\(\int \frac {2+d x}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx\) [289]

Optimal result

Integrand size = 29, antiderivative size = 91 \[ \int \frac {2+d x}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=-\frac {4 \sqrt {d e+2 f} \sqrt {\frac {d (e+f x)}{d e+2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {2-d x}}{\sqrt {d e+2 f}}\right )|\frac {1}{4} \left (2+\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.16 \[ \int \frac {2+d x}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=-\frac {2 \left (f^2 \sqrt {-e-\frac {2 f}{d}} \left (4-d^2 x^2\right )+i d (d e+2 f) \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-2 f}{d e+2 f}\right )-4 i d f \sqrt {\frac {f (-2+d x)}{d (e+f x)}} \sqrt {\frac {f (2+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-e-\frac {2 f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-2 f}{d e+2 f}\right )\right )}{d f^2 \sqrt {-e-\frac {2 f}{d}} \sqrt {e+f x} \sqrt {4-d^2 x^2}} \] Input:

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

Output:

(-2*(f^2*Sqrt[-e - (2*f)/d]*(4 - d^2*x^2) + I*d*(d*e + 2*f)*Sqrt[(f*(-2 + 
d*x))/(d*(e + f*x))]*Sqrt[(f*(2 + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Ell 
ipticE[I*ArcSinh[Sqrt[-e - (2*f)/d]/Sqrt[e + f*x]], (d*e - 2*f)/(d*e + 2*f 
)] - (4*I)*d*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]*Sqrt[e + f*x]* 
Sqrt[4 - d^2*x^2])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {600, 508, 327, 511, 321}

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.

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

\(\Big \downarrow \) 600

\(\displaystyle \frac {d \int \frac {\sqrt {e+f x}}{\sqrt {4-d^2 x^2}}dx}{f}-\frac {(d e-2 f) \int \frac {1}{\sqrt {e+f x} \sqrt {4-d^2 x^2}}dx}{f}\)

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(-2*Sqrt[e + f*x]*EllipticE[ArcSin[Sqrt[2 - d*x]/2], (4*f)/(d*e + 2*f)])/( 
f*Sqrt[(d*(e + f*x))/(d*e + 2*f)]) + (2*(d*e - 2*f)*Sqrt[(d*(e + f*x))/(d* 
e + 2*f)]*EllipticF[ArcSin[Sqrt[2 - d*x]/2], (4*f)/(d*e + 2*f)])/(d*f*Sqrt 
[e + f*x])
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(76)=152\).

Time = 2.74 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {2 \left (d^{2} e^{2}-4 f^{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}{f^{2} d \left (d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e \right )}\)	\(161\)
elliptic	\(\frac {\sqrt {-\left (f x +e \right ) \left (d^{2} x^{2}-4\right )}\, \left (\frac {4 \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}+\frac {2 d \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \left (\left (-\frac {e}{f}+\frac {2}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )-\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+4}}\)	\(443\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (76) = 152\).

Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.10 \[ \int \frac {2+d x}{\sqrt {e+f x} \sqrt {4-d^2 x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-d^{2} f} d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + \sqrt {-d^{2} f} {\left (d e - 6 \, f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )}}{3 \, d^{2} f^{2}} \] Input:

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

Output:

2/3*(3*sqrt(-d^2*f)*d*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 + 1 
2*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) 
) + sqrt(-d^2*f)*(d*e - 6*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))/(d^2*f^ 
2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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