\(\int \frac {F^{c+d x} x^2}{(a+b F^{c+d x})^3} \, dx\) [89]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=-\frac {x}{a^2 b d^2 \log ^2(F)}+\frac {x}{a b d^2 \left (a+b F^{c+d x}\right ) \log ^2(F)}+\frac {x^2}{2 a^2 b d \log (F)}-\frac {x^2}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}+\frac {\log \left (a+b F^{c+d x}\right )}{a^2 b d^3 \log ^3(F)}-\frac {x \log \left (1+\frac {b F^{c+d x}}{a}\right )}{a^2 b d^2 \log ^2(F)}-\frac {\operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{a^2 b d^3 \log ^3(F)} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97 \[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=\frac {b d^2 F^{c+d x} \left (2 a+b F^{c+d x}\right ) x^2 \log ^2(F)+2 \left (a+b F^{c+d x}\right )^2 \log \left (1+\frac {b F^{c+d x}}{a}\right )-2 d \left (a+b F^{c+d x}\right ) x \log (F) \left (b F^{c+d x}+\left (a+b F^{c+d x}\right ) \log \left (1+\frac {b F^{c+d x}}{a}\right )\right )-2 \left (a+b F^{c+d x}\right )^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{2 a^2 b d^3 \left (a+b F^{c+d x}\right )^2 \log ^3(F)} \] Input:

Integrate[(F^(c + d*x)*x^2)/(a + b*F^(c + d*x))^3,x]
 

Output:

(b*d^2*F^(c + d*x)*(2*a + b*F^(c + d*x))*x^2*Log[F]^2 + 2*(a + b*F^(c + d* 
x))^2*Log[1 + (b*F^(c + d*x))/a] - 2*d*(a + b*F^(c + d*x))*x*Log[F]*(b*F^( 
c + d*x) + (a + b*F^(c + d*x))*Log[1 + (b*F^(c + d*x))/a]) - 2*(a + b*F^(c 
 + d*x))^2*PolyLog[2, -((b*F^(c + d*x))/a)])/(2*a^2*b*d^3*(a + b*F^(c + d* 
x))^2*Log[F]^3)
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2621, 2616, 2615, 2620, 2621, 2715, 2720, 47, 14, 16, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 F^{c+d x}}{\left (a+b F^{c+d x}\right )^3} \, dx\)

\(\Big \downarrow \) 2621

\(\displaystyle \frac {\int \frac {x}{\left (b F^{c+d x}+a\right )^2}dx}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2616

\(\displaystyle \frac {\frac {\int \frac {x}{b F^{c+d x}+a}dx}{a}-\frac {b \int \frac {F^{c+d x} x}{\left (b F^{c+d x}+a\right )^2}dx}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \int \frac {F^{c+d x} x}{b F^{c+d x}+a}dx}{a}}{a}-\frac {b \int \frac {F^{c+d x} x}{\left (b F^{c+d x}+a\right )^2}dx}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int \log \left (\frac {b F^{c+d x}}{a}+1\right )dx}{b d \log (F)}\right )}{a}}{a}-\frac {b \int \frac {F^{c+d x} x}{\left (b F^{c+d x}+a\right )^2}dx}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2621

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int \log \left (\frac {b F^{c+d x}}{a}+1\right )dx}{b d \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {1}{b F^{c+d x}+a}dx}{b d \log (F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int F^{-c-d x} \log \left (\frac {b F^{c+d x}}{a}+1\right )dF^{c+d x}}{b d^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {1}{b F^{c+d x}+a}dx}{b d \log (F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int F^{-c-d x} \log \left (\frac {b F^{c+d x}}{a}+1\right )dF^{c+d x}}{b d^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {\int \frac {F^{-c-d x}}{b F^{c+d x}+a}dF^{c+d x}}{b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int F^{-c-d x} \log \left (\frac {b F^{c+d x}}{a}+1\right )dF^{c+d x}}{b d^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\int F^{-c-d x}dF^{c+d x}}{a}-\frac {b \int \frac {1}{b F^{c+d x}+a}dF^{c+d x}}{a}}{b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int F^{-c-d x} \log \left (\frac {b F^{c+d x}}{a}+1\right )dF^{c+d x}}{b d^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (F^{c+d x}\right )}{a}-\frac {b \int \frac {1}{b F^{c+d x}+a}dF^{c+d x}}{a}}{b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {\int F^{-c-d x} \log \left (\frac {b F^{c+d x}}{a}+1\right )dF^{c+d x}}{b d^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (F^{c+d x}\right )}{a}-\frac {\log \left (a+b F^{c+d x}\right )}{a}}{b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {\frac {x^2}{2 a}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}+\frac {x \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {\frac {\log \left (F^{c+d x}\right )}{a}-\frac {\log \left (a+b F^{c+d x}\right )}{a}}{b d^2 \log ^2(F)}-\frac {x}{b d \log (F) \left (a+b F^{c+d x}\right )}\right )}{a}}{b d \log (F)}-\frac {x^2}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2}\)

