\(\int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx\) [396]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=-\frac {e^{e (c+d x)^3}}{b (a+b x)}-\frac {d (b c-a d) e (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac {d e (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}+\frac {3 d (b c-a d)^2 e \text {Int}\left (\frac {e^{e (c+d x)^3}}{a+b x},x\right )}{b^3} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

[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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

[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]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int \frac {{\mathrm e}^{e \left (d x +c \right )^{3}}}{\left (b x +a \right )^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.84 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int { \frac {e^{\left ({\left (d x + c\right )}^{3} e\right )}}{{\left (b x + a\right )}^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.05 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int \frac {e^{e \left (c + d x\right )^{3}}}{\left (a + b x\right )^{2}} \, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int { \frac {e^{\left ({\left (d x + c\right )}^{3} e\right )}}{{\left (b x + a\right )}^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.05 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int { \frac {e^{\left ({\left (d x + c\right )}^{3} e\right )}}{{\left (b x + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

undef

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx=\int \frac {{\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}}{{\left (a+b\,x\right )}^2} \,d x \]

[In]

int(exp(e*(c + d*x)^3)/(a + b*x)^2,x)

[Out]

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