\(\int (d+e (F^{c (a+b x)})^n)^{5/2} (f+g x) \, dx\) [45]

Optimal result

Integrand size = 25, antiderivative size = 535 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{5/2} (f+g x) \, dx=-\frac {92 d^2 \sqrt {d+e \left (F^{c (a+b x)}\right )^n} g}{15 b^2 c^2 n^2 \log ^2(F)}-\frac {32 d \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{3/2} g}{45 b^2 c^2 n^2 \log ^2(F)}-\frac {4 \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{5/2} g}{25 b^2 c^2 n^2 \log ^2(F)}+\frac {92 d^{5/2} g \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right )}{15 b^2 c^2 n^2 \log ^2(F)}+\frac {2 d^{5/2} g \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right )^2}{b^2 c^2 n^2 \log ^2(F)}+\frac {2 d^2 \sqrt {d+e \left (F^{c (a+b x)}\right )^n} (f+g x)}{b c n \log (F)}+\frac {2 d \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{3/2} (f+g x)}{3 b c n \log (F)}+\frac {2 \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{5/2} (f+g x)}{5 b c n \log (F)}-\frac {2 d^{5/2} (f+g x) \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right )}{b c n \log (F)}-\frac {4 d^{5/2} g \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}\right )}{b^2 c^2 n^2 \log ^2(F)}-\frac {2 d^{5/2} g \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}\right )}{b^2 c^2 n^2 \log ^2(F)} \] Output:

-92/15*d^2*(d+e*(F^(c*(b*x+a)))^n)^(1/2)*g/b^2/c^2/n^2/ln(F)^2-32/45*d*(d+ 
e*(F^(c*(b*x+a)))^n)^(3/2)*g/b^2/c^2/n^2/ln(F)^2-4/25*(d+e*(F^(c*(b*x+a))) 
^n)^(5/2)*g/b^2/c^2/n^2/ln(F)^2+92/15*d^(5/2)*g*arctanh((d+e*(F^(c*(b*x+a) 
))^n)^(1/2)/d^(1/2))/b^2/c^2/n^2/ln(F)^2+2*d^(5/2)*g*arctanh((d+e*(F^(c*(b 
*x+a)))^n)^(1/2)/d^(1/2))^2/b^2/c^2/n^2/ln(F)^2+2*d^2*(d+e*(F^(c*(b*x+a))) 
^n)^(1/2)*(g*x+f)/b/c/n/ln(F)+2/3*d*(d+e*(F^(c*(b*x+a)))^n)^(3/2)*(g*x+f)/ 
b/c/n/ln(F)+2/5*(d+e*(F^(c*(b*x+a)))^n)^(5/2)*(g*x+f)/b/c/n/ln(F)-2*d^(5/2 
)*(g*x+f)*arctanh((d+e*(F^(c*(b*x+a)))^n)^(1/2)/d^(1/2))/b/c/n/ln(F)-4*d^( 
5/2)*g*arctanh((d+e*(F^(c*(b*x+a)))^n)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2 
)-(d+e*(F^(c*(b*x+a)))^n)^(1/2)))/b^2/c^2/n^2/ln(F)^2-2*d^(5/2)*g*polylog( 
2,1-2*d^(1/2)/(d^(1/2)-(d+e*(F^(c*(b*x+a)))^n)^(1/2)))/b^2/c^2/n^2/ln(F)^2
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2618

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

\(\Big \downarrow \) 2619

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2618

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Int[(c + d*x)^m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] 
 /; FreeQ[{F, a, b, c, d, g, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v 
, x] && IntegerQ[m]
 

