\(\int \frac {\sqrt {x} \sqrt {c+d x}}{(a+e x)^{3/2}} \, dx\) [467]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+e x)^{3/2}} \, dx=\frac {2 \sqrt {x} \sqrt {c+d x}}{e \sqrt {a+e x}}-\frac {4 \sqrt {a} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{c e}\right )}{e^{3/2} \sqrt {\frac {a (c+d x)}{c (a+e x)}} \sqrt {a+e x}}+\frac {2 \sqrt {a} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{c e}\right )}{e^{3/2} \sqrt {\frac {a (c+d x)}{c (a+e x)}} \sqrt {a+e x}} \] Output:

2*x^(1/2)*(d*x+c)^(1/2)/e/(e*x+a)^(1/2)-4*a^(1/2)*(d*x+c)^(1/2)*EllipticE( 
e^(1/2)*x^(1/2)/a^(1/2)/(1+e*x/a)^(1/2),(1-a*d/c/e)^(1/2))/e^(3/2)/(a*(d*x 
+c)/c/(e*x+a))^(1/2)/(e*x+a)^(1/2)+2*a^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM 
(arctan(e^(1/2)*x^(1/2)/a^(1/2)),(1-a*d/c/e)^(1/2))/e^(3/2)/(a*(d*x+c)/c/( 
e*x+a))^(1/2)/(e*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.52 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+e x)^{3/2}} \, dx=\frac {2 a (c+d x) (2 a+e x)+4 i a d \sqrt {\frac {a}{e}} e \sqrt {1+\frac {c}{d x}} \sqrt {1+\frac {a}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{e}}}{\sqrt {x}}\right )|\frac {c e}{a d}\right )-2 i \sqrt {\frac {a}{e}} e (2 a d-c e) \sqrt {1+\frac {c}{d x}} \sqrt {1+\frac {a}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{e}}}{\sqrt {x}}\right ),\frac {c e}{a d}\right )}{a e^2 \sqrt {x} \sqrt {c+d x} \sqrt {a+e x}} \] Input:

Integrate[(Sqrt[x]*Sqrt[c + d*x])/(a + e*x)^(3/2),x]
 

Output:

(2*a*(c + d*x)*(2*a + e*x) + (4*I)*a*d*Sqrt[a/e]*e*Sqrt[1 + c/(d*x)]*Sqrt[ 
1 + a/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[a/e]/Sqrt[x]], (c*e)/(a*d)] 
- (2*I)*Sqrt[a/e]*e*(2*a*d - c*e)*Sqrt[1 + c/(d*x)]*Sqrt[1 + a/(e*x)]*x^(3 
/2)*EllipticF[I*ArcSinh[Sqrt[a/e]/Sqrt[x]], (c*e)/(a*d)])/(a*e^2*Sqrt[x]*S 
qrt[c + d*x]*Sqrt[a + e*x])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {108, 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.

\(\displaystyle \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[(Sqrt[x]*Sqrt[c + d*x])/(a + e*x)^(3/2),x]
 

Output:

(-2*Sqrt[x]*Sqrt[c + d*x])/(e*Sqrt[a + e*x]) + ((4*Sqrt[-a]*Sqrt[c + d*x]* 
Sqrt[1 + (e*x)/a]*EllipticE[ArcSin[(Sqrt[e]*Sqrt[x])/Sqrt[-a]], (a*d)/(c*e 
)])/(Sqrt[e]*Sqrt[1 + (d*x)/c]*Sqrt[a + e*x]) - (2*Sqrt[-c]*c*Sqrt[1 + (d* 
x)/c]*Sqrt[1 + (e*x)/a]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[x])/Sqrt[-c]], (c*e 
)/(a*d)])/(Sqrt[d]*Sqrt[c + d*x]*Sqrt[a + e*x]))/e
 