Input:

Int[(F^(c + d*x)*x^2)/(a + b*F^(c + d*x))^3,x]
 

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 2615

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x 
_))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ 
b/a   Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] 
, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2616

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[1/a   Int[(c + d*x)^m*(a + b*(F^(g*(e 
+ f*x)))^n)^(p + 1), x], x] - Simp[b/a   Int[(c + d*x)^m*(F^(g*(e + f*x)))^ 
n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n 
}, x] && ILtQ[p, 0] && IGtQ[m, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2621

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] - Simp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.67

method	result	size
risch	\(-\frac {x \left (\ln \left (F \right ) a d x -2 b \,F^{d x +c}-2 a \right )}{2 \ln \left (F \right )^{2} d^{2} b \left (a +b \,F^{d x +c}\right )^{2} a}+\frac {x^{2}}{2 a^{2} b d \ln \left (F \right )}+\frac {c x}{b \,a^{2} d^{2} \ln \left (F \right )}+\frac {c^{2}}{2 b \,a^{2} d^{3} \ln \left (F \right )}-\frac {\ln \left (1+\frac {b \,F^{c} F^{d x}}{a}\right ) x}{b \,a^{2} d^{2} \ln \left (F \right )^{2}}-\frac {\ln \left (1+\frac {b \,F^{c} F^{d x}}{a}\right ) c}{b \,a^{2} d^{3} \ln \left (F \right )^{2}}-\frac {\operatorname {polylog}\left (2, -\frac {b \,F^{c} F^{d x}}{a}\right )}{b \,a^{2} d^{3} \ln \left (F \right )^{3}}-\frac {\ln \left (F^{c} F^{d x}\right )}{b \,a^{2} d^{3} \ln \left (F \right )^{3}}+\frac {\ln \left (F^{d x} F^{c} b +a \right )}{b \,a^{2} d^{3} \ln \left (F \right )^{3}}-\frac {c \ln \left (F^{c} F^{d x}\right )}{b \,a^{2} d^{3} \ln \left (F \right )^{2}}+\frac {c \ln \left (F^{d x} F^{c} b +a \right )}{b \,a^{2} d^{3} \ln \left (F \right )^{2}}\)	\(304\)

Input:

int(F^(d*x+c)*x^2/(a+b*F^(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (177) = 354\).

Time = 0.08 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.08 \[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=-\frac {a^{2} c^{2} \log \left (F\right )^{2} + 2 \, a^{2} c \log \left (F\right ) - {\left ({\left (b^{2} d^{2} x^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )\right )} F^{2 \, d x + 2 \, c} - 2 \, {\left ({\left (a b d^{2} x^{2} - a b c^{2}\right )} \log \left (F\right )^{2} - {\left (a b d x + 2 \, a b c\right )} \log \left (F\right )\right )} F^{d x + c} + 2 \, {\left (2 \, F^{d x + c} a b + F^{2 \, d x + 2 \, c} b^{2} + a^{2}\right )} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) - 2 \, {\left (a^{2} c \log \left (F\right ) + {\left (b^{2} c \log \left (F\right ) + b^{2}\right )} F^{2 \, d x + 2 \, c} + 2 \, {\left (a b c \log \left (F\right ) + a b\right )} F^{d x + c} + a^{2}\right )} \log \left (F^{d x + c} b + a\right ) + 2 \, {\left ({\left (b^{2} d x + b^{2} c\right )} F^{2 \, d x + 2 \, c} \log \left (F\right ) + 2 \, {\left (a b d x + a b c\right )} F^{d x + c} \log \left (F\right ) + {\left (a^{2} d x + a^{2} c\right )} \log \left (F\right )\right )} \log \left (\frac {F^{d x + c} b + a}{a}\right )}{2 \, {\left (2 \, F^{d x + c} a^{3} b^{2} d^{3} \log \left (F\right )^{3} + F^{2 \, d x + 2 \, c} a^{2} b^{3} d^{3} \log \left (F\right )^{3} + a^{4} b d^{3} \log \left (F\right )^{3}\right )}} \] Input:

integrate(F^(d*x+c)*x^2/(a+b*F^(d*x+c))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=\frac {2 F^{c + d x} b x - a d x^{2} \log {\left (F \right )} + 2 a x}{4 F^{c + d x} a^{2} b^{2} d^{2} \log {\left (F \right )}^{2} + 2 F^{2 c + 2 d x} a b^{3} d^{2} \log {\left (F \right )}^{2} + 2 a^{3} b d^{2} \log {\left (F \right )}^{2}} + \frac {\int \frac {d x \log {\left (F \right )}}{a + b e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\, dx + \int \left (- \frac {1}{a + b e^{c \log {\left (F \right )}} e^{d x \log {\left (F \right )}}}\right )\, dx}{a b d^{2} \log {\left (F \right )}^{2}} \] Input:

