Integrand size = 56, antiderivative size = 365 \[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\frac {\left (d^2-a f^2\right )^5 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-5+n}}{32 e f^4 (5-n)}-\frac {5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-3+n}}{32 e f^4 (3-n)}+\frac {5 \left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-1+n}}{16 e f^4 (1-n)}+\frac {5 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{16 e f^4 (1+n)}-\frac {5 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{3+n}}{32 e f^4 (3+n)}+\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{5+n}}{32 e f^4 (5+n)} \]
1/32*(-a*f^2+d^2)^5*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(-5+n)/e/f ^4/(5-n)-5/32*(-a*f^2+d^2)^4*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^( -3+n)/e/f^4/(3-n)+5/16*(-a*f^2+d^2)^3*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2) ^(1/2))^(-1+n)/e/f^4/(1-n)+5/16*(-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))^(1+n)/e/f^4/(1+n)-5/32*(-a*f^2+d^2)*(d+e*x+f*(a+2*d*e*x/f ^2+e^2*x^2/f^2)^(1/2))^(3+n)/e/f^4/(3+n)+1/32*(d+e*x+f*(a+2*d*e*x/f^2+e^2* x^2/f^2)^(1/2))^(5+n)/e/f^4/(5+n)
Time = 4.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.77 \[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{-5+n} \left (-\frac {\left (d^2-a f^2\right )^5}{-5+n}+\frac {5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{-3+n}-\frac {10 \left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^4}{-1+n}+\frac {10 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^6}{1+n}-\frac {5 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^8}{3+n}+\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{10}}{5+n}\right )}{32 e f^4} \]
Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2*(d + e*x + f*Sqrt[a + (2*d *e*x)/f^2 + (e^2*x^2)/f^2])^n,x]
((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-5 + n)*(-((d^2 - a*f^2)^5 /(-5 + n)) + (5*(d^2 - a*f^2)^4*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^ 2])^2)/(-3 + n) - (10*(d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (e*x*(2*d + e* x))/f^2])^4)/(-1 + n) + (10*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (e*x*(2* d + e*x))/f^2])^6)/(1 + n) - (5*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*( 2*d + e*x))/f^2])^8)/(3 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2 ])^10/(5 + n)))/(32*e*f^4)
Time = 0.50 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {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.
| \(\displaystyle \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n \, dx\) |
| \(\Big \downarrow \) 2546 |
| \(\displaystyle \frac {2 \int -\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-6} \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^5}{64 e}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{f^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-6} \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^5d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{32 e f^4}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {\int \left (\left (d^2-a f^2\right )^5 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-6}-5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-4}+10 \left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n-2}-10 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^n+5 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+2}-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+4}\right )d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{32 e f^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {\left (d^2-a f^2\right )^5 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{5-n}+\frac {5 \left (d^2-a f^2\right )^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{3-n}-\frac {10 \left (d^2-a f^2\right )^3 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{1-n}-\frac {10 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{n+1}+\frac {5 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{n+3}-\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{n+5}}{32 e f^4}\) |
Int[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/ f^2 + (e^2*x^2)/f^2])^n,x]
-1/32*(-(((d^2 - a*f^2)^5*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/ f^2])^(-5 + n))/(5 - n)) + (5*(d^2 - a*f^2)^4*(d + e*x + f*Sqrt[a + (2*d*e *x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))/(3 - n) - (10*(d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(1 - n) - (10*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n)) /(1 + n) + (5*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2 )/f^2])^(3 + n))/(3 + n) - (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2) /f^2])^(5 + n)/(5 + n))/(e*f^4)
