3.1.85 \(\int \frac {F^{c+d x}}{(a+b F^{c+d x})^2} \, dx\) [85]

Optimal. Leaf size=25 \[ -\frac {1}{b d \left (a+b F^{c+d x}\right ) \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 32} \begin {gather*} -\frac {1}{b d \log (F) \left (a+b F^{c+d x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*d*(a + b*F^(c + d*x))*Log[F]))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2278

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{a b d \log (F)+b^2 d F^{c+d x} \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(c + d*x)/(a + b*F^(c + d*x))^2,x]

[Out]

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

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Maple [A]
time = 0.01, size = 26, normalized size = 1.04

method	result	size
derivativedivides	\(-\frac {1}{b d \left (a +b \,F^{d x +c}\right ) \ln \left (F \right )}\)	\(26\)
default	\(-\frac {1}{b d \left (a +b \,F^{d x +c}\right ) \ln \left (F \right )}\)	\(26\)
risch	\(-\frac {1}{b d \left (a +b \,F^{d x +c}\right ) \ln \left (F \right )}\)	\(26\)
norman	\(\frac {{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}}{\ln \left (F \right ) a d \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(d*x+c)/(a+b*F^(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (F^{d x + c} b + a\right )} b d \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/((F^(d*x + c)*b + a)*b*d*log(F))

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Fricas [A]
time = 0.40, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{F^{d x + c} b^{2} d \log \left (F\right ) + a b d \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.04, size = 26, normalized size = 1.04 \begin {gather*} - \frac {1}{F^{c + d x} b^{2} d \log {\left (F \right )} + a b d \log {\left (F \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(d*x+c)/(a+b*F**(d*x+c))**2,x)

[Out]

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

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Giac [A]
time = 2.38, size = 26, normalized size = 1.04 \begin {gather*} -\frac {1}{{\left (F^{d x} F^{c} b + a\right )} b d \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(d*x+c)/(a+b*F^(d*x+c))^2,x, algorithm="giac")

[Out]

-1/((F^(d*x)*F^c*b + a)*b*d*log(F))

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Mupad [B]
time = 3.46, size = 31, normalized size = 1.24 \begin {gather*} \frac {F^{c+d\,x}}{a^2\,d\,\ln \left (F\right )+F^{c+d\,x}\,a\,b\,d\,\ln \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(c + d*x)/(a^2*d*log(F) + F^(c + d*x)*a*b*d*log(F))

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