\(\int \frac {g+h x}{(a+b \log (c (d (e+f x)^p)^q))^3} \, dx\) [475]

Optimal result

Integrand size = 26, antiderivative size = 322 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}+\frac {2 e^{-\frac {2 a}{b p q}} h (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^2 p^3 q^3}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=-\frac {e^{-\frac {2 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (-e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-4 h (e+f x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b e^{\frac {2 a}{b p q}} p q \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \left (b f p q (g+h x)+a (f g+e h+2 f h x)+b (e h+f (g+2 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{2 b^3 f^2 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 3.97 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.65, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2895, 2847, 2836, 2734, 2737, 2609, 2847, 2836, 2737, 2609, 2846, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2847

\(\displaystyle -\frac {(f g-e h) \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{2 b f p q}+\frac {\int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2836

\(\displaystyle -\frac {(f g-e h) \int \frac {1}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}d(e+f x)}{2 b f^2 p q}+\frac {\int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2734

\(\displaystyle -\frac {(f g-e h) \left (\frac {\int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d(e+f x)}{b p q}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}+\frac {\int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2737

\(\displaystyle -\frac {(f g-e h) \left (\frac {(e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d\log \left (c d^q (e+f x)^{p q}\right )}{b p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}+\frac {\int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2847

\(\displaystyle \frac {-\frac {(f g-e h) \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2836

\(\displaystyle \frac {-\frac {(f g-e h) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d(e+f x)}{b f^2 p q}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2737

\(\displaystyle \frac {-\frac {(e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d\log \left (c d^q (e+f x)^{p q}\right )}{b f^2 p^2 q^2}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2846

\(\displaystyle \frac {\frac {2 \int \left (\frac {f g-e h}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )dx}{b p q}-\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(f g-e h) \left (\frac {(e+f x) e^{-\frac {a}{b p q}} \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 p^2 q^2}-\frac {e+f x}{b p q \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right )}{2 b f^2 p q}+\frac {-\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}+\frac {2 \left (\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b f^2 p q}+\frac {h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b f^2 p q}\right )}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}}{b p q}-\frac {(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\)

Input:

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

Output:

-1/2*((f*g - e*h)*(((e + f*x)*ExpIntegralEi[(a + b*Log[c*d^q*(e + f*x)^(p* 
q)])/(b*p*q)])/(b^2*E^(a/(b*p*q))*p^2*q^2*(c*d^q*(e + f*x)^(p*q))^(1/(p*q) 
)) - (e + f*x)/(b*p*q*(a + b*Log[c*d^q*(e + f*x)^(p*q)]))))/(b*f^2*p*q) - 
((e + f*x)*(g + h*x))/(2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) + (-( 
((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*d^q*(e + f*x)^(p*q)])/(b 
*p*q)])/(b^2*E^(a/(b*p*q))*f^2*p^2*q^2*(c*d^q*(e + f*x)^(p*q))^(1/(p*q)))) 
 + (2*(((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q 
])/(b*p*q)])/(b*E^(a/(b*p*q))*f^2*p*q*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + ( 
h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)]) 
/(b*E^((2*a)/(b*p*q))*f^2*p*q*(c*(d*(e + f*x)^p)^q)^(2/(p*q)))))/(b*p*q) - 
 ((e + f*x)*(g + h*x))/(b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])))/(b*p*q)
 

Defintions of rubi rules used

rule 2009

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

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2734

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b 
*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[(a + b 
*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int 
egerQ[2*p]
 

rule 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x 
^n)^(1/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ 
[{a, b, c, n, p}, x]
 

rule 2836

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

rule 2846

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) 
]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* 
x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & 
& IGtQ[q, 0]
 

rule 2847

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e 
*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[( 
f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) 
/(b*e*n*(p + 1)))   Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt 
Q[p, -1] && GtQ[q, 0]
 

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

Maple [F]

\[\int \frac {h x +g}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 931 vs. \(2 (317) = 634\).