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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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 = 0.66 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.43

method	result	size
default	\(-\frac {2 \sqrt {x d +c}\, \sqrt {e x +a}\, \left (\sqrt {\frac {e x +a}{a}}\, \sqrt {-\frac {\left (x d +c \right ) e}{a d -c e}}\, \sqrt {-\frac {e x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +a}{a}}, \sqrt {\frac {a d}{a d -c e}}\right ) a c e +2 \sqrt {\frac {e x +a}{a}}\, \sqrt {-\frac {\left (x d +c \right ) e}{a d -c e}}\, \sqrt {-\frac {e x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +a}{a}}, \sqrt {\frac {a d}{a d -c e}}\right ) a^{2} d -2 \sqrt {\frac {e x +a}{a}}\, \sqrt {-\frac {\left (x d +c \right ) e}{a d -c e}}\, \sqrt {-\frac {e x}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +a}{a}}, \sqrt {\frac {a d}{a d -c e}}\right ) a c e +e^{2} d \,x^{2}+x c \,e^{2}\right )}{\sqrt {x}\, \left (d e \,x^{2}+a d x +c e x +a c \right ) e^{3}}\)	\(274\)
elliptic	\(\frac {\sqrt {\left (e x +a \right ) x \left (x d +c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e x \right )}{e^{2} \sqrt {\left (x +\frac {a}{e}\right ) \left (d e \,x^{2}+c e x \right )}}+\frac {2 c a \sqrt {\frac {\left (x +\frac {a}{e}\right ) e}{a}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a}{e}+\frac {c}{d}}}\, \sqrt {-\frac {e x}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{e}\right ) e}{a}}, \sqrt {-\frac {a}{e \left (-\frac {a}{e}+\frac {c}{d}\right )}}\right )}{e^{2} \sqrt {d e \,x^{3}+a d \,x^{2}+c e \,x^{2}+a c x}}+\frac {4 d a \sqrt {\frac {\left (x +\frac {a}{e}\right ) e}{a}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a}{e}+\frac {c}{d}}}\, \sqrt {-\frac {e x}{a}}\, \left (\left (-\frac {a}{e}+\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{e}\right ) e}{a}}, \sqrt {-\frac {a}{e \left (-\frac {a}{e}+\frac {c}{d}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{e}\right ) e}{a}}, \sqrt {-\frac {a}{e \left (-\frac {a}{e}+\frac {c}{d}\right )}}\right )}{d}\right )}{e^{2} \sqrt {d e \,x^{3}+a d \,x^{2}+c e \,x^{2}+a c x}}\right )}{\sqrt {e x +a}\, \sqrt {x}\, \sqrt {x d +c}}\)	\(360\)

Input:

int(x^(1/2)*(d*x+c)^(1/2)/(e*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (166) = 332\).

Time = 0.13 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+e x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d x + c} \sqrt {e x + a} d e^{2} \sqrt {x} + {\left (2 \, a^{2} d - a c e + {\left (2 \, a d e - c e^{2}\right )} x\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a c d e + c^{2} e^{2}\right )}}{3 \, d^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} c d^{2} e - 3 \, a c^{2} d e^{2} + 2 \, c^{3} e^{3}\right )}}{27 \, d^{3} e^{3}}, \frac {3 \, d e x + a d + c e}{3 \, d e}\right ) + 6 \, {\left (d e^{2} x + a d e\right )} \sqrt {d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a c d e + c^{2} e^{2}\right )}}{3 \, d^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} c d^{2} e - 3 \, a c^{2} d e^{2} + 2 \, c^{3} e^{3}\right )}}{27 \, d^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a c d e + c^{2} e^{2}\right )}}{3 \, d^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} c d^{2} e - 3 \, a c^{2} d e^{2} + 2 \, c^{3} e^{3}\right )}}{27 \, d^{3} e^{3}}, \frac {3 \, d e x + a d + c e}{3 \, d e}\right )\right )\right )}}{3 \, {\left (d e^{4} x + a d e^{3}\right )}} \] Input:

integrate(x^(1/2)*(d*x+c)^(1/2)/(e*x+a)^(3/2),x, algorithm="fricas")
 

Output:

-2/3*(3*sqrt(d*x + c)*sqrt(e*x + a)*d*e^2*sqrt(x) + (2*a^2*d - a*c*e + (2* 
a*d*e - c*e^2)*x)*sqrt(d*e)*weierstrassPInverse(4/3*(a^2*d^2 - a*c*d*e + c 
^2*e^2)/(d^2*e^2), -4/27*(2*a^3*d^3 - 3*a^2*c*d^2*e - 3*a*c^2*d*e^2 + 2*c^ 
3*e^3)/(d^3*e^3), 1/3*(3*d*e*x + a*d + c*e)/(d*e)) + 6*(d*e^2*x + a*d*e)*s 
qrt(d*e)*weierstrassZeta(4/3*(a^2*d^2 - a*c*d*e + c^2*e^2)/(d^2*e^2), -4/2 
7*(2*a^3*d^3 - 3*a^2*c*d^2*e - 3*a*c^2*d*e^2 + 2*c^3*e^3)/(d^3*e^3), weier 
strassPInverse(4/3*(a^2*d^2 - a*c*d*e + c^2*e^2)/(d^2*e^2), -4/27*(2*a^3*d 
^3 - 3*a^2*c*d^2*e - 3*a*c^2*d*e^2 + 2*c^3*e^3)/(d^3*e^3), 1/3*(3*d*e*x + 
a*d + c*e)/(d*e))))/(d*e^4*x + a*d*e^3)
 

Sympy [F]

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

integrate(x**(1/2)*(d*x+c)**(1/2)/(e*x+a)**(3/2),x)
 

Output:

Integral(sqrt(x)*sqrt(c + d*x)/(a + e*x)**(3/2), x)
 

Maxima [F]

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

integrate(x^(1/2)*(d*x+c)^(1/2)/(e*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x + c)*sqrt(x)/(e*x + a)^(3/2), x)
 

Giac [F]

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

integrate(x^(1/2)*(d*x+c)^(1/2)/(e*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x + c)*sqrt(x)/(e*x + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((x^(1/2)*(c + d*x)^(1/2))/(a + e*x)^(3/2),x)
 

Output:

int((x^(1/2)*(c + d*x)^(1/2))/(a + e*x)^(3/2), x)
 

Reduce [F]

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

int(x^(1/2)*(d*x+c)^(1/2)/(e*x+a)^(3/2),x)
 

Output:

(2*sqrt(x)*sqrt(c + d*x)*sqrt(a + e*x)*c - int((sqrt(c + d*x)*sqrt(a + e*x 
))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*c*e*x + 2*sqrt(x)*a*d* 
e*x**2 + sqrt(x)*c*e**2*x**2 + sqrt(x)*d*e**2*x**3),x)*a**2*c**2 - int((sq 
rt(c + d*x)*sqrt(a + e*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)* 
a*c*e*x + 2*sqrt(x)*a*d*e*x**2 + sqrt(x)*c*e**2*x**2 + sqrt(x)*d*e**2*x**3 
),x)*a*c**2*e*x + 2*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + e*x)*x)/(a**2*c + 
a**2*d*x + 2*a*c*e*x + 2*a*d*e*x**2 + c*e**2*x**2 + d*e**2*x**3),x)*a**2*d 
**2 - int((sqrt(x)*sqrt(c + d*x)*sqrt(a + e*x)*x)/(a**2*c + a**2*d*x + 2*a 
*c*e*x + 2*a*d*e*x**2 + c*e**2*x**2 + d*e**2*x**3),x)*a*c*d*e + 2*int((sqr 
t(x)*sqrt(c + d*x)*sqrt(a + e*x)*x)/(a**2*c + a**2*d*x + 2*a*c*e*x + 2*a*d 
*e*x**2 + c*e**2*x**2 + d*e**2*x**3),x)*a*d**2*e*x - int((sqrt(x)*sqrt(c + 
 d*x)*sqrt(a + e*x)*x)/(a**2*c + a**2*d*x + 2*a*c*e*x + 2*a*d*e*x**2 + c*e 
**2*x**2 + d*e**2*x**3),x)*c*d*e**2*x)/(2*a*d*(a + e*x))