\(\int (f+g x)^2 (a+b \log (c (d+e x)^n))^{5/2} \, dx\) [117]

Optimal result

Integrand size = 26, antiderivative size = 660 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=-\frac {15 b^{5/2} e^{-\frac {a}{b n}} (e f-d g)^2 n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^3}-\frac {15 b^{5/2} e^{-\frac {2 a}{b n}} g (e f-d g) n^{5/2} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{32 e^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b n}} g^2 n^{5/2} \sqrt {\frac {\pi }{3}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{72 e^3}+\frac {15 b^2 (e f-d g)^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^3}+\frac {15 b^2 g (e f-d g) n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{16 e^3}+\frac {5 b^2 g^2 n^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{36 e^3}-\frac {5 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^3}-\frac {5 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{4 e^3}-\frac {5 b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{18 e^3}+\frac {(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{3 e^3} \] Output:

-15/8*b^(5/2)*(-d*g+e*f)^2*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^3/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))-15/64*b^( 
5/2)*g*(-d*g+e*f)*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^3/exp(2*a/b/n)/((c*(e*x+d)^n)^(2/n) 
)-5/216*b^(5/2)*g^2*n^(5/2)*3^(1/2)*Pi^(1/2)*(e*x+d)^3*erfi(3^(1/2)*(a+b*l 
n(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))/e^3/exp(3*a/b/n)/((c*(e*x+d)^n)^(3/ 
n))+15/4*b^2*(-d*g+e*f)^2*n^2*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(1/2)/e^3+15/1 
6*b^2*g*(-d*g+e*f)*n^2*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^(1/2)/e^3+5/36*b^2* 
g^2*n^2*(e*x+d)^3*(a+b*ln(c*(e*x+d)^n))^(1/2)/e^3-5/2*b*(-d*g+e*f)^2*n*(e* 
x+d)*(a+b*ln(c*(e*x+d)^n))^(3/2)/e^3-5/4*b*g*(-d*g+e*f)*n*(e*x+d)^2*(a+b*l 
n(c*(e*x+d)^n))^(3/2)/e^3-5/18*b*g^2*n*(e*x+d)^3*(a+b*ln(c*(e*x+d)^n))^(3/ 
2)/e^3+(-d*g+e*f)^2*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(5/2)/e^3+g*(-d*g+e*f)*( 
e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^(5/2)/e^3+1/3*g^2*(e*x+d)^3*(a+b*ln(c*(e*x+ 
d)^n))^(5/2)/e^3
 

Mathematica [A] (verified)

Time = 2.16 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.77 \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\frac {(d+e x) \left (1728 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}+1728 g (e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}+576 g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-1080 b (e f-d g)^2 n \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )-40 b g^2 n (d+e x)^2 \left (b^{3/2} e^{-\frac {3 a}{b n}} n^{3/2} \sqrt {3 \pi } \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+6 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-b n+2 b \log \left (c (d+e x)^n\right )\right )\right )-135 b g (e f-d g) n (d+e x) \left (3 b^{3/2} e^{-\frac {2 a}{b n}} n^{3/2} \sqrt {2 \pi } \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+4 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (4 a-3 b n+4 b \log \left (c (d+e x)^n\right )\right )\right )\right )}{1728 e^3} \] Input:

Integrate[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2),x]
 

Output:

((d + e*x)*(1728*(e*f - d*g)^2*(a + b*Log[c*(d + e*x)^n])^(5/2) + 1728*g*( 
e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2) + 576*g^2*(d + e*x)^ 
2*(a + b*Log[c*(d + e*x)^n])^(5/2) - 1080*b*(e*f - d*g)^2*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])) - 40*b*g^2*n*(d + e*x)^2*((b^(3/2)*n^(3 
/2)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt 
[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + 6*Sqrt[a + b*Log[c*(d + e 
*x)^n]]*(2*a - b*n + 2*b*Log[c*(d + e*x)^n])) - 135*b*g*(e*f - d*g)*n*(d + 
 e*x)*((3*b^(3/2)*n^(3/2)*Sqrt[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*S 
qrt[a + b*Log[c*(d + e*x)^n]]*(4*a - 3*b*n + 4*b*Log[c*(d + e*x)^n]))))/(1 
728*e^3)
 

Rubi [A] (verified)

Time = 2.61 (sec) , antiderivative size = 660, 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.