Time = 0.14 (sec) , antiderivative size = 931, normalized size of antiderivative = 2.89 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(((b^2*f*g - b^2*e*h)*p^2*q^2*log(f*x + e)^2 + (b^2*f*g - b^2*e*h)*q^2 
*log(d)^2 + a^2*f*g - a^2*e*h + (b^2*f*g - b^2*e*h)*log(c)^2 + 2*((b^2*f*g 
 - b^2*e*h)*p*q^2*log(d) + (b^2*f*g - b^2*e*h)*p*q*log(c) + (a*b*f*g - a*b 
*e*h)*p*q)*log(f*x + e) + 2*(a*b*f*g - a*b*e*h)*log(c) + 2*((b^2*f*g - b^2 
*e*h)*q*log(c) + (a*b*f*g - a*b*e*h)*q)*log(d))*e^((b*q*log(d) + b*log(c) 
+ a)/(b*p*q))*log_integral((f*x + e)*e^((b*q*log(d) + b*log(c) + a)/(b*p*q 
))) - (b^2*e*f*g*p^2*q^2 + (a*b*e*f*g + a*b*e^2*h)*p*q + (b^2*f^2*h*p^2*q^ 
2 + 2*a*b*f^2*h*p*q)*x^2 + ((b^2*f^2*g + b^2*e*f*h)*p^2*q^2 + (a*b*f^2*g + 
 3*a*b*e*f*h)*p*q)*x + (2*b^2*f^2*h*p^2*q^2*x^2 + (b^2*f^2*g + 3*b^2*e*f*h 
)*p^2*q^2*x + (b^2*e*f*g + b^2*e^2*h)*p^2*q^2)*log(f*x + e) + (2*b^2*f^2*h 
*p*q*x^2 + (b^2*f^2*g + 3*b^2*e*f*h)*p*q*x + (b^2*e*f*g + b^2*e^2*h)*p*q)* 
log(c) + (2*b^2*f^2*h*p*q^2*x^2 + (b^2*f^2*g + 3*b^2*e*f*h)*p*q^2*x + (b^2 
*e*f*g + b^2*e^2*h)*p*q^2)*log(d))*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q 
)) + 4*(b^2*h*p^2*q^2*log(f*x + e)^2 + b^2*h*q^2*log(d)^2 + b^2*h*log(c)^2 
 + 2*a*b*h*log(c) + a^2*h + 2*(b^2*h*p*q^2*log(d) + b^2*h*p*q*log(c) + a*b 
*h*p*q)*log(f*x + e) + 2*(b^2*h*q*log(c) + a*b*h*q)*log(d))*log_integral(( 
f^2*x^2 + 2*e*f*x + e^2)*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q))))*e^(-2 
*(b*q*log(d) + b*log(c) + a)/(b*p*q))/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + b^ 
5*f^2*p^3*q^5*log(d)^2 + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^3*lo 
g(c) + a^2*b^3*f^2*p^3*q^3 + 2*(b^5*f^2*p^4*q^5*log(d) + b^5*f^2*p^4*q^...
 

Sympy [F]

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

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

Output:

Integral((g + h*x)/(a + b*log(c*(d*(e + f*x)**p)**q))**3, x)
 

Maxima [F]

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

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

Output:

-1/2*((2*a*f^2*h + (f^2*h*p*q + 2*f^2*h*q*log(d) + 2*f^2*h*log(c))*b)*x^2 
+ (e*f*g + e^2*h)*a + (e*f*g*p*q + (e*f*g + e^2*h)*log(c) + (e*f*g*q + e^2 
*h*q)*log(d))*b + ((f^2*g + 3*e*f*h)*a + (f^2*g*p*q + e*f*h*p*q + (f^2*g + 
 3*e*f*h)*log(c) + (f^2*g*q + 3*e*f*h*q)*log(d))*b)*x + (2*b*f^2*h*x^2 + ( 
f^2*g + 3*e*f*h)*b*x + (e*f*g + e^2*h)*b)*log(((f*x + e)^p)^q))/(b^4*f^2*p 
^2*q^2*log(((f*x + e)^p)^q)^2 + a^2*b^2*f^2*p^2*q^2 + 2*(f^2*p^2*q^3*log(d 
) + f^2*p^2*q^2*log(c))*a*b^3 + (f^2*p^2*q^4*log(d)^2 + 2*f^2*p^2*q^3*log( 
c)*log(d) + f^2*p^2*q^2*log(c)^2)*b^4 + 2*(a*b^3*f^2*p^2*q^2 + (f^2*p^2*q^ 
3*log(d) + f^2*p^2*q^2*log(c))*b^4)*log(((f*x + e)^p)^q)) + integrate(1/2* 
(4*f*h*x + f*g + 3*e*h)/(b^3*f*p^2*q^2*log(((f*x + e)^p)^q) + a*b^2*f*p^2* 
q^2 + (f*p^2*q^3*log(d) + f*p^2*q^2*log(c))*b^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11278 vs. \(2 (317) = 634\).

Time = 0.31 (sec) , antiderivative size = 11278, normalized size of antiderivative = 35.02 \[ \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(x**2/(log(d**q*(e + f*x)**(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)** 
(p*q)*c)**3*b**3*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log( 
d**q*(e + f*x)**(p*q)*c)**2*a*b**2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a* 
*2*b*e + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x) 
*log(d**q*(e + f*x)**(p*q)*c)**2*b**3*f**2*h*p*q + 4*int(x**2/(log(d**q*(e 
 + f*x)**(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3 
*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c) 
**2*a*b**2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + 
 f*x)**(p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x)*log(d**q*(e + f*x)**(p* 
q)*c)*a*b**2*f**2*h*p*q + 2*int(x**2/(log(d**q*(e + f*x)**(p*q)*c)**3*b**3 
*e + log(d**q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3*log(d**q*(e + f*x)**(p*q 
)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*f*x + 3*log(d* 
*q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*f* 
x + a**3*e + a**3*f*x),x)*a**2*b*f**2*h*p*q + 2*int(x/(log(d**q*(e + f*x)* 
*(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3*log(d** 
q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b* 
*2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + f*x)**( 
p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x)*log(d**q*(e + f*x)**(p*q)*c)**2 
*b**3*e*f*h*p*q + 2*int(x/(log(d**q*(e + f*x)**(p*q)*c)**3*b**3*e + log(d* 
*q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)**2*...