Integrand size = 26, antiderivative size = 283 \[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {e^{-\frac {3 a}{b n}} g^2 \sqrt {\frac {\pi }{3}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}} \]
1/3*g^2*(e*x+d)^3*erfi(3^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2) )*3^(1/2)*Pi^(1/2)/e^3/exp(3*a/b/n)/((c*(e*x+d)^n)^(3/n))/b^(1/2)/n^(1/2)+ (-d*g+e*f)^2*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^ (1/2)/e^3/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))/b^(1/2)/n^(1/2)+g*(-d*g+e*f)*(e *x+d)^2*erfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)* Pi^(1/2)/e^3/exp(2*a/b/n)/((c*(e*x+d)^n)^(2/n))/b^(1/2)/n^(1/2)
Time = 0.18 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {3 a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (3 e^{\frac {2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+3 \sqrt {2} e^{\frac {a}{b n}} g (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {3} g^2 (d+e x)^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{3 \sqrt {b} e^3 \sqrt {n}} \]
(Sqrt[Pi]*(d + e*x)*(3*E^((2*a)/(b*n))*(e*f - d*g)^2*(c*(d + e*x)^n)^(2/n) *Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])] + 3*Sqrt[2]*E^(a/( b*n))*g*(e*f - d*g)*(d + e*x)*(c*(d + e*x)^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])] + Sqrt[3]*g^2*(d + e*x)^2*Erfi [(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]))/(3*Sqrt[b]* e^3*E^((3*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(3/n))
Time = 0.83 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {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.
\(\displaystyle \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx\) |
\(\Big \downarrow \) 2848 |
\(\displaystyle \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 g (d+e x) (e f-d g)}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^2 (d+e x)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}+\frac {\sqrt {\frac {\pi }{3}} g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^3 \sqrt {n}}\) |
((e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqr t[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(2/n)) + (g^2*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Lo g[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((3*a)/(b*n))*Sqrt[n ]*(c*(d + e*x)^n)^(3/n))
3.2.24.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (g x +f \right )^{2}}{\sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}d x\]
Exception generated. \[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {\left (f + g x\right )^{2}}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \]
\[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
\[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]