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

Optimal. Leaf size=153 \[ \frac {3 d e (b c-a d)^2 \text {Int}\left (\frac {e^{e (c+d x)^3}}{a+b x},x\right )}{b^3}-\frac {d e (c+d x) (b c-a d) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac {e^{e (c+d x)^3}}{b (a+b x)}-\frac {d e (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}} \]

[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

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Rubi [A]  time = 0.35, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

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*}

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Mathematica [A]  time = 2.06, size = 0, normalized size = 0.00 \[ \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

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

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\left (d x +c \right )^{3} e}}{\left (b x +a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left ({\left (d x + c\right )}^{3} e\right )}}{{\left (b x + a\right )}^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}}{{\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c^{3} e} \int \frac {e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}}{a^{2} + 2 a b x + b^{2} x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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