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

Optimal. Leaf size=65 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b F^{c+d x}\right )^2},x\right )}{2 b d \log (F)}-\frac {1}{2 b d x \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

-1/2/b/d/(a+b*F^(d*x+c))^2/x/ln(F)-1/2*Unintegrable(1/(a+b*F^(d*x+c))^2/x^2,x)/b/d/ln(F)

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-1/(2*b*d*(a + b*F^(c + d*x))^2*x*Log[F]) - Defer[Int][1/((a + b*F^(c + d*x))^2*x^2), x]/(2*b*d*Log[F])

Rubi steps

\begin {align*} \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x} \, dx &=-\frac {1}{2 b d \left (a+b F^{c+d x}\right )^2 x \log (F)}-\frac {\int \frac {1}{\left (a+b F^{c+d x}\right )^2 x^2} \, dx}{2 b d \log (F)}\\ \end {align*}

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Mathematica [A]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3 x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{d x + c}}{3 \, F^{d x + c} a^{2} b x + 3 \, F^{2 \, d x + 2 \, c} a b^{2} x + F^{3 \, d x + 3 \, c} b^{3} x + a^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^3/x,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {F^{d x +c}}{\left (b \,F^{d x +c}+a \right )^{3} x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^3/x,x, algorithm="maxima")

[Out]

-1/2*(a*d*x*log(F) + F^(d*x)*F^c*b + a)/(2*F^(d*x)*F^c*a^2*b^2*d^2*x^2*log(F)^2 + F^(2*d*x)*F^(2*c)*a*b^3*d^2*
x^2*log(F)^2 + a^3*b*d^2*x^2*log(F)^2) - integrate(1/2*(d*x*log(F) + 2)/(F^(d*x)*F^c*a*b^2*d^2*x^3*log(F)^2 +
a^2*b*d^2*x^3*log(F)^2), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{c+d\,x}}{x\,{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- F^{c + d x} b - a d x \log {\relax (F )} - a}{4 F^{c + d x} a^{2} b^{2} d^{2} x^{2} \log {\relax (F )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} x^{2} \log {\relax (F )}^{2} + 2 a^{3} b d^{2} x^{2} \log {\relax (F )}^{2}} - \frac {\int \frac {d x \log {\relax (F )}}{a x^{3} + b x^{3} e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx + \int \frac {2}{a x^{3} + b x^{3} e^{c \log {\relax (F )}} e^{d x \log {\relax (F )}}}\, dx}{2 a b d^{2} \log {\relax (F )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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