\(\int (g+h x) \sqrt {a+b \log (c (d (e+f x)^p)^q)} \, dx\) [480]

Optimal result

Integrand size = 28, antiderivative size = 311 \[ \int (g+h x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b p q}} (f g-e h) \sqrt {p} \sqrt {\pi } \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b p q}} h \sqrt {p} \sqrt {\frac {\pi }{2}} \sqrt {q} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {(f g-e h) (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2} \] Output:

-1/2*b^(1/2)*(-e*h+f*g)*p^(1/2)*Pi^(1/2)*q^(1/2)*(f*x+e)*erfi((a+b*ln(c*(d 
*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))/exp(a/b/p/q)/f^2/((c*(d*(f* 
x+e)^p)^q)^(1/p/q))-1/8*b^(1/2)*h*p^(1/2)*2^(1/2)*Pi^(1/2)*q^(1/2)*(f*x+e) 
^2*erfi(2^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2)) 
/exp(2*a/b/p/q)/f^2/((c*(d*(f*x+e)^p)^q)^(2/p/q))+(-e*h+f*g)*(f*x+e)*(a+b* 
ln(c*(d*(f*x+e)^p)^q))^(1/2)/f^2+1/2*h*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q 
))^(1/2)/f^2
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.96 \[ \int (g+h x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=-\frac {e^{-\frac {2 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (4 \sqrt {b} e^{\frac {a}{b p q}} (f g-e h) \sqrt {p} \sqrt {\pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )+\sqrt {b} h \sqrt {p} \sqrt {2 \pi } \sqrt {q} (e+f x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-4 e^{\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} (2 f g-e h+f h x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )}{8 f^2} \] Input:

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

Output:

-1/8*((e + f*x)*(4*Sqrt[b]*E^(a/(b*p*q))*(f*g - e*h)*Sqrt[p]*Sqrt[Pi]*Sqrt 
[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^ 
q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])] + Sqrt[b]*h*Sqrt[p]*Sqrt[2*Pi]*Sqrt[q]*(e + 
 f*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p] 
*Sqrt[q])] - 4*E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(2*f*g - 
e*h + f*h*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]))/(E^((2*a)/(b*p*q))*f^2 
*(c*(d*(e + f*x)^p)^q)^(2/(p*q)))
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2895, 2848, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\pi } \sqrt {b} \sqrt {p} \sqrt {q} (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{2 f^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} h \sqrt {p} \sqrt {q} (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {(e+f x) (f g-e h) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac {h (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2848

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

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

Input:

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2),x)
 

Output:

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2),x)
 

Fricas [F(-2)]

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

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="fricas" 
)
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

integrate((h*x+g)*(a+b*ln(c*(d*(f*x+e)**p)**q))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="maxima" 
)
 

Output:

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

Giac [F]

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

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c 
)*b*e**2*h - 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a*e**2*h + 24*sqrt 
(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a*f**2*g*x + 12*sqrt(log(d**q*(e + f* 
x)**(p*q)*c)*b + a)*a*f**2*h*x**2 + 6*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b 
+ a)*b*e*f*h*p*q*x + 6*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*b*f**2*g*p 
*q*x + 24*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x) 
**(p*q)*c)*x**2)/(4*log(d**q*(e + f*x)**(p*q)*c)*a*b*e + 4*log(d**q*(e + f 
*x)**(p*q)*c)*a*b*f*x + log(d**q*(e + f*x)**(p*q)*c)*b**2*e*p*q + log(d**q 
*(e + f*x)**(p*q)*c)*b**2*f*p*q*x + 4*a**2*e + 4*a**2*f*x + a*b*e*p*q + a* 
b*f*p*q*x),x)*a*b**2*f**3*h*p*q + 6*int((sqrt(log(d**q*(e + f*x)**(p*q)*c) 
*b + a)*log(d**q*(e + f*x)**(p*q)*c)*x**2)/(4*log(d**q*(e + f*x)**(p*q)*c) 
*a*b*e + 4*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*x + log(d**q*(e + f*x)**(p*q 
)*c)*b**2*e*p*q + log(d**q*(e + f*x)**(p*q)*c)*b**2*f*p*q*x + 4*a**2*e + 4 
*a**2*f*x + a*b*e*p*q + a*b*f*p*q*x),x)*b**3*f**3*h*p**2*q**2 - 48*int((sq 
rt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x)/(4*log(d**q*(e + f*x)**(p*q)*c)* 
a*b*e + 4*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*x + log(d**q*(e + f*x)**(p*q) 
*c)*b**2*e*p*q + log(d**q*(e + f*x)**(p*q)*c)*b**2*f*p*q*x + 4*a**2*e + 4* 
a**2*f*x + a*b*e*p*q + a*b*f*p*q*x),x)*a**2*b*f**3*g*p*q - 12*int((sqrt(lo 
g(d**q*(e + f*x)**(p*q)*c)*b + a)*x)/(4*log(d**q*(e + f*x)**(p*q)*c)*a*b*e 
 + 4*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*x + log(d**q*(e + f*x)**(p*q)*c...