3.21 \(\int \frac{1}{\left (a+b e^{c+d x}\right )^3 x} \, dx\)

Optimal. Leaf size=20 \[ \text{Int}\left (\frac{1}{x \left (a+b e^{c+d x}\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0712577, 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. \[ \text{Int}\left (\frac{1}{\left (a+b e^{c+d x}\right )^3 x},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[1/((a + b*E^(c + d*x))^3*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b e^{c + d x}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(d*x+c))**3/x,x)

[Out]

Integral(1/(x*(a + b*exp(c + d*x))**3), x)

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Mathematica [A]  time = 0.983702, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b e^{c+d x}\right )^3 x} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/((a + b*E^(c + d*x))^3*x),x]

[Out]

Integrate[1/((a + b*E^(c + d*x))^3*x), x]

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Maple [A]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{{\rm e}^{dx+c}} \right ) ^{3}x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(d*x+c))^3/x,x)

[Out]

int(1/(a+b*exp(d*x+c))^3/x,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \, a d x +{\left (2 \, b d x e^{c} + b e^{c}\right )} e^{\left (d x\right )} + a}{2 \,{\left (a^{2} b^{2} d^{2} x^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} x^{2} e^{\left (d x + c\right )} + a^{4} d^{2} x^{2}\right )}} + \int \frac{2 \, d^{2} x^{2} + 3 \, d x + 2}{2 \,{\left (a^{2} b d^{2} x^{3} e^{\left (d x + c\right )} + a^{3} d^{2} x^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)^3*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{3} x e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b^{2} x e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} b x e^{\left (d x + c\right )} + a^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)^3*x),x, algorithm="fricas")

[Out]

integral(1/(b^3*x*e^(3*d*x + 3*c) + 3*a*b^2*x*e^(2*d*x + 2*c) + 3*a^2*b*x*e^(d*x
+ c) + a^3*x), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a d x + a + \left (2 b d x + b\right ) e^{c + d x}}{2 a^{4} d^{2} x^{2} + 4 a^{3} b d^{2} x^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} x^{2} e^{2 c + 2 d x}} + \frac{\int \frac{3 d x}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac{2 d^{2} x^{2}}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac{2}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx}{2 a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*exp(d*x+c))**3/x,x)

[Out]

(3*a*d*x + a + (2*b*d*x + b)*exp(c + d*x))/(2*a**4*d**2*x**2 + 4*a**3*b*d**2*x**
2*exp(c + d*x) + 2*a**2*b**2*d**2*x**2*exp(2*c + 2*d*x)) + (Integral(3*d*x/(a*x*
*3 + b*x**3*exp(c)*exp(d*x)), x) + Integral(2*d**2*x**2/(a*x**3 + b*x**3*exp(c)*
exp(d*x)), x) + Integral(2/(a*x**3 + b*x**3*exp(c)*exp(d*x)), x))/(2*a**2*d**2)

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*e^(d*x + c) + a)^3*x),x, algorithm="giac")

[Out]

integrate(1/((b*e^(d*x + c) + a)^3*x), x)