\(\int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx\) [72]

Optimal result

Integrand size = 31, antiderivative size = 186 \[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=-\frac {2 \sqrt {2-d x} \sqrt {2+d x} \sqrt {e+f x}}{3 f}+\frac {4 (d e-6 f) \sqrt {e+f x} E\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right )|\frac {4 f}{d e+2 f}\right )}{3 f^2 \sqrt {\frac {d (e+f x)}{d e+2 f}}}-\frac {4 \left (\frac {8}{d}+\frac {e (d e-6 f)}{f^2}\right ) \sqrt {\frac {d (e+f x)}{d e+2 f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {2-d x}\right ),\frac {4 f}{d e+2 f}\right )}{3 \sqrt {e+f x}} \] Output:

-2/3*(-d*x+2)^(1/2)*(d*x+2)^(1/2)*(f*x+e)^(1/2)/f+4/3*(d*e-6*f)*(f*x+e)^(1 
/2)*EllipticE(1/2*(-d*x+2)^(1/2),2*(f/(d*e+2*f))^(1/2))/f^2/(d*(f*x+e)/(d* 
e+2*f))^(1/2)-4/3*(8/d+e*(d*e-6*f)/f^2)*(d*(f*x+e)/(d*e+2*f))^(1/2)*Ellipt 
icF(1/2*(-d*x+2)^(1/2),2*(f/(d*e+2*f))^(1/2))/(f*x+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.82 \[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\frac {2 \sqrt {4-d^2 x^2} \left (-e-f x+\frac {2 \left ((d e-6 f) f^2 \sqrt {-e-\frac {2 f}{d}} \left (-4+d^2 x^2\right )-i d \left (d^2 e^2-4 d e f-12 f^2\right ) \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-2 f}{d e+2 f}\right )+2 i d (d e-10 f) f \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-2 f}{d e+2 f}\right )\right )}{d f^2 \sqrt {-e-\frac {2 f}{d}} \left (-4+d^2 x^2\right )}\right )}{3 f \sqrt {e+f x}} \] Input:

Integrate[(2 + d*x)^2/(Sqrt[e + f*x]*Sqrt[4 - d^2*x^2]),x]
 

Output:

(2*Sqrt[4 - d^2*x^2]*(-e - f*x + (2*((d*e - 6*f)*f^2*Sqrt[-e - (2*f)/d]*(- 
4 + d^2*x^2) - I*d*(d^2*e^2 - 4*d*e*f - 12*f^2)*Sqrt[(f*(-2 + d*x))/(d*(e 
+ f*x))]*Sqrt[(f*(2 + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticE[I*Arc 
Sinh[Sqrt[-e - (2*f)/d]/Sqrt[e + f*x]], (d*e - 2*f)/(d*e + 2*f)] + (2*I)*d 
*(d*e - 10*f)*f*Sqrt[(f*(-2 + d*x))/(d*(e + f*x))]*Sqrt[(f*(2 + d*x))/(d*( 
e + f*x))]*(e + f*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-e - (2*f)/d]/Sqrt[e + 
 f*x]], (d*e - 2*f)/(d*e + 2*f)]))/(d*f^2*Sqrt[-e - (2*f)/d]*(-4 + d^2*x^2 
))))/(3*f*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {717, 113, 25, 27, 176, 124, 27, 123, 131, 129}

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.

\(\displaystyle \int \frac {(d x+2)^2}{\sqrt {4-d^2 x^2} \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 717

\(\displaystyle \int \frac {(d x+2)^{3/2}}{\sqrt {2-d x} \sqrt {e+f x}}dx\)

