\(\int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx\) [358]

Optimal result

Integrand size = 26, antiderivative size = 366 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\frac {d^2 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{2 h (d g-c h)^2}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{2 h (g+h x)^2}+\frac {d (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{2 (d g-c h)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{2 (d g-c h)^2 (g+h x)}+\frac {d (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^3}+\frac {(b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{2 (d g-c h)^4} \] Output:

1/2*d^2*F^(e+b*f/d-(-a*d+b*c)*f/d/(d*x+c))/h/(-c*h+d*g)^2-1/2*F^(e+f*(b*x+ 
a)/(d*x+c))/h/(h*x+g)^2+1/2*d*(-a*d+b*c)*f*F^(e+b*f/d-(-a*d+b*c)*f/d/(d*x+ 
c))*ln(F)/(-c*h+d*g)^3-1/2*(-a*d+b*c)*f*F^(e+f*(b*x+a)/(d*x+c))*ln(F)/(-c* 
h+d*g)^2/(h*x+g)+d*(-a*d+b*c)*f*F^(e+f*(-a*h+b*g)/(-c*h+d*g))*Ei(-(-a*d+b* 
c)*f*(h*x+g)*ln(F)/(-c*h+d*g)/(d*x+c))*ln(F)/(-c*h+d*g)^3+1/2*(-a*d+b*c)^2 
*f^2*F^(e+f*(-a*h+b*g)/(-c*h+d*g))*h*Ei(-(-a*d+b*c)*f*(h*x+g)*ln(F)/(-c*h+ 
d*g)/(d*x+c))*ln(F)^2/(-c*h+d*g)^4
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 6.20 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2662, 7293, 2009}

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.

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

\(\Big \downarrow \) 2662

\(\displaystyle \frac {f \log (F) (b c-a d) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)^2}dx}{2 h}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {f \log (F) (b c-a d) \left (\frac {d^2 F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{f \log (F) (b c-a d) (d g-c h)^2}+\frac {f h^2 \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^4}+\frac {2 d h F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^3}+\frac {d h F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{(d g-c h)^3}-\frac {h F^{\frac {f (a+b x)}{c+d x}+e}}{(g+h x) (d g-c h)^2}\right )}{2 h}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}\)

Input:

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

Output:

-1/2*F^(e + (f*(a + b*x))/(c + d*x))/(h*(g + h*x)^2) + ((b*c - a*d)*f*Log[ 
F]*((d*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*h)/(d*g - c*h)^3 - 
(F^(e + (f*(a + b*x))/(c + d*x))*h)/((d*g - c*h)^2*(g + h*x)) + (2*d*F^(e 
+ (f*(b*g - a*h))/(d*g - c*h))*h*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)* 
Log[F])/((d*g - c*h)*(c + d*x)))])/(d*g - c*h)^3 + (d^2*F^(e + (b*f)/d - ( 
(b*c - a*d)*f)/(d*(c + d*x))))/((b*c - a*d)*f*(d*g - c*h)^2*Log[F]) + ((b* 
c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g - c*h))*h^2*ExpIntegralEi[-(((b*c - 
 a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F])/(d*g - c*h)^4) 
)/(2*h)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2662

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2018\) vs. \(2(356)=712\).

Time = 0.66 (sec) , antiderivative size = 2019, normalized size of antiderivative = 5.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(2019\)

Input:

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

Output:

