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

Optimal. Leaf size=240 \[ \frac{b d^2 e^2 e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{2 (b c-a d)^4}+\frac{d^2 e e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{(b c-a d)^3}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}+\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2} \]

[Out]

(d^2*E^(e/(c + d*x)))/(2*b*(b*c - a*d)^2) + (d^2*e*E^(e/(c + d*x)))/(2*(b*c - a*
d)^3) - E^(e/(c + d*x))/(2*b*(a + b*x)^2) + (d*e*E^(e/(c + d*x)))/(2*(b*c - a*d)
^2*(a + b*x)) + (d^2*e*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b
*c - a*d)*(c + d*x)))])/(b*c - a*d)^3 + (b*d^2*e^2*E^((b*e)/(b*c - a*d))*ExpInte
gralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(2*(b*c - a*d)^4)

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Rubi [A]  time = 1.71362, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{b d^2 e^2 e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{2 (b c-a d)^4}+\frac{d^2 e e^{\frac{b e}{b c-a d}} \text{ExpIntegralEi}\left (-\frac{d e (a+b x)}{(c+d x) (b c-a d)}\right )}{(b c-a d)^3}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}+\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*E^(e/(c + d*x)))/(2*b*(b*c - a*d)^2) + (d^2*e*E^(e/(c + d*x)))/(2*(b*c - a*
d)^3) - E^(e/(c + d*x))/(2*b*(a + b*x)^2) + (d*e*E^(e/(c + d*x)))/(2*(b*c - a*d)
^2*(a + b*x)) + (d^2*e*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b
*c - a*d)*(c + d*x)))])/(b*c - a*d)^3 + (b*d^2*e^2*E^((b*e)/(b*c - a*d))*ExpInte
gralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(2*(b*c - a*d)^4)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.336642, size = 0, normalized size = 0. \[ \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.013, size = 240, normalized size = 1. \[ -{\frac{e}{d} \left ({\frac{{d}^{3}}{ \left ( ad-cb \right ) ^{3}} \left ( -{1{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-cb}} \right ) ^{-1}}-{{\rm e}^{-{\frac{be}{ad-cb}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-cb}} \right ) \right ) }-{\frac{be{d}^{3}}{ \left ( ad-cb \right ) ^{4}} \left ( -{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-cb}} \right ) ^{-2}}-{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-cb}} \right ) ^{-1}}-{\frac{1}{2}{{\rm e}^{-{\frac{be}{ad-cb}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-cb}} \right ) } \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c))/(b*x+a)^3,x)

[Out]

-1/d*e*(d^3/(a*d-b*c)^3*(-exp(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))-exp(-b*e/(a*d
-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*d-b*c)))-b*e/(a*d-b*c)^4*d^3*(-1/2*exp(e/(d*x+c))/
(e/(d*x+c)+b*e/(a*d-b*c))^2-1/2*exp(e/(d*x+c))/(e/(d*x+c)+b*e/(a*d-b*c))-1/2*exp
(-b*e/(a*d-b*c))*Ei(1,-e/(d*x+c)-b*e/(a*d-b*c))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256916, size = 698, normalized size = 2.91 \[ \frac{{\left (a^{2} b d^{2} e^{2} +{\left (b^{3} d^{2} e^{2} + 2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \,{\left (a b^{2} d^{2} e^{2} + 2 \,{\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\right )}{\rm Ei}\left (-\frac{b d e x + a d e}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac{b e}{b c - a d}\right )} -{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} -{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} -{\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e -{\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} +{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{2 \,{\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} +{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((a^2*b*d^2*e^2 + (b^3*d^2*e^2 + 2*(b^3*c*d^2 - a*b^2*d^3)*e)*x^2 + 2*(a^2*b
*c*d^2 - a^3*d^3)*e + 2*(a*b^2*d^2*e^2 + 2*(a*b^2*c*d^2 - a^2*b*d^3)*e)*x)*Ei(-(
b*d*e*x + a*d*e)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x))*e^(b*e/(b*c - a*d)) - (b^3
*c^4 - 4*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 - 2*a^3*c*d^3 - (b^3*c^2*d^2 - 2*a*b^2*c*
d^3 + a^2*b*d^4 + (b^3*c*d^2 - a*b^2*d^3)*e)*x^2 - (a*b^2*c^2*d - a^2*b*c*d^2)*e
- (2*a*b^2*c^2*d^2 - 4*a^2*b*c*d^3 + 2*a^3*d^4 + (b^3*c^2*d - a^2*b*d^3)*e)*x)*
e^(e/(d*x + c)))/(a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*
d^3 + a^6*d^4 + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 +
a^4*b^2*d^4)*x^2 + 2*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b
^2*c*d^3 + a^5*b*d^4)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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