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

Output:

(2*F**(c + d*x)*b*x - a*d*x**2*log(F) + 2*a*x)/(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*l 
og(F)**2) + (Integral(d*x*log(F)/(a + b*exp(c*log(F))*exp(d*x*log(F))), x) 
 + Integral(-1/(a + b*exp(c*log(F))*exp(d*x*log(F))), x))/(a*b*d**2*log(F) 
**2)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.11 \[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=-\frac {a d x^{2} \log \left (F\right ) - 2 \, F^{d x} F^{c} b x - 2 \, a x}{2 \, {\left (2 \, F^{d x} F^{c} a^{2} b^{2} d^{2} \log \left (F\right )^{2} + F^{2 \, d x} F^{2 \, c} a b^{3} d^{2} \log \left (F\right )^{2} + a^{3} b d^{2} \log \left (F\right )^{2}\right )}} + \frac {x^{2}}{2 \, a^{2} b d \log \left (F\right )} - \frac {x}{a^{2} b d^{2} \log \left (F\right )^{2}} - \frac {d x \log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right )}{a^{2} b d^{3} \log \left (F\right )^{3}} + \frac {\log \left (F^{d x} F^{c} b + a\right )}{a^{2} b d^{3} \log \left (F\right )^{3}} \] Input:

integrate(F^(d*x+c)*x^2/(a+b*F^(d*x+c))^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=\int { \frac {F^{d x + c} x^{2}}{{\left (F^{d x + c} b + a\right )}^{3}} \,d x } \] Input:

integrate(F^(d*x+c)*x^2/(a+b*F^(d*x+c))^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=\int \frac {F^{c+d\,x}\,x^2}{{\left (a+F^{c+d\,x}\,b\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {F^{c+d x} x^2}{\left (a+b F^{c+d x}\right )^3} \, dx=\frac {4 f^{2 d x +2 c} \left (\int \frac {x}{f^{3 d x +3 c} b^{3}+3 f^{2 d x +2 c} a \,b^{2}+3 f^{d x +c} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{2} a^{3} b^{2} d^{2}-2 f^{2 d x +2 c} \mathrm {log}\left (f^{d x +c} b +a \right ) b^{2}+2 f^{2 d x +2 c} \mathrm {log}\left (f \right ) b^{2} d x -f^{2 d x +2 c} b^{2}+8 f^{d x +c} \left (\int \frac {x}{f^{3 d x +3 c} b^{3}+3 f^{2 d x +2 c} a \,b^{2}+3 f^{d x +c} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{2} a^{4} b \,d^{2}-4 f^{d x +c} \mathrm {log}\left (f^{d x +c} b +a \right ) a b +4 f^{d x +c} \mathrm {log}\left (f \right ) a b d x +4 \left (\int \frac {x}{f^{3 d x +3 c} b^{3}+3 f^{2 d x +2 c} a \,b^{2}+3 f^{d x +c} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right )^{2} a^{5} d^{2}-2 \,\mathrm {log}\left (f^{d x +c} b +a \right ) a^{2}-2 \mathrm {log}\left (f \right )^{2} a^{2} d^{2} x^{2}+a^{2}}{4 \mathrm {log}\left (f \right )^{3} a^{2} b \,d^{3} \left (f^{2 d x +2 c} b^{2}+2 f^{d x +c} a b +a^{2}\right )} \] Input:

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

Output:

(4*f**(2*c + 2*d*x)*int(x/(f**(3*c + 3*d*x)*b**3 + 3*f**(2*c + 2*d*x)*a*b* 
*2 + 3*f**(c + d*x)*a**2*b + a**3),x)*log(f)**2*a**3*b**2*d**2 - 2*f**(2*c 
 + 2*d*x)*log(f**(c + d*x)*b + a)*b**2 + 2*f**(2*c + 2*d*x)*log(f)*b**2*d* 
x - f**(2*c + 2*d*x)*b**2 + 8*f**(c + d*x)*int(x/(f**(3*c + 3*d*x)*b**3 + 
3*f**(2*c + 2*d*x)*a*b**2 + 3*f**(c + d*x)*a**2*b + a**3),x)*log(f)**2*a** 
4*b*d**2 - 4*f**(c + d*x)*log(f**(c + d*x)*b + a)*a*b + 4*f**(c + d*x)*log 
(f)*a*b*d*x + 4*int(x/(f**(3*c + 3*d*x)*b**3 + 3*f**(2*c + 2*d*x)*a*b**2 + 
 3*f**(c + d*x)*a**2*b + a**3),x)*log(f)**2*a**5*d**2 - 2*log(f**(c + d*x) 
*b + a)*a**2 - 2*log(f)**2*a**2*d**2*x**2 + a**2)/(4*log(f)**3*a**2*b*d**3 
*(f**(2*c + 2*d*x)*b**2 + 2*f**(c + d*x)*a*b + a**2))