-1/2*ln(F)^2*d^2*h*f^2/(c*h-d*g)^4*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/ 
d)/(f*ln(F)/(d*x+c)*a-b*c*f*ln(F)/d/(d*x+c)+ln(F)/d*b*f+e*ln(F)-1/(c*h-d*g 
)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)* 
ln(F)*d*e*g)^2*a^2+ln(F)^2*d*h*f^2/(c*h-d*g)^4*F^(f*(a*d-b*c)/d/(d*x+c))*F 
^((b*f+d*e)/d)/(f*ln(F)/(d*x+c)*a-b*c*f*ln(F)/d/(d*x+c)+ln(F)/d*b*f+e*ln(F 
)-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+ 
1/(c*h-d*g)*ln(F)*d*e*g)^2*a*b*c-1/2*ln(F)^2*h*f^2/(c*h-d*g)^4*F^(f*(a*d-b 
*c)/d/(d*x+c))*F^((b*f+d*e)/d)/(f*ln(F)/(d*x+c)*a-b*c*f*ln(F)/d/(d*x+c)+ln 
(F)/d*b*f+e*ln(F)-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d 
*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g)^2*b^2*c^2-1/2*ln(F)^2*d^2*h*f^2/( 
c*h-d*g)^4*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/d)/(f*ln(F)/(d*x+c)*a-b* 
c*f*ln(F)/d/(d*x+c)+ln(F)/d*b*f+e*ln(F)-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g 
)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g)*a^2+ln(F)^2 
*d*h*f^2/(c*h-d*g)^4*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/d)/(f*ln(F)/(d 
*x+c)*a-b*c*f*ln(F)/d/(d*x+c)+ln(F)/d*b*f+e*ln(F)-1/(c*h-d*g)*ln(F)*a*f*h+ 
1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g)*a 
*b*c-1/2*ln(F)^2*h*f^2/(c*h-d*g)^4*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/ 
d)/(f*ln(F)/(d*x+c)*a-b*c*f*ln(F)/d/(d*x+c)+ln(F)/d*b*f+e*ln(F)-1/(c*h-d*g 
)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)* 
ln(F)*d*e*g)*b^2*c^2-1/2*ln(F)^2*d^2*h*f^2/(c*h-d*g)^4*F^((a*f*h-b*f*g+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (356) = 712\).

Time = 0.10 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.06 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\frac {{\left ({\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} h^{3} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} g h^{2} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} h\right )} \log \left (F\right )^{2} + 2 \, {\left ({\left (b c d^{2} - a d^{3}\right )} f g^{3} - {\left (b c^{2} d - a c d^{2}\right )} f g^{2} h + {\left ({\left (b c d^{2} - a d^{3}\right )} f g h^{2} - {\left (b c^{2} d - a c d^{2}\right )} f h^{3}\right )} x^{2} + 2 \, {\left ({\left (b c d^{2} - a d^{3}\right )} f g^{2} h - {\left (b c^{2} d - a c d^{2}\right )} f g h^{2}\right )} x\right )} \log \left (F\right )\right )} F^{\frac {{\left (d e + b f\right )} g - {\left (c e + a f\right )} h}{d g - c h}} {\rm Ei}\left (-\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} \log \left (F\right )}{c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x}\right ) + {\left (2 \, c d^{3} g^{3} - 5 \, c^{2} d^{2} g^{2} h + 4 \, c^{3} d g h^{2} - c^{4} h^{3} + {\left (d^{4} g^{2} h - 2 \, c d^{3} g h^{2} + c^{2} d^{2} h^{3}\right )} x^{2} + 2 \, {\left (d^{4} g^{3} - 2 \, c d^{3} g^{2} h + c^{2} d^{2} g h^{2}\right )} x + {\left ({\left (b c^{2} d - a c d^{2}\right )} f g^{2} h - {\left (b c^{3} - a c^{2} d\right )} f g h^{2} + {\left ({\left (b c d^{2} - a d^{3}\right )} f g h^{2} - {\left (b c^{2} d - a c d^{2}\right )} f h^{3}\right )} x^{2} + {\left ({\left (b c d^{2} - a d^{3}\right )} f g^{2} h - {\left (b c^{3} - a c^{2} d\right )} f h^{3}\right )} x\right )} \log \left (F\right )\right )} F^{\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}}}{2 \, {\left (d^{4} g^{6} - 4 \, c d^{3} g^{5} h + 6 \, c^{2} d^{2} g^{4} h^{2} - 4 \, c^{3} d g^{3} h^{3} + c^{4} g^{2} h^{4} + {\left (d^{4} g^{4} h^{2} - 4 \, c d^{3} g^{3} h^{3} + 6 \, c^{2} d^{2} g^{2} h^{4} - 4 \, c^{3} d g h^{5} + c^{4} h^{6}\right )} x^{2} + 2 \, {\left (d^{4} g^{5} h - 4 \, c d^{3} g^{4} h^{2} + 6 \, c^{2} d^{2} g^{3} h^{3} - 4 \, c^{3} d g^{2} h^{4} + c^{4} g h^{5}\right )} x\right )}} \] Input:

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

Output:

