Integrand size = 28, antiderivative size = 279 \[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b f^3 p q}+\frac {2 e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b f^3 p q}+\frac {e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b f^3 p q} \]
(-e*h+f*g)^2*(f*x+e)*Ei((a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b/exp(a/b/p/q)/ f^3/p/q/((c*(d*(f*x+e)^p)^q)^(1/p/q))+2*h*(-e*h+f*g)*(f*x+e)^2*Ei(2*(a+b*l n(c*(d*(f*x+e)^p)^q))/b/p/q)/b/exp(2*a/b/p/q)/f^3/p/q/((c*(d*(f*x+e)^p)^q) ^(2/p/q))+h^2*(f*x+e)^3*Ei(3*(a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b/exp(3*a/ b/p/q)/f^3/p/q/((c*(d*(f*x+e)^p)^q)^(3/p/q))
Time = 0.55 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.90 \[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {e^{-\frac {3 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (e^{\frac {2 a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )-h (e+f x) \left (-2 e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )-h (e+f x) \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )\right )\right )}{b f^3 p q} \]
((e + f*x)*(E^((2*a)/(b*p*q))*(f*g - e*h)^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q) )*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)] - h*(e + f*x)*(- 2*E^(a/(b*p*q))*(f*g - e*h)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*ExpIntegralEi[ (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)] - h*(e + f*x)*ExpIntegralEi[ (3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])))/(b*E^((3*a)/(b*p*q))*f^3* p*q*(c*(d*(e + f*x)^p)^q)^(3/(p*q)))
Time = 1.10 (sec) , antiderivative size = 279, 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, 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.
| \(\displaystyle \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx\) |
| \(\Big \downarrow \) 2846 |
| \(\displaystyle \int \left (\frac {(f g-e h)^2}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {2 h (e+f x) (f g-e h)}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {h^2 (e+f x)^2}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 h (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b f^3 p q}+\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b f^3 p q}+\frac {h^2 (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b f^3 p q}\) |
((f*g - e*h)^2*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b *p*q)])/(b*E^(a/(b*p*q))*f^3*p*q*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*h*( f*g - e*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^3*p*q*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (h^2*(e + f*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q )])/(b*E^((3*a)/(b*p*q))*f^3*p*q*(c*(d*(e + f*x)^p)^q)^(3/(p*q)))
3.5.45.3.1 Defintions of rubi rules used
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]
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]]
\[\int \frac {\left (h x +g \right )^{2}}{a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}d x\]
Time = 0.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.87 \[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {{\left (h^{2} \operatorname {log\_integral}\left ({\left (f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}\right )} e^{\left (\frac {3 \, {\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}\right ) + 2 \, {\left (f g h - e h^{2}\right )} e^{\left (\frac {b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \operatorname {log\_integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} e^{\left (\frac {2 \, {\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}\right ) + {\left (f^{2} g^{2} - 2 \, e f g h + e^{2} h^{2}\right )} e^{\left (\frac {2 \, {\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )} \operatorname {log\_integral}\left ({\left (f x + e\right )} e^{\left (\frac {b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )}\right )\right )} e^{\left (-\frac {3 \, {\left (b q \log \left (d\right ) + b \log \left (c\right ) + a\right )}}{b p q}\right )}}{b f^{3} p q} \]
(h^2*log_integral((f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3)*e^(3*(b*q*log( d) + b*log(c) + a)/(b*p*q))) + 2*(f*g*h - e*h^2)*e^((b*q*log(d) + b*log(c) + a)/(b*p*q))*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))) + (f^2*g^2 - 2*e*f*g*h + e^2*h^2)*e^(2*(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))))*e^(-3*(b*q*log(d) + b*log(c) + a)/(b*p*q))/(b*f^3*p*q)
\[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {\left (g + h x\right )^{2}}{a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}\, dx \]
\[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \frac {{\left (h x + g\right )}^{2}}{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.85 \[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\frac {g^{2} {\rm Ei}\left (\frac {\log \left (d\right )}{p} + \frac {\log \left (c\right )}{p q} + \frac {a}{b p q} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b p q}\right )}}{b c^{\frac {1}{p q}} d^{\left (\frac {1}{p}\right )} f p q} - \frac {2 \, e g h {\rm Ei}\left (\frac {\log \left (d\right )}{p} + \frac {\log \left (c\right )}{p q} + \frac {a}{b p q} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b p q}\right )}}{b c^{\frac {1}{p q}} d^{\left (\frac {1}{p}\right )} f^{2} p q} + \frac {e^{2} h^{2} {\rm Ei}\left (\frac {\log \left (d\right )}{p} + \frac {\log \left (c\right )}{p q} + \frac {a}{b p q} + \log \left (f x + e\right )\right ) e^{\left (-\frac {a}{b p q}\right )}}{b c^{\frac {1}{p q}} d^{\left (\frac {1}{p}\right )} f^{3} p q} + \frac {2 \, g h {\rm Ei}\left (\frac {2 \, \log \left (d\right )}{p} + \frac {2 \, \log \left (c\right )}{p q} + \frac {2 \, a}{b p q} + 2 \, \log \left (f x + e\right )\right ) e^{\left (-\frac {2 \, a}{b p q}\right )}}{b c^{\frac {2}{p q}} d^{\frac {2}{p}} f^{2} p q} - \frac {2 \, e h^{2} {\rm Ei}\left (\frac {2 \, \log \left (d\right )}{p} + \frac {2 \, \log \left (c\right )}{p q} + \frac {2 \, a}{b p q} + 2 \, \log \left (f x + e\right )\right ) e^{\left (-\frac {2 \, a}{b p q}\right )}}{b c^{\frac {2}{p q}} d^{\frac {2}{p}} f^{3} p q} + \frac {h^{2} {\rm Ei}\left (\frac {3 \, \log \left (d\right )}{p} + \frac {3 \, \log \left (c\right )}{p q} + \frac {3 \, a}{b p q} + 3 \, \log \left (f x + e\right )\right ) e^{\left (-\frac {3 \, a}{b p q}\right )}}{b c^{\frac {3}{p q}} d^{\frac {3}{p}} f^{3} p q} \]
g^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))/ (b*c^(1/(p*q))*d^(1/p)*f*p*q) - 2*e*g*h*Ei(log(d)/p + log(c)/(p*q) + a/(b* p*q) + log(f*x + e))*e^(-a/(b*p*q))/(b*c^(1/(p*q))*d^(1/p)*f^2*p*q) + e^2* h^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))/ (b*c^(1/(p*q))*d^(1/p)*f^3*p*q) + 2*g*h*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2 *a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))/(b*c^(2/(p*q))*d^(2/p)*f^2*p *q) - 2*e*h^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e ))*e^(-2*a/(b*p*q))/(b*c^(2/(p*q))*d^(2/p)*f^3*p*q) + h^2*Ei(3*log(d)/p + 3*log(c)/(p*q) + 3*a/(b*p*q) + 3*log(f*x + e))*e^(-3*a/(b*p*q))/(b*c^(3/(p *q))*d^(3/p)*f^3*p*q)
Timed out. \[ \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \frac {{\left (g+h\,x\right )}^2}{a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \]