time = 0.185527, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 b (c+d x)^2 (b c-a d) \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}}-\frac{(c+d x) (b c-a d)^2 \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac{b^2 e^{e (c+d x)^3}}{3 d^3 e} \]

Antiderivative was successfully verified.

[In]  Int[E^(e*(c + d*x)^3)*(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 24.2999, size = 114, normalized size = 0.9 \[ \frac{b^{2} e^{e \left (c + d x\right )^{3}}}{3 d^{3} e} - \frac{2 b \left (c + d x\right )^{2} \left (a d - b c\right ) \Gamma{\left (\frac{2}{3},- e \left (c + d x\right )^{3} \right )}}{3 d^{3} \left (- e \left (c + d x\right )^{3}\right )^{\frac{2}{3}}} - \frac{\left (c + d x\right ) \left (a d - b c\right )^{2} \Gamma{\left (\frac{1}{3},- e \left (c + d x\right )^{3} \right )}}{3 d^{3} \sqrt [3]{- e \left (c + d x\right )^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e*(d*x+c)**3)*(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.457718, size = 0, normalized size = 0. \[ \int e^{e (c+d x)^3} (a+b x)^2 \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[E^(e*(c + d*x)^3)*(a + b*x)^2,x]

[Out]

Integrate[E^(e*(c + d*x)^3)*(a + b*x)^2, x]

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{{\rm e}^{e \left ( dx+c \right ) ^{3}}} \left ( bx+a \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e*(d*x+c)^3)*(b*x+a)^2,x)

[Out]

int(exp(e*(d*x+c)^3)*(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{2} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*e^((d*x + c)^3*e),x, algorithm="maxima")

[Out]

integrate((b*x + a)^2*e^((d*x + c)^3*e), x)

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Fricas [A]  time = 0.244782, size = 254, normalized size = 2.02 \[ \frac{\left (-d^{3} e\right )^{\frac{2}{3}} b^{2} e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )} -{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \left (-d^{3} e\right )^{\frac{1}{3}} e \Gamma \left (\frac{1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) + 2 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e \Gamma \left (\frac{2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, \left (-d^{3} e\right )^{\frac{2}{3}} d^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*e^((d*x + c)^3*e),x, algorithm="fricas")

[Out]

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

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{2} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*e^((d*x + c)^3*e),x, algorithm="giac")

[Out]

integrate((b*x + a)^2*e^((d*x + c)^3*e), x)