\(\int \frac {\sqrt {e x} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx\) [441]

Optimal result

Integrand size = 26, antiderivative size = 277 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 (b c-2 a d) \sqrt {e x} \sqrt {c+d x}}{3 b d \sqrt {a+b x}}+\frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {2 \sqrt {a} (b c-2 a d) \sqrt {e} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 a^{3/2} \sqrt {e} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2/3*(-2*a*d+b*c)*(e*x)^(1/2)*(d*x+c)^(1/2)/b/d/(b*x+a)^(1/2)+2/3*(e*x)^(1/ 
2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b-2/3*a^(1/2)*(-2*a*d+b*c)*e^(1/2)*(d*x+c)^ 
(1/2)*EllipticE(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^(3/2)/d/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)-2/3*a^(3/ 
2)*e^(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^(3/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^ 
(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.04 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 e \left (-((a+b x) (c+d x) (2 a d-b (c+d x)))-i \sqrt {\frac {a}{b}} b d (-b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d (-b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{3 b^2 d \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

(2*e*(-((a + b*x)*(c + d*x)*(2*a*d - b*(c + d*x))) - I*Sqrt[a/b]*b*d*(-(b* 
c) + 2*a*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticE[I*ArcSin 
h[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] + (2*I)*Sqrt[a/b]*b*d*(-(b*c) + a*d)*Sq 
rt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sq 
rt[x]], (b*c)/(a*d)]))/(3*b^2*d*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {112, 27, 176, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {2 \int \frac {e (a c-(b c-2 a d) x)}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \int \frac {a c-(b c-2 a d) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 b}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \left (\frac {c (b c-a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {(b c-2 a d) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}\right )}{3 b}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \left (\frac {c (b c-a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \left (\frac {c (b c-a d) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \left (\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b}-\frac {e \left (\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (b c-2 a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}\right )}{3 b}\)

Input:

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

Output:

(2*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*b) - (e*((-2*Sqrt[-a]*(b*c - 
2*a*d)*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]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt 
[1 + (d*x)/c]) + (2*Sqrt[-a]*c*(b*c - a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x 
)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c) 
])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[c + d*x])))/(3*b)
 

Defintions of rubi rules used

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 112

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

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])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[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] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

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

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]
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.43

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 b}-\frac {2 a \,c^{2} e \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{3 b d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (c e -\frac {2 \left (a d e +b c e \right )}{3 b}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(397\)
default	\(\frac {2 \sqrt {e x}\, \sqrt {x d +c}\, \sqrt {b x +a}\, \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3}+b^{2} d^{3} x^{3}+a b \,d^{3} x^{2}+b^{2} c \,d^{2} x^{2}+a b c \,d^{2} x \right )}{3 x \left (b d \,x^{2}+a d x +b c x +a c \right ) b^{2} d^{2}}\)	\(467\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b^{2} d^{2} - {\left (b^{2} c^{2} + 2 \, a b c d - 2 \, a^{2} d^{2}\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) - 3 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{9 \, b^{3} d^{2}} \] Input:

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

Output:

2/9*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)*b^2*d^2 - (b^2*c^2 + 2*a*b*c* 
d - 2*a^2*d^2)*sqrt(b*d*e)*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)) - 3*(b^2*c*d - 2*a*b*d^2 
)*sqrt(b*d*e)*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))))/(b^3*d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {e x} \sqrt {c+d x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, c +2 \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{2} d^{3}+\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a b c \,d^{2}-\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}\, x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) b^{2} c^{2} d -\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{a b \,d^{2} x^{3}+b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+2 a b c d \,x^{2}+b^{2} c^{2} x^{2}+a^{2} c d x +a b \,c^{2} x}d x \right ) a^{2} c^{2} d -\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{a b \,d^{2} x^{3}+b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+2 a b c d \,x^{2}+b^{2} c^{2} x^{2}+a^{2} c d x +a b \,c^{2} x}d x \right ) a b \,c^{3}\right )}{2 a d +2 b c} \] Input:

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

Output:

(sqrt(e)*(2*sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*c + 2*int((sqrt(x)*sqrt(c 
+ d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + 
 a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a**2*d**3 + int((sqrt(x)* 
sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b* 
c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a*b*c*d**2 - int(( 
sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 
+ 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*b**2*c**2* 
d - int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a**2*c*d*x + a**2*d**2*x**2 
 + a*b*c**2*x + 2*a*b*c*d*x**2 + a*b*d**2*x**3 + b**2*c**2*x**2 + b**2*c*d 
*x**3),x)*a**2*c**2*d - int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a**2*c* 
d*x + a**2*d**2*x**2 + a*b*c**2*x + 2*a*b*c*d*x**2 + a*b*d**2*x**3 + b**2* 
c**2*x**2 + b**2*c*d*x**3),x)*a*b*c**3))/(2*(a*d + b*c))