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