\(\Big \downarrow \) 113

\(\displaystyle -\frac {2 \int -\frac {d (8 f-d (d e-6 f) x)}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx}{3 d f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {d (8 f-d (d e-6 f) x)}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx}{3 d f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {8 f-d (d e-6 f) x}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 \left ((d e-6 f) \int \frac {\sqrt {2-d x}}{\sqrt {d x+2} \sqrt {e+f x}}dx-2 (d e-10 f) \int \frac {1}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {2 \left (\frac {2 (d e-6 f) \sqrt {\frac {d (e+f x)}{d e-2 f}} \int \frac {\sqrt {2-d x}}{2 \sqrt {d x+2} \sqrt {\frac {d e}{d e-2 f}+\frac {d f x}{d e-2 f}}}dx}{\sqrt {e+f x}}-2 (d e-10 f) \int \frac {1}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {(d e-6 f) \sqrt {\frac {d (e+f x)}{d e-2 f}} \int \frac {\sqrt {2-d x}}{\sqrt {d x+2} \sqrt {\frac {d e}{d e-2 f}+\frac {d f x}{d e-2 f}}}dx}{\sqrt {e+f x}}-2 (d e-10 f) \int \frac {1}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 \left (\frac {4 (d e-6 f) \sqrt {2 f-d e} \sqrt {\frac {d (e+f x)}{d e-2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {d x+2}}{\sqrt {2 f-d e}}\right )|\frac {1}{4} \left (2-\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}}-2 (d e-10 f) \int \frac {1}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}dx\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {2 \left (\frac {4 (d e-6 f) \sqrt {2 f-d e} \sqrt {\frac {d (e+f x)}{d e-2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {d x+2}}{\sqrt {2 f-d e}}\right )|\frac {1}{4} \left (2-\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}}-\frac {2 (d e-10 f) \sqrt {\frac {d (e+f x)}{d e-2 f}} \int \frac {1}{\sqrt {2-d x} \sqrt {d x+2} \sqrt {\frac {d e}{d e-2 f}+\frac {d f x}{d e-2 f}}}dx}{\sqrt {e+f x}}\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2 \left (\frac {4 (d e-6 f) \sqrt {2 f-d e} \sqrt {\frac {d (e+f x)}{d e-2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {d x+2}}{\sqrt {2 f-d e}}\right )|\frac {1}{4} \left (2-\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}}-\frac {4 (d e-10 f) \sqrt {\frac {d (e+f x)}{d e-2 f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {d x+2}\right ),-\frac {4 f}{d e-2 f}\right )}{d \sqrt {e+f x}}\right )}{3 f}-\frac {2 \sqrt {2-d x} \sqrt {d x+2} \sqrt {e+f x}}{3 f}\)

Input:

Int[(2 + d*x)^2/(Sqrt[e + f*x]*Sqrt[4 - d^2*x^2]),x]
 

Output:

(-2*Sqrt[2 - d*x]*Sqrt[2 + d*x]*Sqrt[e + f*x])/(3*f) + (2*((4*(d*e - 6*f)* 
Sqrt[-(d*e) + 2*f]*Sqrt[(d*(e + f*x))/(d*e - 2*f)]*EllipticE[ArcSin[(Sqrt[ 
f]*Sqrt[2 + d*x])/Sqrt[-(d*e) + 2*f]], (2 - (d*e)/f)/4])/(d*Sqrt[f]*Sqrt[e 
 + f*x]) - (4*(d*e - 10*f)*Sqrt[(d*(e + f*x))/(d*e - 2*f)]*EllipticF[ArcSi 
n[Sqrt[2 + d*x]/2], (-4*f)/(d*e - 2*f)])/(d*Sqrt[e + f*x])))/(3*f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 113

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

rule 131

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x]   Int[1/(Sq 
rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x 
], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && Simpler 
Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

rule 717

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, 
 x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a 
, 0] && GtQ[d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(162)=324\).

Time = 2.94 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.62

method	result	size
elliptic	\(\frac {\sqrt {-\left (f x +e \right ) \left (d^{2} x^{2}-4\right )}\, \left (-\frac {2 \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}{3 f}+\frac {32 \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{3 \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}+\frac {2 \left (4 d -\frac {2 d^{2} e}{3 f}\right ) \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \left (\left (-\frac {e}{f}+\frac {2}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )-\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}\)	\(488\)
risch	\(\frac {2 \sqrt {f x +e}\, \left (d^{2} x^{2}-4\right ) \sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+4\right )}}{3 f \sqrt {-\left (f x +e \right ) \left (d^{2} x^{2}-4\right )}\, \sqrt {-d^{2} x^{2}+4}}-\frac {2 \left (\frac {2 d \left (d e -6 f \right ) \left (\frac {e}{f}-\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \left (\left (-\frac {e}{f}-\frac {2}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\right )+\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}-\frac {16 f \left (\frac {e}{f}-\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}\right ) \sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+4\right )}}{3 f \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}\)	\(525\)
default	\(-\frac {2 \left (2 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) d^{3} e^{3}+4 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) d^{2} e^{2} f -12 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) d^{2} e^{2} f +d^{3} f^{3} x^{3}-8 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) d e \,f^{2}+d^{3} e \,f^{2} x^{2}-16 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) f^{3}+48 \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) f^{3}-4 d \,f^{3} x -4 d e \,f^{2}\right ) \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{3 d \,f^{3} \left (d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e \right )}\)	\(697\)

