\(\int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx\) [76]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)} \] Output:

x^3*ln(1+b*F^(d*x+c)/a)/b/d/ln(F)+3*x^2*polylog(2,-b*F^(d*x+c)/a)/b/d^2/ln 
(F)^2-6*x*polylog(3,-b*F^(d*x+c)/a)/b/d^3/ln(F)^3+6*polylog(4,-b*F^(d*x+c) 
/a)/b/d^4/ln(F)^4
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx=\frac {x^3 \log \left (1+\frac {b F^{c+d x}}{a}\right )}{b d \log (F)}+\frac {3 x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{b d^2 \log ^2(F)}-\frac {6 x \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{b d^3 \log ^3(F)}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {b F^{c+d x}}{a}\right )}{b d^4 \log ^4(F)} \] Input:

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

Output:

(x^3*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + (3*x^2*PolyLog[2, -((b*F^( 
c + d*x))/a)])/(b*d^2*Log[F]^2) - (6*x*PolyLog[3, -((b*F^(c + d*x))/a)])/( 
b*d^3*Log[F]^3) + (6*PolyLog[4, -((b*F^(c + d*x))/a)])/(b*d^4*Log[F]^4)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2620, 3011, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 2620

\(\displaystyle \frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {3 \int x^2 \log \left (\frac {b F^{c+d x}}{a}+1\right )dx}{b d \log (F)}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {3 \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )dx}{d \log (F)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}\right )}{b d \log (F)}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )dx}{d \log (F)}\right )}{d \log (F)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}\right )}{b d \log (F)}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}-\frac {\int F^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )dF^{c+d x}}{d^2 \log ^2(F)}\right )}{d \log (F)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}\right )}{b d \log (F)}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {x^3 \log \left (\frac {b F^{c+d x}}{a}+1\right )}{b d \log (F)}-\frac {3 \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b F^{c+d x}}{a}\right )}{d^2 \log ^2(F)}\right )}{d \log (F)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b F^{c+d x}}{a}\right )}{d \log (F)}\right )}{b d \log (F)}\)

Input:

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

Output:

(x^3*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) - (3*(-((x^2*PolyLog[2, -((b 
*F^(c + d*x))/a)])/(d*Log[F])) + (2*((x*PolyLog[3, -((b*F^(c + d*x))/a)])/ 
(d*Log[F]) - PolyLog[4, -((b*F^(c + d*x))/a)]/(d^2*Log[F]^2)))/(d*Log[F])) 
)/(b*d*Log[F])
 

Defintions of rubi rules used

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 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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.96

method	result	size
risch	\(\frac {\ln \left (1+\frac {b \,F^{c} F^{d x}}{a}\right ) c^{3}}{d^{4} \ln \left (F \right ) b}+\frac {c^{3} \ln \left (F^{c} F^{d x}\right )}{d^{4} \ln \left (F \right ) b}-\frac {c^{3} \ln \left (F^{d x} F^{c} b +a \right )}{d^{4} \ln \left (F \right ) b}-\frac {c^{3} x}{d^{3} b}+\frac {\ln \left (1+\frac {b \,F^{c} F^{d x}}{a}\right ) x^{3}}{d \ln \left (F \right ) b}+\frac {3 \operatorname {polylog}\left (2, -\frac {b \,F^{c} F^{d x}}{a}\right ) x^{2}}{d^{2} \ln \left (F \right )^{2} b}-\frac {6 \operatorname {polylog}\left (3, -\frac {b \,F^{c} F^{d x}}{a}\right ) x}{d^{3} \ln \left (F \right )^{3} b}+\frac {6 \operatorname {polylog}\left (4, -\frac {b \,F^{c} F^{d x}}{a}\right )}{d^{4} \ln \left (F \right )^{4} b}-\frac {3 c^{4}}{4 d^{4} b}\)	\(225\)

Input:

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

Output:

1/d^4/ln(F)/b*ln(1+b*F^c*F^(d*x)/a)*c^3+1/d^4/ln(F)/b*c^3*ln(F^c*F^(d*x))- 
1/d^4/ln(F)/b*c^3*ln(F^(d*x)*F^c*b+a)-1/d^3/b*c^3*x+1/d/ln(F)/b*ln(1+b*F^c 
*F^(d*x)/a)*x^3+3/d^2/ln(F)^2/b*polylog(2,-b*F^c*F^(d*x)/a)*x^2-6/d^3/ln(F 
)^3/b*polylog(3,-b*F^c*F^(d*x)/a)*x+6/d^4/ln(F)^4/b*polylog(4,-b*F^c*F^(d* 
x)/a)-3/4/d^4/b*c^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.17 \[ \int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx=\frac {3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {F^{d x + c} b + a}{a} + 1\right ) \log \left (F\right )^{2} - c^{3} \log \left (F^{d x + c} b + a\right ) \log \left (F\right )^{3} + {\left (d^{3} x^{3} + c^{3}\right )} \log \left (F\right )^{3} \log \left (\frac {F^{d x + c} b + a}{a}\right ) - 6 \, d x \log \left (F\right ) {\rm polylog}\left (3, -\frac {F^{d x + c} b}{a}\right ) + 6 \, {\rm polylog}\left (4, -\frac {F^{d x + c} b}{a}\right )}{b d^{4} \log \left (F\right )^{4}} \] Input:

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

Output:

(3*d^2*x^2*dilog(-(F^(d*x + c)*b + a)/a + 1)*log(F)^2 - c^3*log(F^(d*x + c 
)*b + a)*log(F)^3 + (d^3*x^3 + c^3)*log(F)^3*log((F^(d*x + c)*b + a)/a) - 
6*d*x*log(F)*polylog(3, -F^(d*x + c)*b/a) + 6*polylog(4, -F^(d*x + c)*b/a) 
)/(b*d^4*log(F)^4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.92 \[ \int \frac {F^{c+d x} x^3}{a+b F^{c+d x}} \, dx=\frac {d^{3} x^{3} \log \left (\frac {F^{d x} F^{c} b}{a} + 1\right ) \log \left (F\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {F^{d x} F^{c} b}{a}\right ) \log \left (F\right )^{2} - 6 \, d x \log \left (F\right ) {\rm Li}_{3}(-\frac {F^{d x} F^{c} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{d x} F^{c} b}{a})}{b d^{4} \log \left (F\right )^{4}} \] Input:

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

Output:

(d^3*x^3*log(F^(d*x)*F^c*b/a + 1)*log(F)^3 + 3*d^2*x^2*dilog(-F^(d*x)*F^c* 
b/a)*log(F)^2 - 6*d*x*log(F)*polylog(3, -F^(d*x)*F^c*b/a) + 6*polylog(4, - 
F^(d*x)*F^c*b/a))/(b*d^4*log(F)^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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