Integrand size = 24, antiderivative size = 413 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=-\frac {15 b^{5/2} e^{-\frac {a}{b n}} (e f-d g) n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b n}} g n^{5/2} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{64 e^2}+\frac {15 b^2 (e f-d g) n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac {15 b^2 g n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac {5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac {5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2} \]
-5/2*b*(-d*g+e*f)*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(3/2)/e^2-5/8*b*g*n*(e*x +d)^2*(a+b*ln(c*(e*x+d)^n))^(3/2)/e^2+(-d*g+e*f)*(e*x+d)*(a+b*ln(c*(e*x+d) ^n))^(5/2)/e^2+1/2*g*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^(5/2)/e^2-15/128*b^(5 /2)*g*n^(5/2)*(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^2/exp(2*a/b/n)/((c*(e*x+d)^n)^(2/n))-15/8*b^(5/ 2)*(-d*g+e*f)*n^(5/2)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^( 1/2))*Pi^(1/2)/e^2/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))+15/4*b^2*(-d*g+e*f)*n^ 2*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(1/2)/e^2+15/32*b^2*g*n^2*(e*x+d)^2*(a+b*l n(c*(e*x+d)^n))^(1/2)/e^2
Time = 0.50 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.79 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\frac {(d+e x) \left (128 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}+64 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-80 b (e f-d g) n \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )-5 b g n (d+e x) \left (3 b^{3/2} e^{-\frac {2 a}{b n}} n^{3/2} \sqrt {2 \pi } \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+4 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (4 a-3 b n+4 b \log \left (c (d+e x)^n\right )\right )\right )\right )}{128 e^2} \]
((d + e*x)*(128*(e*f - d*g)*(a + b*Log[c*(d + e*x)^n])^(5/2) + 64*g*(d + e *x)*(a + b*Log[c*(d + e*x)^n])^(5/2) - 80*b*(e*f - d*g)*n*((3*b^(3/2)*n^(3 /2)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a /(b*n))*(c*(d + e*x)^n)^n^(-1)) + 2*Sqrt[a + b*Log[c*(d + e*x)^n]]*(2*a - 3*b*n + 2*b*Log[c*(d + e*x)^n])) - 5*b*g*n*(d + e*x)*((3*b^(3/2)*n^(3/2)*S qrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])] )/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) + 4*Sqrt[a + b*Log[c*(d + e*x)^n ]]*(4*a - 3*b*n + 4*b*Log[c*(d + e*x)^n]))))/(128*e^2)
Time = 0.90 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \int \left (\frac {(e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}+\frac {g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {15 \sqrt {\pi } b^{5/2} n^{5/2} e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} g n^{5/2} e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{64 e^2}+\frac {15 b^2 n^2 (d+e x) (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac {15 b^2 g n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{32 e^2}+\frac {(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}-\frac {5 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac {5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}\) |
(-15*b^(5/2)*(e*f - d*g)*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c* (d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(8*e^2*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1 )) - (15*b^(5/2)*g*n^(5/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b *Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(64*e^2*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) + (15*b^2*(e*f - d*g)*n^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e *x)^n]])/(4*e^2) + (15*b^2*g*n^2*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n] ])/(32*e^2) - (5*b*(e*f - d*g)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2 ))/(2*e^2) - (5*b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(8*e^2 ) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2))/(2*e^2)
3.2.18.3.1 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]
\[\int \left (g x +f \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}d x\]
Exception generated. \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \]
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2} \,d x \]