Input:

int((d*x+2)^2/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(-(f*x+e)*(d^2*x^2-4))^(1/2)/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2)*(-2/3/f*(-d^ 
2*f*x^3-d^2*e*x^2+4*f*x+4*e)^(1/2)+32/3*(e/f+2/d)*((x+e/f)/(e/f+2/d))^(1/2 
)*((x+2/d)/(-e/f+2/d))^(1/2)*((x-2/d)/(-e/f-2/d))^(1/2)/(-d^2*f*x^3-d^2*e* 
x^2+4*f*x+4*e)^(1/2)*EllipticF(((x+e/f)/(e/f+2/d))^(1/2),((-e/f-2/d)/(-e/f 
+2/d))^(1/2))+2*(4*d-2/3*d^2*e/f)*(e/f+2/d)*((x+e/f)/(e/f+2/d))^(1/2)*((x+ 
2/d)/(-e/f+2/d))^(1/2)*((x-2/d)/(-e/f-2/d))^(1/2)/(-d^2*f*x^3-d^2*e*x^2+4* 
f*x+4*e)^(1/2)*((-e/f+2/d)*EllipticE(((x+e/f)/(e/f+2/d))^(1/2),((-e/f-2/d) 
/(-e/f+2/d))^(1/2))-2/d*EllipticF(((x+e/f)/(e/f+2/d))^(1/2),((-e/f-2/d)/(- 
e/f+2/d))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.30 \[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {-d^{2} x^{2} + 4} \sqrt {f x + e} d^{2} f^{2} + 2 \, {\left (d^{2} e^{2} - 6 \, d e f + 24 \, f^{2}\right )} \sqrt {-d^{2} f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right ) + 6 \, {\left (d^{2} e f - 6 \, d f^{2}\right )} \sqrt {-d^{2} f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )\right )}}{9 \, d^{2} f^{3}} \] Input:

integrate((d*x+2)^2/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2),x, algorithm="fricas" 
)
 

Output:

-2/9*(3*sqrt(-d^2*x^2 + 4)*sqrt(f*x + e)*d^2*f^2 + 2*(d^2*e^2 - 6*d*e*f + 
24*f^2)*sqrt(-d^2*f)*weierstrassPInverse(4/3*(d^2*e^2 + 12*f^2)/(d^2*f^2), 
 -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f) + 6*(d^2*e*f - 6 
*d*f^2)*sqrt(-d^2*f)*weierstrassZeta(4/3*(d^2*e^2 + 12*f^2)/(d^2*f^2), -8/ 
27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), weierstrassPInverse(4/3*(d^2*e^2 + 12*f 
^2)/(d^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f)))/ 
(d^2*f^3)
 

Sympy [F]

\[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\int \frac {\left (d x + 2\right )^{2}}{\sqrt {- \left (d x - 2\right ) \left (d x + 2\right )} \sqrt {e + f x}}\, dx \] Input:

integrate((d*x+2)**2/(f*x+e)**(1/2)/(-d**2*x**2+4)**(1/2),x)
 

Output:

Integral((d*x + 2)**2/(sqrt(-(d*x - 2)*(d*x + 2))*sqrt(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 2\right )}^{2}}{\sqrt {-d^{2} x^{2} + 4} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+2)^2/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((d*x + 2)^2/(sqrt(-d^2*x^2 + 4)*sqrt(f*x + e)), x)
 

Giac [F]

\[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 2\right )}^{2}}{\sqrt {-d^{2} x^{2} + 4} \sqrt {f x + e}} \,d x } \] Input:

integrate((d*x+2)^2/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2),x, algorithm="giac")
 

Output:

integrate((d*x + 2)^2/(sqrt(-d^2*x^2 + 4)*sqrt(f*x + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\int \frac {{\left (d\,x+2\right )}^2}{\sqrt {e+f\,x}\,\sqrt {4-d^2\,x^2}} \,d x \] Input:

int((d*x + 2)^2/((e + f*x)^(1/2)*(4 - d^2*x^2)^(1/2)),x)
 

Output:

int((d*x + 2)^2/((e + f*x)^(1/2)*(4 - d^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(2+d x)^2}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\frac {-4 \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}-\left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}\, x^{2}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e}d x \right ) d^{3} e +6 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}\, x^{2}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e}d x \right ) d^{2} f -4 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e}d x \right ) d e -8 \left (\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e}d x \right ) f}{d e} \] Input:

int((d*x+2)^2/(f*x+e)^(1/2)/(-d^2*x^2+4)^(1/2),x)
 

Output:

( - 4*sqrt(e + f*x)*sqrt( - d**2*x**2 + 4) - int((sqrt(e + f*x)*sqrt( - d* 
*2*x**2 + 4)*x**2)/(d**2*e*x**2 + d**2*f*x**3 - 4*e - 4*f*x),x)*d**3*e + 6 
*int((sqrt(e + f*x)*sqrt( - d**2*x**2 + 4)*x**2)/(d**2*e*x**2 + d**2*f*x** 
3 - 4*e - 4*f*x),x)*d**2*f - 4*int((sqrt(e + f*x)*sqrt( - d**2*x**2 + 4))/ 
(d**2*e*x**2 + d**2*f*x**3 - 4*e - 4*f*x),x)*d*e - 8*int((sqrt(e + f*x)*sq 
rt( - d**2*x**2 + 4))/(d**2*e*x**2 + d**2*f*x**3 - 4*e - 4*f*x),x)*f)/(d*e 
)