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

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

[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

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Rubi [A]
time = 0.24, 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 = {} \begin {gather*} \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \end {gather*}

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 = 2.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{e \left (d x +c \right )^{3}}}{\left (b x +a \right )^{2}}\, dx\]

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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