Integrand size = 62, antiderivative size = 177 \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\frac {4 f^3 \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{2+n} \operatorname {Hypergeometric2F1}\left (2,\frac {2+n}{2},\frac {4+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^2 g (2+n) \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \]
4*f^3*hypergeom([2, 1+1/2*n],[2+1/2*n],(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2 )^(1/2))^2/(-a*f^2+d^2))*(a+2*d*e*x/f^2+e^2*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))^(2+n)/e/(-a*f^2+d^2)^2/g/(2+n)/(a*g+2*d*e*g*x /f^2+e^2*g*x^2/f^2)^(1/2)
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\frac {4 f^3 \left (a+\frac {e x (2 d+e x)}{f^2}\right )^{3/2} \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{2+n} \operatorname {Hypergeometric2F1}\left (2,\frac {2+n}{2},\frac {4+n}{2},\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^2 (2+n) \left (g \left (a+\frac {e x (2 d+e x)}{f^2}\right )\right )^{3/2}} \]
Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a*g + ( 2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2)^(3/2),x]
(4*f^3*(a + (e*x*(2*d + e*x))/f^2)^(3/2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)^2*( 2 + n)*(g*(a + (e*x*(2*d + e*x))/f^2))^(3/2))
Time = 0.66 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2550, 2546, 27, 278}
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.
| \(\displaystyle \int \frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2550 |
| \(\displaystyle \frac {\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \int \frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^n}{\left (\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a\right )^{3/2}}dx}{g \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\) |
| \(\Big \downarrow \) 2546 |
| \(\displaystyle \frac {2 f^3 \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \int \frac {2 \left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+1}}{e \left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^2}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{g \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 f^3 \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \int \frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^{n+1}}{\left (d^2-a f^2-\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2\right )^2}d\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )}{e g \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {4 f^3 \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+2} \operatorname {Hypergeometric2F1}\left (2,\frac {n+2}{2},\frac {n+4}{2},\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e g (n+2) \left (d^2-a f^2\right )^2 \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\) |
Int[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/(a*g + (2*d*e* g*x)/f^2 + (e^2*g*x^2)/f^2)^(3/2),x]
(4*f^3*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2]*(d + e*x + f*Sqrt[a + (2*d* e*x)/f^2 + (e^2*x^2)/f^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n) /2, (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^2/(d^2 - a*f^2)] )/(e*(d^2 - a*f^2)^2*g*(2 + n)*Sqrt[a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^ 2])
3.6.21.3.1 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] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 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[((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[a + b*x + c*x^2]/Sqrt[g + h*x + i*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] && ILtQ[m - 1/2, 0] && !GtQ[i/c, 0]
\[\int \frac {{\left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )}^{n}}{\left (a g +\frac {2 d e g x}{f^{2}}+\frac {e^{2} g \,x^{2}}{f^{2}}\right )^{\frac {3}{2}}}d x\]
int((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)^(3/2),x)
\[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\int { \frac {{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}}{{\left (\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((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)^(3/2),x, algorithm="fricas")
integral((e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2) + d)^n*f^4*sqrt((e ^2*g*x^2 + a*f^2*g + 2*d*e*g*x)/f^2)/(e^4*g^2*x^4 + 4*d*e^3*g^2*x^3 + a^2* f^4*g^2 + 4*a*d*e*f^2*g^2*x + 2*(a*e^2*f^2 + 2*d^2*e^2)*g^2*x^2), x)
Exception generated. \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
integrate((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)**(3/2),x)
\[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\int { \frac {{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}}{{\left (\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((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)^(3/2),x, algorithm="maxima")
integrate((e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n/(e^2*g*x^2/f ^2 + a*g + 2*d*e*g*x/f^2)^(3/2), x)
\[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\int { \frac {{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}}{{\left (\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
integrate((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)^(3/2),x, algorithm="giac")
integrate((e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n/(e^2*g*x^2/f ^2 + a*g + 2*d*e*g*x/f^2)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\left (a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}\right )^{3/2}} \, dx=\int \frac {{\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n}{{\left (a\,g+\frac {e^2\,g\,x^2}{f^2}+\frac {2\,d\,e\,g\,x}{f^2}\right )}^{3/2}} \,d x \]
int((d + f*(a + (e^2*x^2)/f^2 + (2*d*e*x)/f^2)^(1/2) + e*x)^n/(a*g + (e^2* g*x^2)/f^2 + (2*d*e*g*x)/f^2)^(3/2),x)