\(\int (g+h x)^m (a+b \log (c (d (e+f x)^p)^q)) \, dx\) [525]

Optimal result

Integrand size = 26, antiderivative size = 99 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {b f p q (g+h x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h) (1+m) (2+m)}+\frac {(g+h x)^{1+m} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)} \] Output:

b*f*p*q*(h*x+g)^(2+m)*hypergeom([1, 2+m],[3+m],f*(h*x+g)/(-e*h+f*g))/h/(-e 
*h+f*g)/(1+m)/(2+m)+(h*x+g)^(1+m)*(a+b*ln(c*(d*(f*x+e)^p)^q))/h/(1+m)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {(g+h x)^{1+m} \left (a+\frac {b f p q (g+h x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {f (g+h x)}{f g-e h}\right )}{(f g-e h) (2+m)}+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)} \] Input:

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

Output:

((g + h*x)^(1 + m)*(a + (b*f*p*q*(g + h*x)*Hypergeometric2F1[1, 2 + m, 3 + 
 m, (f*(g + h*x))/(f*g - e*h)])/((f*g - e*h)*(2 + m)) + b*Log[c*(d*(e + f* 
x)^p)^q]))/(h*(1 + m))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2895, 2842, 78}

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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2842

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

\(\Big \downarrow \) 78

\(\displaystyle \frac {(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (m+1)}+\frac {b f p q (g+h x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {f (g+h x)}{f g-e h}\right )}{h (m+1) (m+2) (f g-e h)}\)

Input:

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

Output:

(b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x) 
)/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + ((g + h*x)^(1 + m)*(a + 
b*Log[c*(d*(e + f*x)^p)^q]))/(h*(1 + m))
 

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 2842

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((h*x + g)^m*b*log(((f*x + e)^p*d)^q*c) + (h*x + g)^m*a, x)
 

Sympy [F(-2)]

Exception generated. \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate((b*log(((f*x + e)^p*d)^q*c) + a)*(h*x + g)^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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