Integrand size = 24, antiderivative size = 413 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=-\frac {15 b^{5/2} e^{-\frac {a}{b n}} (e f-d g) n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b n}} g n^{5/2} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{64 e^2}+\frac {15 b^2 (e f-d g) n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac {15 b^2 g n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac {5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac {5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2} \] Output:
-15/8*b^(5/2)*(-d*g+e*f)*n^(5/2)*Pi^(1/2)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n ))^(1/2)/b^(1/2)/n^(1/2))/e^2/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))-15/128*b^(5 /2)*g*n^(5/2)*2^(1/2)*Pi^(1/2)*(e*x+d)^2*erfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n) )^(1/2)/b^(1/2)/n^(1/2))/e^2/exp(2*a/b/n)/((c*(e*x+d)^n)^(2/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*ln(c*(e*x+d)^n))^(1/2)/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*l n(c*(e*x+d)^n))^(5/2)/e^2
Time = 0.81 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.79 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\frac {(d+e x) \left (128 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}+64 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-80 b (e f-d g) n \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )-5 b g n (d+e x) \left (3 b^{3/2} e^{-\frac {2 a}{b n}} n^{3/2} \sqrt {2 \pi } \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+4 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (4 a-3 b n+4 b \log \left (c (d+e x)^n\right )\right )\right )\right )}{128 e^2} \] Input:
Integrate[(f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2),x]
Output:
((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 = 1.58 (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 |
\(\displaystyle -\frac {15 \sqrt {\pi } b^{5/2} n^{5/2} e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} g n^{5/2} e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{64 e^2}+\frac {15 b^2 n^2 (d+e x) (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac {15 b^2 g n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{32 e^2}+\frac {(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}-\frac {5 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac {5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}\) |
Input:
Int[(f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2),x]
Output:
(-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)
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]
\[\int \left (g x +f \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}d x\]
Input:
int((g*x+f)*(a+b*ln(c*(e*x+d)^n))^(5/2),x)
Output:
int((g*x+f)*(a+b*ln(c*(e*x+d)^n))^(5/2),x)
Exception generated. \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \] Input:
integrate((g*x+f)*(a+b*ln(c*(e*x+d)**n))**(5/2),x)
Output:
Integral((a + b*log(c*(d + e*x)**n))**(5/2)*(f + g*x), x)
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="maxima")
Output:
integrate((g*x + f)*(b*log((e*x + d)^n*c) + a)^(5/2), x)
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*x+f)*(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="giac")
Output:
integrate((g*x + f)*(b*log((e*x + d)^n*c) + a)^(5/2), x)
Timed out. \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2} \,d x \] Input:
int((f + g*x)*(a + b*log(c*(d + e*x)^n))^(5/2),x)
Output:
int((f + g*x)*(a + b*log(c*(d + e*x)^n))^(5/2), x)
\[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {too large to display} \] Input:
int((g*x+f)*(a+b*log(c*(e*x+d)^n))^(5/2),x)
Output:
( - 32*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*d**2* g + 64*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*d*e*f + 64*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*e**2*f *x + 32*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*e**2 *g*x**2 - 8*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*d* *2*g*n + 16*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*d* e*f*n + 16*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*e** 2*f*n*x + 8*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*e* *2*g*n*x**2 - 64*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2* b*d**2*g + 128*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b* d*e*f + 128*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b*e** 2*f*x + 64*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b*e**2 *g*x**2 + 104*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b**2*d **2*g*n - 128*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b**2*d *e*f*n + 80*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b**2*d*e *g*n*x - 128*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b**2*e* *2*f*n*x - 24*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b**2*e **2*g*n*x**2 + 20*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*b**3 *d**2*g*n**2 - 40*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*b**3 *d*e*f*n**2 + 20*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*b*...