\(\displaystyle \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx\)

\(\Big \downarrow \) 2848

\(\displaystyle \int \left (\frac {(e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac {2 g (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac {g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} g n^{5/2} e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{32 e^3}-\frac {15 \sqrt {\pi } b^{5/2} n^{5/2} e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} g^2 n^{5/2} e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{72 e^3}+\frac {15 b^2 g n^2 (d+e x)^2 (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{16 e^3}+\frac {15 b^2 n^2 (d+e x) (e f-d g)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^3}+\frac {5 b^2 g^2 n^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{36 e^3}+\frac {g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^3}+\frac {(d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^3}-\frac {5 b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{4 e^3}-\frac {5 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^3}+\frac {g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{3 e^3}-\frac {5 b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{18 e^3}\)

Input:

Int[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2),x]
 

Output:

(-15*b^(5/2)*(e*f - d*g)^2*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[ 
c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(8*e^3*E^(a/(b*n))*(c*(d + e*x)^n)^n^( 
-1)) - (15*b^(5/2)*g*(e*f - d*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])])/(32*e^3*E^((2*a)/( 
b*n))*(c*(d + e*x)^n)^(2/n)) - (5*b^(5/2)*g^2*n^(5/2)*Sqrt[Pi/3]*(d + e*x) 
^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(72*e 
^3*E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + (15*b^2*(e*f - d*g)^2*n^2*(d + 
 e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e^3) + (15*b^2*g*(e*f - d*g)*n^2* 
(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(16*e^3) + (5*b^2*g^2*n^2*(d + 
 e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(36*e^3) - (5*b*(e*f - d*g)^2*n*(d 
 + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e^3) - (5*b*g*(e*f - d*g)*n*( 
d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(4*e^3) - (5*b*g^2*n*(d + e*x 
)^3*(a + b*Log[c*(d + e*x)^n])^(3/2))/(18*e^3) + ((e*f - d*g)^2*(d + e*x)* 
(a + b*Log[c*(d + e*x)^n])^(5/2))/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b* 
Log[c*(d + e*x)^n])^(5/2))/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n 
])^(5/2))/(3*e^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2848

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]
 

Maple [F]

\[\int \left (g x +f \right )^{2} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}d x\]

Input:

int((g*x+f)^2*(a+b*ln(c*(e*x+d)^n))^(5/2),x)
 

Output:

int((g*x+f)^2*(a+b*ln(c*(e*x+d)^n))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((g*x+f)^2*(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)
 

Sympy [F]

\[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}} \left (f + g x\right )^{2}\, dx \] Input:

integrate((g*x+f)**2*(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)**2, x)
 

Maxima [F]

\[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="maxima")
 

Output:

integrate((g*x + f)^2*(b*log((e*x + d)^n*c) + a)^(5/2), x)
 

Giac [F]

\[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int { {\left (g x + f\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="giac")
 

Output:

integrate((g*x + f)^2*(b*log((e*x + d)^n*c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\int {\left (f+g\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2} \,d x \] Input:

int((f + g*x)^2*(a + b*log(c*(d + e*x)^n))^(5/2),x)
 

Output:

int((f + g*x)^2*(a + b*log(c*(d + e*x)^n))^(5/2), x)
 

Reduce [F]

\[ \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx=\text {too large to display} \] Input:

int((g*x+f)^2*(a+b*log(c*(e*x+d)^n))^(5/2),x)
 

Output:

(576*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*d**3*g* 
*2 - 1728*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b**2*d* 
*2*e*f*g + 1728*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*a*b 
**2*d*e**2*f**2 + 1728*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c) 
**2*a*b**2*e**3*f**2*x + 1728*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x 
)**n*c)**2*a*b**2*e**3*f*g*x**2 + 576*sqrt(log((d + e*x)**n*c)*b + a)*log( 
(d + e*x)**n*c)**2*a*b**2*e**3*g**2*x**3 + 96*sqrt(log((d + e*x)**n*c)*b + 
 a)*log((d + e*x)**n*c)**2*b**3*d**3*g**2*n - 288*sqrt(log((d + e*x)**n*c) 
*b + a)*log((d + e*x)**n*c)**2*b**3*d**2*e*f*g*n + 288*sqrt(log((d + e*x)* 
*n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*d*e**2*f**2*n + 288*sqrt(log((d + 
 e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*e**3*f**2*n*x + 288*sqrt(lo 
g((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*e**3*f*g*n*x**2 + 96* 
sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)**2*b**3*e**3*g**2*n*x* 
*3 + 1152*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b*d**3* 
g**2 - 3456*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b*d** 
2*e*f*g + 3456*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a**2*b* 
d*e**2*f**2 + 3456*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a** 
2*b*e**3*f**2*x + 3456*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c) 
*a**2*b*e**3*f*g*x**2 + 1152*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x) 
**n*c)*a**2*b*e**3*g**2*x**3 - 2448*sqrt(log((d + e*x)**n*c)*b + a)*log...