1/2*((((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*h^3*x^2 + 2*(b^2*c^2 - 2*a*b*c* 
d + a^2*d^2)*f^2*g*h^2*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*g^2*h)*log( 
F)^2 + 2*((b*c*d^2 - a*d^3)*f*g^3 - (b*c^2*d - a*c*d^2)*f*g^2*h + ((b*c*d^ 
2 - a*d^3)*f*g*h^2 - (b*c^2*d - a*c*d^2)*f*h^3)*x^2 + 2*((b*c*d^2 - a*d^3) 
*f*g^2*h - (b*c^2*d - a*c*d^2)*f*g*h^2)*x)*log(F))*F^(((d*e + b*f)*g - (c* 
e + a*f)*h)/(d*g - c*h))*Ei(-((b*c - a*d)*f*h*x + (b*c - a*d)*f*g)*log(F)/ 
(c*d*g - c^2*h + (d^2*g - c*d*h)*x)) + (2*c*d^3*g^3 - 5*c^2*d^2*g^2*h + 4* 
c^3*d*g*h^2 - c^4*h^3 + (d^4*g^2*h - 2*c*d^3*g*h^2 + c^2*d^2*h^3)*x^2 + 2* 
(d^4*g^3 - 2*c*d^3*g^2*h + c^2*d^2*g*h^2)*x + ((b*c^2*d - a*c*d^2)*f*g^2*h 
 - (b*c^3 - a*c^2*d)*f*g*h^2 + ((b*c*d^2 - a*d^3)*f*g*h^2 - (b*c^2*d - a*c 
*d^2)*f*h^3)*x^2 + ((b*c*d^2 - a*d^3)*f*g^2*h - (b*c^3 - a*c^2*d)*f*h^3)*x 
)*log(F))*F^((c*e + a*f + (d*e + b*f)*x)/(d*x + c)))/(d^4*g^6 - 4*c*d^3*g^ 
5*h + 6*c^2*d^2*g^4*h^2 - 4*c^3*d*g^3*h^3 + c^4*g^2*h^4 + (d^4*g^4*h^2 - 4 
*c*d^3*g^3*h^3 + 6*c^2*d^2*g^2*h^4 - 4*c^3*d*g*h^5 + c^4*h^6)*x^2 + 2*(d^4 
*g^5*h - 4*c*d^3*g^4*h^2 + 6*c^2*d^2*g^3*h^3 - 4*c^3*d*g^2*h^4 + c^4*g*h^5 
)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\text {Timed out} \] Input:

integrate(F**(e+f*(b*x+a)/(d*x+c))/(h*x+g)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3, x)
 

Giac [F]

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

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

Output:

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3, x)
 

Mupad [F(-1)]

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

int(F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^3,x)
 

Output:

int(F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^3, x)
 

Reduce [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\text {too large to display} \] Input:

int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^3,x)
                                                                                    
                                                                                    
 

Output:

( - 4*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*c**3*d**4* 
f**3*h**4 + 6*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*c* 
*2*d**5*f**3*g*h**3 - 6*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)* 
*3*a**3*c**2*d**5*f**3*h**4*x - 4*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x 
))*log(f)**3*a**3*c*d**6*f**3*g**2*h**2 + 4*f**((a*f + b*f*x + c*e + d*e*x 
)/(c + d*x))*log(f)**3*a**3*c*d**6*f**3*g*h**3*x - 4*f**((a*f + b*f*x + c* 
e + d*e*x)/(c + d*x))*log(f)**3*a**3*c*d**6*f**3*h**4*x**2 + f**((a*f + b* 
f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*d**7*f**3*g**3*h - f**((a*f + 
 b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*d**7*f**3*g**2*h**2*x + f* 
*((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*d**7*f**3*g*h**3*x 
**2 - f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**3*d**7*f**3* 
h**4*x**3 + 12*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**2*b 
*c**4*d**3*f**3*h**4 - 18*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f 
)**3*a**2*b*c**3*d**4*f**3*g*h**3 + 18*f**((a*f + b*f*x + c*e + d*e*x)/(c 
+ d*x))*log(f)**3*a**2*b*c**3*d**4*f**3*h**4*x + 12*f**((a*f + b*f*x + c*e 
 + d*e*x)/(c + d*x))*log(f)**3*a**2*b*c**2*d**5*f**3*g**2*h**2 - 12*f**((a 
*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**2*b*c**2*d**5*f**3*g*h** 
3*x + 12*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)**3*a**2*b*c**2* 
d**5*f**3*h**4*x**2 - 3*f**((a*f + b*f*x + c*e + d*e*x)/(c + d*x))*log(f)* 
*3*a**2*b*c*d**6*f**3*g**3*h + 3*f**((a*f + b*f*x + c*e + d*e*x)/(c + d...