Integrand size = 22, antiderivative size = 298 \[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=-\frac {1}{2} \sqrt {b} d^2 e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{2} \sqrt {b} d e e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{6} \sqrt {b} e^2 e^{-\frac {3 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d^2 x \sqrt {a+b \log \left (c x^n\right )}+d e x^2 \sqrt {a+b \log \left (c x^n\right )}+\frac {1}{3} e^2 x^3 \sqrt {a+b \log \left (c x^n\right )} \]
-1/18*e^2*x^3*erfi(3^(1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)* n^(1/2)*3^(1/2)*Pi^(1/2)/exp(3*a/b/n)/((c*x^n)^(3/n))-1/4*d*e*x^2*erfi(2^( 1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*2^(1/2)*Pi^(1/ 2)/exp(2*a/b/n)/((c*x^n)^(2/n))-1/2*d^2*x*erfi((a+b*ln(c*x^n))^(1/2)/b^(1/ 2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi^(1/2)/exp(a/b/n)/((c*x^n)^(1/n))+d^2*x*(a+b *ln(c*x^n))^(1/2)+d*e*x^2*(a+b*ln(c*x^n))^(1/2)+1/3*e^2*x^3*(a+b*ln(c*x^n) )^(1/2)
Time = 0.26 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.96 \[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\frac {1}{36} x \left (-18 \sqrt {b} d^2 e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-9 \sqrt {b} d e e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {2 \pi } x \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-2 \sqrt {b} e^2 e^{-\frac {3 a}{b n}} \sqrt {n} \sqrt {3 \pi } x^2 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+36 d^2 \sqrt {a+b \log \left (c x^n\right )}+36 d e x \sqrt {a+b \log \left (c x^n\right )}+12 e^2 x^2 \sqrt {a+b \log \left (c x^n\right )}\right ) \]
(x*((-18*Sqrt[b]*d^2*Sqrt[n]*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b] *Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) - (9*Sqrt[b]*d*e*Sqrt[n]*Sqrt[2*P i]*x*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/( b*n))*(c*x^n)^(2/n)) - (2*Sqrt[b]*e^2*Sqrt[n]*Sqrt[3*Pi]*x^2*Erfi[(Sqrt[3] *Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*x^n)^(3/n )) + 36*d^2*Sqrt[a + b*Log[c*x^n]] + 36*d*e*x*Sqrt[a + b*Log[c*x^n]] + 12* e^2*x^2*Sqrt[a + b*Log[c*x^n]]))/36
Time = 0.71 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2767, 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.
\(\displaystyle \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx\) |
\(\Big \downarrow \) 2767 |
\(\displaystyle \int \left (d^2 \sqrt {a+b \log \left (c x^n\right )}+2 d e x \sqrt {a+b \log \left (c x^n\right )}+e^2 x^2 \sqrt {a+b \log \left (c x^n\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \sqrt {\pi } \sqrt {b} d^2 \sqrt {n} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d^2 x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} d e \sqrt {n} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d e x^2 \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{6} \sqrt {\frac {\pi }{3}} \sqrt {b} e^2 \sqrt {n} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\frac {1}{3} e^2 x^3 \sqrt {a+b \log \left (c x^n\right )}\) |
-1/2*(Sqrt[b]*d^2*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]* Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) - (Sqrt[b]*d*e*Sqrt[n]*Sqrt[Pi/2]* x^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(2*E^((2*a)/ (b*n))*(c*x^n)^(2/n)) - (Sqrt[b]*e^2*Sqrt[n]*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]* Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/ n)) + d^2*x*Sqrt[a + b*Log[c*x^n]] + d*e*x^2*Sqrt[a + b*Log[c*x^n]] + (e^2 *x^3*Sqrt[a + b*Log[c*x^n]])/3
3.2.25.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x ^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
\[\int \left (e x +d \right )^{2} \sqrt {a +b \ln \left (c \,x^{n}\right )}d x\]
Exception generated. \[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\int \sqrt {a + b \log {\left (c x^{n} \right )}} \left (d + e x\right )^{2}\, dx \]
\[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\int { {\left (e x + d\right )}^{2} \sqrt {b \log \left (c x^{n}\right ) + a} \,d x } \]
\[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\int { {\left (e x + d\right )}^{2} \sqrt {b \log \left (c x^{n}\right ) + a} \,d x } \]
Timed out. \[ \int (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )} \, dx=\int \sqrt {a+b\,\ln \left (c\,x^n\right )}\,{\left (d+e\,x\right )}^2 \,d x \]