\(\int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx\) [442]

Optimal result

Integrand size = 26, antiderivative size = 217 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {e x} \sqrt {c+d x}}{e \sqrt {a+b x}}-\frac {2 \sqrt {a} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {a} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*(e*x)^(1/2)*(d*x+c)^(1/2)/e/(b*x+a)^(1/2)-2*a^(1/2)*(d*x+c)^(1/2)*Ellipt 
icE(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2)) 
/b^(1/2)/e^(1/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+2*a^(1/2)*(d*x+ 
c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)),(1-a* 
d/b/c)^(1/2))/b^(1/2)/e^(1/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 4.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=-\frac {2 \left (b+\frac {a}{x}\right ) \sqrt {x} \sqrt {c+d x} \left (-\sqrt {x}+\frac {c \sqrt {1+\frac {c}{d x}} E\left (\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{\sqrt {x}}\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {c}{d}} \sqrt {1+\frac {a}{b x}} \left (d+\frac {c}{x}\right )}\right )}{b \sqrt {e x} \sqrt {a+b x}} \] Input:

Integrate[Sqrt[c + d*x]/(Sqrt[e*x]*Sqrt[a + b*x]),x]
 

Output:

(-2*(b + a/x)*Sqrt[x]*Sqrt[c + d*x]*(-Sqrt[x] + (c*Sqrt[1 + c/(d*x)]*Ellip 
ticE[ArcSin[Sqrt[-(c/d)]/Sqrt[x]], (a*d)/(b*c)])/(Sqrt[-(c/d)]*Sqrt[1 + a/ 
(b*x)]*(d + c/x))))/(b*Sqrt[e*x]*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {122, 120}

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.

\(\displaystyle \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{\sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\)

\(\Big \downarrow \) 120

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

Input:

Int[Sqrt[c + d*x]/(Sqrt[e*x]*Sqrt[a + b*x]),x]
 

Output:

(2*Sqrt[-a]*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt 
[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*S 
qrt[1 + (d*x)/c])
 

Defintions of rubi rules used

rule 120

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

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {2 \sqrt {x d +c}\, \sqrt {b x +a}\, \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, c \left (a \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) d -\operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c -\operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d +\operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c \right )}{b d \sqrt {e x}\, \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(215\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 c^{2} \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{\sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(331\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - \sqrt {b d e} {\left (2 \, b c - a d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d e} \] Input:

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

Output:

-2/3*(3*sqrt(b*d*e)*b*d*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/ 
(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/( 
b^3*d^3), weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), 
 -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 
1/3*(3*b*d*x + b*c + a*d)/(b*d))) - sqrt(b*d*e)*(2*b*c - a*d)*weierstrassP 
Inverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3* 
a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a 
*d)/(b*d)))/(b^2*d*e)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{\sqrt {e x} \sqrt {a + b x}}\, dx \] Input:

integrate((d*x+c)**(1/2)/(e*x)**(1/2)/(b*x+a)**(1/2),x)
 

Output:

Integral(sqrt(c + d*x)/(sqrt(e*x)*sqrt(a + b*x)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c}}{\sqrt {b x + a} \sqrt {e x}} \,d x } \] Input:

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

Output:

integrate(sqrt(d*x + c)/(sqrt(b*x + a)*sqrt(e*x)), x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c}}{\sqrt {b x + a} \sqrt {e x}} \,d x } \] Input:

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

Output:

integrate(sqrt(d*x + c)/(sqrt(b*x + a)*sqrt(e*x)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c+d\,x}}{\sqrt {e\,x}\,\sqrt {a+b\,x}} \,d x \] Input:

int((c + d*x)^(1/2)/((e*x)^(1/2)*(a + b*x)^(1/2)),x)
 

Output:

int((c + d*x)^(1/2)/((e*x)^(1/2)*(a + b*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a +\sqrt {x}\, b x}d x \right )}{e} \] Input:

int((d*x+c)^(1/2)/(e*x)^(1/2)/(b*x+a)^(1/2),x)
 

Output:

(sqrt(e)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a + sqrt(x)*b*x),x))/e