3.6.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
\[\int \left (a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}\right )^{2} {\left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )}^{n}d x\]
Time = 0.31 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.79 \[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^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 (5 \, a^{2} d f^{4} n^{4} + 225 \, a^{2} d f^{4} - 300 \, a d^{3} f^{2} + 5 \, {\left (e^{5} n^{4} - 10 \, e^{5} n^{2} + 9 \, e^{5}\right )} x^{5} + 120 \, d^{5} + 25 \, {\left (d e^{4} n^{4} - 10 \, d e^{4} n^{2} + 9 \, d e^{4}\right )} x^{4} + 10 \, {\left (15 \, a e^{3} f^{2} + 30 \, d^{2} e^{3} + {\left (a e^{3} f^{2} + 4 \, d^{2} e^{3}\right )} n^{4} - 2 \, {\left (8 \, a e^{3} f^{2} + 17 \, d^{2} e^{3}\right )} n^{2}\right )} x^{3} - 10 \, {\left (11 \, a^{2} d f^{4} - 6 \, a d^{3} f^{2}\right )} n^{2} + 10 \, {\left (45 \, a d e^{2} f^{2} + {\left (3 \, a d e^{2} f^{2} + 2 \, d^{3} e^{2}\right )} n^{4} - 2 \, {\left (24 \, a d e^{2} f^{2} + d^{3} e^{2}\right )} n^{2}\right )} x^{2} + 5 \, {\left (45 \, a^{2} e f^{4} + {\left (a^{2} e f^{4} + 4 \, a d^{2} e f^{2}\right )} n^{4} - 2 \, {\left (11 \, a^{2} e f^{4} + 26 \, a d^{2} e f^{2} - 12 \, d^{4} e\right )} n^{2}\right )} x - {\left (a^{2} f^{5} n^{5} + {\left (e^{4} f n^{5} - 10 \, e^{4} f n^{3} + 9 \, e^{4} f n\right )} x^{4} - 10 \, {\left (3 \, a^{2} f^{5} - 2 \, a d^{2} f^{3}\right )} n^{3} + 4 \, {\left (d e^{3} f n^{5} - 10 \, d e^{3} f n^{3} + 9 \, d e^{3} f n\right )} x^{3} + 2 \, {\left ({\left (a e^{2} f^{3} + 2 \, d^{2} e^{2} f\right )} n^{5} - 10 \, {\left (2 \, a e^{2} f^{3} + d^{2} e^{2} f\right )} n^{3} + {\left (19 \, a e^{2} f^{3} + 8 \, d^{2} e^{2} f\right )} n\right )} x^{2} + {\left (149 \, a^{2} f^{5} - 260 \, a d^{2} f^{3} + 120 \, d^{4} f\right )} n + 4 \, {\left (a d e f^{3} n^{5} - 10 \, {\left (2 \, a d e f^{3} - d^{3} e f\right )} n^{3} + {\left (19 \, a d e f^{3} - 10 \, d^{3} e f\right )} n\right )} x\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}}{e f^{4} n^{6} - 35 \, e f^{4} n^{4} + 259 \, e f^{4} n^{2} - 225 \, e f^{4}} \]
integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)^2*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^ 2)^(1/2))^n,x, algorithm="fricas")
-(5*a^2*d*f^4*n^4 + 225*a^2*d*f^4 - 300*a*d^3*f^2 + 5*(e^5*n^4 - 10*e^5*n^ 2 + 9*e^5)*x^5 + 120*d^5 + 25*(d*e^4*n^4 - 10*d*e^4*n^2 + 9*d*e^4)*x^4 + 1 0*(15*a*e^3*f^2 + 30*d^2*e^3 + (a*e^3*f^2 + 4*d^2*e^3)*n^4 - 2*(8*a*e^3*f^ 2 + 17*d^2*e^3)*n^2)*x^3 - 10*(11*a^2*d*f^4 - 6*a*d^3*f^2)*n^2 + 10*(45*a* d*e^2*f^2 + (3*a*d*e^2*f^2 + 2*d^3*e^2)*n^4 - 2*(24*a*d*e^2*f^2 + d^3*e^2) *n^2)*x^2 + 5*(45*a^2*e*f^4 + (a^2*e*f^4 + 4*a*d^2*e*f^2)*n^4 - 2*(11*a^2* e*f^4 + 26*a*d^2*e*f^2 - 12*d^4*e)*n^2)*x - (a^2*f^5*n^5 + (e^4*f*n^5 - 10 *e^4*f*n^3 + 9*e^4*f*n)*x^4 - 10*(3*a^2*f^5 - 2*a*d^2*f^3)*n^3 + 4*(d*e^3* f*n^5 - 10*d*e^3*f*n^3 + 9*d*e^3*f*n)*x^3 + 2*((a*e^2*f^3 + 2*d^2*e^2*f)*n ^5 - 10*(2*a*e^2*f^3 + d^2*e^2*f)*n^3 + (19*a*e^2*f^3 + 8*d^2*e^2*f)*n)*x^ 2 + (149*a^2*f^5 - 260*a*d^2*f^3 + 120*d^4*f)*n + 4*(a*d*e*f^3*n^5 - 10*(2 *a*d*e*f^3 - d^3*e*f)*n^3 + (19*a*d*e*f^3 - 10*d^3*e*f)*n)*x)*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/(e*f^4*n^6 - 35*e*f^4*n^4 + 259*e*f^4*n^2 - 225*e*f^4)
Exception generated. \[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
integrate((a+2*d*e*x/f**2+e**2*x**2/f**2)**2*(d+e*x+f*(a+2*d*e*x/f**2+e**2 *x**2/f**2)**(1/2))**n,x)
\[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int { {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )}^{2} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n} \,d x } \]
integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)^2*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^ 2)^(1/2))^n,x, algorithm="maxima")
integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)^2*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n, x)
\[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int { {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )}^{2} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n} \,d x } \]
integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)^2*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^ 2)^(1/2))^n,x, algorithm="giac")
integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)^2*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n, x)
Timed out. \[ \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx=\int {\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n\,{\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right )}^2 \,d x \]
int((d + f*(a + (e^2*x^2)/f^2 + (2*d*e*x)/f^2)^(1/2) + e*x)^n*(a + (e^2*x^ 2)/f^2 + (2*d*e*x)/f^2)^2,x)