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

Optimal. Leaf size=106 \[ -\frac{\log \left (a+b F^{c+d x}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{x}{2 a^2 b d \log (F)}+\frac{1}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

[Out]

1/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x/(2*a^2*b*d*Log[F]) - x/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) - L
og[a + b*F^(c + d*x)]/(2*a^2*b*d^2*Log[F]^2)

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Rubi [A]  time = 0.0885608, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2191, 2282, 44} \[ -\frac{\log \left (a+b F^{c+d x}\right )}{2 a^2 b d^2 \log ^2(F)}+\frac{x}{2 a^2 b d \log (F)}+\frac{1}{2 a b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )}-\frac{x}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x/(2*a^2*b*d*Log[F]) - x/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) - L
og[a + b*F^(c + d*x)]/(2*a^2*b*d^2*Log[F]^2)

Rule 2191

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p +
1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0853442, size = 98, normalized size = 0.92 \[ \frac{b d x \log (F) F^{c+d x} \left (2 a+b F^{c+d x}\right )-\left (a+b F^{c+d x}\right ) \left (\left (a+b F^{c+d x}\right ) \log \left (a+b F^{c+d x}\right )-a\right )}{2 a^2 b d^2 \log ^2(F) \left (a+b F^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.019, size = 127, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) ^{2}} \left ({\frac{{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }}}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}a{d}^{2}}}+{\frac{x{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }}}{\ln \left ( F \right ) ad}}+{\frac{bx \left ({{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) ^{2}}{2\,\ln \left ( F \right ){a}^{2}d}}+{\frac{1}{2\, \left ( \ln \left ( F \right ) \right ) ^{2}b{d}^{2}}} \right ) }-{\frac{\ln \left ( a+b{{\rm e}^{ \left ( dx+c \right ) \ln \left ( F \right ) }} \right ) }{2\, \left ( \ln \left ( F \right ) \right ) ^{2}{a}^{2}b{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16584, size = 203, normalized size = 1.92 \begin{align*} \frac{F^{2 \, d x} F^{2 \, c} b^{2} d x \log \left (F\right ) +{\left (2 \, F^{c} a b d x \log \left (F\right ) + F^{c} a b\right )} F^{d x} + a^{2}}{2 \,{\left (2 \, F^{d x} F^{c} a^{3} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a^{2} b^{3} d^{2} \log \left (F\right )^{2} + a^{4} b d^{2} \log \left (F\right )^{2}\right )}} - \frac{\log \left (\frac{F^{d x} F^{c} b + a}{F^{c} b}\right )}{2 \, a^{2} b d^{2} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(F^(2*d*x)*F^(2*c)*b^2*d*x*log(F) + (2*F^c*a*b*d*x*log(F) + F^c*a*b)*F^(d*x) + a^2)/(2*F^(d*x)*F^c*a^3*b^2
*d^2*log(F)^2 + F^(2*d*x)*F^(2*c)*a^2*b^3*d^2*log(F)^2 + a^4*b*d^2*log(F)^2) - 1/2*log((F^(d*x)*F^c*b + a)/(F^
c*b))/(a^2*b*d^2*log(F)^2)

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Fricas [A]  time = 1.54996, size = 342, normalized size = 3.23 \begin{align*} \frac{F^{2 \, d x + 2 \, c} b^{2} d x \log \left (F\right ) +{\left (2 \, a b d x \log \left (F\right ) + a b\right )} F^{d x + c} + a^{2} -{\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} \log \left (F^{d x + c} b + a\right )}{2 \,{\left (2 \, F^{d x + c} a^{3} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{2} \log \left (F\right )^{2} + a^{4} b d^{2} \log \left (F\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(F^(2*d*x + 2*c)*b^2*d*x*log(F) + (2*a*b*d*x*log(F) + a*b)*F^(d*x + c) + a^2 - (2*F^(d*x + c)*a*b + F^(2*d
*x + 2*c)*b^2 + a^2)*log(F^(d*x + c)*b + a))/(2*F^(d*x + c)*a^3*b^2*d^2*log(F)^2 + F^(2*d*x + 2*c)*a^2*b^3*d^2
*log(F)^2 + a^4*b*d^2*log(F)^2)

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Sympy [A]  time = 0.204384, size = 122, normalized size = 1.15 \begin{align*} \frac{F^{c + d x} b - a d x \log{\left (F \right )} + a}{4 F^{c + d x} a^{2} b^{2} d^{2} \log{\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log{\left (F \right )}^{2} + 2 a^{3} b d^{2} \log{\left (F \right )}^{2}} + \frac{x}{2 a^{2} b d \log{\left (F \right )}} - \frac{\log{\left (F^{c + d x} + \frac{a}{b} \right )}}{2 a^{2} b d^{2} \log{\left (F \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{d x + c} x}{{\left (F^{d x + c} b + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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