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

Optimal. Leaf size=152 \[ \frac{3 d e (b c-a d)^2 \text{Unintegrable}\left (\frac{e^{e (c+d x)^3}}{a+b x},x\right )}{b^3}-\frac{d e (c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac{d e (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}-\frac{e^{e (c+d x)^3}}{b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.350316, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(e*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{(3 d e) \int \frac{e^{e (c+d x)^3} (c+d x)^2}{a+b x} \, dx}{b}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{(3 d e) \int \left (\frac{d (b c-a d) e^{e (c+d x)^3}}{b^2}+\frac{(b c-a d)^2 e^{e (c+d x)^3}}{b^2 (a+b x)}+\frac{d e^{e (c+d x)^3} (c+d x)}{b}\right ) \, dx}{b}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{\left (3 d^2 e\right ) \int e^{e (c+d x)^3} (c+d x) \, dx}{b^2}+\frac{\left (3 d^2 (b c-a d) e\right ) \int e^{e (c+d x)^3} \, dx}{b^3}+\frac{\left (3 d (b c-a d)^2 e\right ) \int \frac{e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}-\frac{d (b c-a d) e (c+d x) \Gamma \left (\frac{1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac{d e (c+d x)^2 \Gamma \left (\frac{2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}+\frac{\left (3 d (b c-a d)^2 e\right ) \int \frac{e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ \end{align*}

Mathematica [A]  time = 2.05486, size = 0, normalized size = 0. \[ \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

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 [A]  time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{e \left ( dx+c \right ) ^{3}}}}{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}

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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left ({\left (d x + c\right )}^{3} e\right )}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e*(d*x+c)^3)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e*(d*x+c)^3)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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

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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e*(d*x+c)^3)/(b*x+a)^2,x, algorithm="giac")

[Out]

undef