rule 2619

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + 
 (d_.)*(x_))^(m_.), x_Symbol] :> Unintegrable[(a + b*(F^(g*(e + f*x)))^n)^p 
*(c + d*x)^m, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/15*f*(15*d^(5/2)*log((sqrt(F^(b*c*n*x + a*c*n)*e + d) - sqrt(d))/(sqrt(F 
^(b*c*n*x + a*c*n)*e + d) + sqrt(d)))/(b*c*n*log(F)) + 2*(3*(F^(b*c*n*x + 
a*c*n)*e + d)^(5/2) + 5*(F^(b*c*n*x + a*c*n)*e + d)^(3/2)*d + 15*sqrt(F^(b 
*c*n*x + a*c*n)*e + d)*d^2)/(b*c*n*log(F))) + g*integrate((2*F^(b*c*n*x)*F 
^(a*c*n)*d*e*x + F^(2*b*c*n*x)*F^(2*a*c*n)*e^2*x + d^2*x)*sqrt(F^(b*c*n*x) 
*F^(a*c*n)*e + d), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^{5/2} (f+g x) \, dx=\frac {90 f^{2 b c n x +2 a c n} \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c \,e^{2} f n +90 f^{2 b c n x +2 a c n} \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c \,e^{2} g n x -36 f^{2 b c n x +2 a c n} \sqrt {f^{b c n x +a c n} e +d}\, e^{2} g +330 f^{b c n x +a c n} \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c d e f n +330 f^{b c n x +a c n} \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c d e g n x -232 f^{b c n x +a c n} \sqrt {f^{b c n x +a c n} e +d}\, d e g +690 \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c \,d^{2} f n +690 \sqrt {f^{b c n x +a c n} e +d}\, \mathrm {log}\left (f \right ) b c \,d^{2} g n x -1576 \sqrt {f^{b c n x +a c n} e +d}\, d^{2} g +225 \left (\int \frac {\sqrt {f^{b c n x +a c n} e +d}}{f^{b c n x +a c n} e +d}d x \right ) \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{3} f \,n^{2}-690 \left (\int \frac {\sqrt {f^{b c n x +a c n} e +d}}{f^{b c n x +a c n} e +d}d x \right ) \mathrm {log}\left (f \right ) b c \,d^{3} g n +225 \left (\int \frac {\sqrt {f^{b c n x +a c n} e +d}\, x}{f^{b c n x +a c n} e +d}d x \right ) \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{3} g \,n^{2}}{225 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} n^{2}} \] Input:

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

Output:

(90*f**(2*a*c*n + 2*b*c*n*x)*sqrt(f**(a*c*n + b*c*n*x)*e + d)*log(f)*b*c*e 
**2*f*n + 90*f**(2*a*c*n + 2*b*c*n*x)*sqrt(f**(a*c*n + b*c*n*x)*e + d)*log 
(f)*b*c*e**2*g*n*x - 36*f**(2*a*c*n + 2*b*c*n*x)*sqrt(f**(a*c*n + b*c*n*x) 
*e + d)*e**2*g + 330*f**(a*c*n + b*c*n*x)*sqrt(f**(a*c*n + b*c*n*x)*e + d) 
*log(f)*b*c*d*e*f*n + 330*f**(a*c*n + b*c*n*x)*sqrt(f**(a*c*n + b*c*n*x)*e 
 + d)*log(f)*b*c*d*e*g*n*x - 232*f**(a*c*n + b*c*n*x)*sqrt(f**(a*c*n + b*c 
*n*x)*e + d)*d*e*g + 690*sqrt(f**(a*c*n + b*c*n*x)*e + d)*log(f)*b*c*d**2* 
f*n + 690*sqrt(f**(a*c*n + b*c*n*x)*e + d)*log(f)*b*c*d**2*g*n*x - 1576*sq 
rt(f**(a*c*n + b*c*n*x)*e + d)*d**2*g + 225*int(sqrt(f**(a*c*n + b*c*n*x)* 
e + d)/(f**(a*c*n + b*c*n*x)*e + d),x)*log(f)**2*b**2*c**2*d**3*f*n**2 - 6 
90*int(sqrt(f**(a*c*n + b*c*n*x)*e + d)/(f**(a*c*n + b*c*n*x)*e + d),x)*lo 
g(f)*b*c*d**3*g*n + 225*int((sqrt(f**(a*c*n + b*c*n*x)*e + d)*x)/(f**(a*c* 
n + b*c*n*x)*e + d),x)*log(f)**2*b**2*c**2*d**3*g*n**2)/(225*log(f)**2*b** 
2*c**2*n**2)