\(\int \frac {c+d x}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx\) [60]

Optimal result

Integrand size = 31, antiderivative size = 137 \[ \int \frac {c+d x}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {2 \sqrt {2} \sqrt {d e+c f} \sqrt {\frac {d (e+f x)}{d e+c f}} \sqrt {c^2-d^2 x^2} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c-d x}}{\sqrt {d e+c f}}\right )|\frac {1}{2} \left (1+\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {c-d x} \sqrt {1+\frac {d x}{c}} \sqrt {e+f x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.14 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.24 \[ \int \frac {c+d x}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {2 \left (f^2 \sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )+i d (d e+c f) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-c f}{d e+c f}\right )-2 i c d f \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-c f}{d e+c f}\right )\right )}{d f^2 \sqrt {-\frac {d e+c f}{d}} \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \] Input:

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

Output:

(-2*(f^2*Sqrt[-((d*e + c*f)/d)]*(c^2 - d^2*x^2) + I*d*(d*e + c*f)*Sqrt[(f* 
(-c + d*x))/(d*(e + f*x))]*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/ 
2)*EllipticE[I*ArcSinh[Sqrt[-((d*e + c*f)/d)]/Sqrt[e + f*x]], (d*e - c*f)/ 
(d*e + c*f)] - (2*I)*c*d*f*Sqrt[(f*(-c + d*x))/(d*(e + f*x))]*Sqrt[(f*(c + 
 d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-((d*e + c* 
f)/d)]/Sqrt[e + f*x]], (d*e - c*f)/(d*e + c*f)]))/(d*f^2*Sqrt[-((d*e + c*f 
)/d)]*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.58, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {600, 509, 508, 327, 512, 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.

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(-2*c*Sqrt[e + f*x]*Sqrt[1 - (d^2*x^2)/c^2]*EllipticE[ArcSin[Sqrt[1 - (d*x 
)/c]/Sqrt[2]], (2*c*f)/(d*e + c*f)])/(f*Sqrt[(d*(e + f*x))/(d*e + c*f)]*Sq 
rt[c^2 - d^2*x^2]) + (2*c*(d*e - c*f)*Sqrt[(d*(e + f*x))/(d*e + c*f)]*Sqrt 
[1 - (d^2*x^2)/c^2]*EllipticF[ArcSin[Sqrt[1 - (d*x)/c]/Sqrt[2]], (2*c*f)/( 
d*e + c*f)])/(d*f*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2])
 

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 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], 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 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], 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. \(317\) vs. \(2(115)=230\).

Time = 1.74 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.32

method	result	size
default	\(-\frac {2 \left (2 \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c^{2} f^{2}-2 \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c d e f -\operatorname {EllipticE}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c^{2} f^{2}+\operatorname {EllipticE}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) d^{2} e^{2}\right ) \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, \sqrt {\frac {f \left (-d x +c \right )}{c f +d e}}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{f^{2} d \left (-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}\right )}\)	\(318\)
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\frac {2 c \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {2 d \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}\)	\(471\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53 \[ \int \frac {c+d x}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-d^{2} f} d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + \sqrt {-d^{2} f} {\left (d e - 3 \, c f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )}}{3 \, d^{2} f^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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