3.6.19 \(\int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}})^n \, dx\) [519]

3.6.19.1 Optimal result

Integrand size = 62, antiderivative size = 327 \[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=-\frac {\left (d^2-a f^2\right )^2 \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-2+n}}{4 e f (2-n) \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}-\frac {\left (d^2-a f^2\right ) \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{2 e f n \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}+\frac {\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{2+n}}{4 e f (2+n) \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}} \]

output

-1/4*(-a*f^2+d^2)^2*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(-2+n)*(a* 
g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)/e/f/(2-n)/(a+2*d*e*x/f^2+e^2*x^2/f^2) 
^(1/2)-1/2*(-a*f^2+d^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n*(a*g 
+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)/e/f/n/(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2 
)+1/4*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(2+n)*(a*g+2*d*e*g*x/f^2 
+e^2*g*x^2/f^2)^(1/2)/e/f/(2+n)/(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2)
 

3.6.19.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.54 \[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\frac {\sqrt {g \left (a+\frac {e x (2 d+e x)}{f^2}\right )} \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^n \left (\frac {2 \left (-d^2+a f^2\right )}{n}+\frac {\left (d^2-a f^2\right )^2}{(-2+n) \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}+\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{2+n}\right )}{4 e f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}} \]

input

Integrate[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[ 
a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]
 

output

(Sqrt[g*(a + (e*x*(2*d + e*x))/f^2)]*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x 
))/f^2])^n*((2*(-d^2 + a*f^2))/n + (d^2 - a*f^2)^2/((-2 + n)*(d + e*x + f* 
Sqrt[a + (e*x*(2*d + e*x))/f^2])^2) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e* 
x))/f^2])^2/(2 + n)))/(4*e*f*Sqrt[a + (e*x*(2*d + e*x))/f^2])
 

3.6.19.3 Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.64, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2548, 2546, 27, 244, 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.

\(\displaystyle \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n \, dx\)

\(\Big \downarrow \) 2548

\(\displaystyle \frac {\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \int \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a} \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^ndx}{\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}\)

\(\Big \downarrow \) 2546

\(\displaystyle \frac {2 \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \int \frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-3} \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^2}{8 e}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \int \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-3} \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^2d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{4 e f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \int \left (\left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-3}-2 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-1}+\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+1}\right )d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{4 e f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (-\frac {\left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-2}}{2-n}-\frac {2 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{n}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+2}}{n+2}\right )}{4 e f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}}\)

input

Int[Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2 
*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]
 

output

(Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2]*(-(((d^2 - a*f^2)^2*(d + e* 
x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-2 + n))/(2 - n)) - (2*(d^ 
2 - a*f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n)/n + (d 
 + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(2 + n)/(2 + n)))/(4*e 
*f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])
 

3.6.19.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 2546

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*S 
qrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Simp[(2/f^(2*m) 
)*(i/c)^m   Subst[Int[x^n*((d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e 
*x^2)^(2*m + 1)/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1))), x], x, d + e*x + f*S 
qrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n}, x] && Eq 
Q[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && IntegerQ[2*m 
] && (IntegerQ[m] || GtQ[i/c, 0])
 

rule 2548

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*S 
qrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Simp[(i/c)^(m - 
 1/2)*(Sqrt[g + h*x + i*x^2]/Sqrt[a + b*x + c*x^2])   Int[(a + b*x + c*x^2) 
^m*(d + e*x + f*Sqrt[a + b*x + c*x^2])^n, x], x] /; FreeQ[{a, b, c, d, e, f 
, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - 
b*i, 0] && IGtQ[m + 1/2, 0] &&  !GtQ[i/c, 0]
 

3.6.19.4 Maple [F]

\[\int \sqrt {a g +\frac {2 d e g x}{f^{2}}+\frac {e^{2} g \,x^{2}}{f^{2}}}\, {\left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )}^{n}d x\]

input

int((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^ 
2/f^2)^(1/2))^n,x)
 

output

int((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^ 
2/f^2)^(1/2))^n,x)
 

3.6.19.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=-\frac {{\left (2 \, e^{3} n x^{3} + 6 \, d e^{2} n x^{2} + 2 \, a d f^{2} n + 2 \, {\left (a e f^{2} + 2 \, d^{2} e\right )} n x - {\left (e^{2} f n^{2} x^{2} + a f^{3} n^{2} + 2 \, d e f n^{2} x - 2 \, a f^{3} + 2 \, d^{2} f\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )} {\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} \sqrt {\frac {e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}}}{a e f^{2} n^{3} - 4 \, a e f^{2} n + {\left (e^{3} n^{3} - 4 \, e^{3} n\right )} x^{2} + 2 \, {\left (d e^{2} n^{3} - 4 \, d e^{2} n\right )} x} \]

input

integrate((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+ 
e^2*x^2/f^2)^(1/2))^n,x, algorithm="fricas")
 

output

-(2*e^3*n*x^3 + 6*d*e^2*n*x^2 + 2*a*d*f^2*n + 2*(a*e*f^2 + 2*d^2*e)*n*x - 
(e^2*f*n^2*x^2 + a*f^3*n^2 + 2*d*e*f*n^2*x - 2*a*f^3 + 2*d^2*f)*sqrt((e^2* 
x^2 + a*f^2 + 2*d*e*x)/f^2))*(e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2 
) + d)^n*sqrt((e^2*g*x^2 + a*f^2*g + 2*d*e*g*x)/f^2)/(a*e*f^2*n^3 - 4*a*e* 
f^2*n + (e^3*n^3 - 4*e^3*n)*x^2 + 2*(d*e^2*n^3 - 4*d*e^2*n)*x)
 

3.6.19.6 Sympy [F(-2)]

Exception generated. \[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]

input

integrate((a*g+2*d*e*g*x/f**2+e**2*g*x**2/f**2)**(1/2)*(d+e*x+f*(a+2*d*e*x 
/f**2+e**2*x**2/f**2)**(1/2))**n,x)
 

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.6.19.7 Maxima [F]

\[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int { \sqrt {\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n} \,d x } \]

input

integrate((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+ 
e^2*x^2/f^2)^(1/2))^n,x, algorithm="maxima")
 

output

integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^ 
2 + a + 2*d*e*x/f^2)*f + d)^n, x)
 

3.6.19.8 Giac [F]

\[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int { \sqrt {\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n} \,d x } \]

input

integrate((a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*(a+2*d*e*x/f^2+ 
e^2*x^2/f^2)^(1/2))^n,x, algorithm="giac")
 

output

integrate(sqrt(e^2*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2)*(e*x + sqrt(e^2*x^2/f^ 
2 + a + 2*d*e*x/f^2)*f + d)^n, x)
 

3.6.19.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int {\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n\,\sqrt {a\,g+\frac {e^2\,g\,x^2}{f^2}+\frac {2\,d\,e\,g\,x}{f^2}} \,d x \]

input

int((d + f*(a + (e^2*x^2)/f^2 + (2*d*e*x)/f^2)^(1/2) + e*x)^n*(a*g + (e^2* 
g*x^2)/f^2 + (2*d*e*g*x)/f^2)^(1/2),x)
 

output

int((d + f*(a + (e^2*x^2)/f^2 + (2*d*e*x)/f^2)^(1/2) + e*x)^n*(a*g + (e^2* 
g*x^2)/f^2 + (2*d*e*g*x)/f^2)^(1/2), x)