\(\int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx\) [455]

Optimal result

Integrand size = 26, antiderivative size = 288 \[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {4 (b c+a d) \sqrt {c+d x}}{3 a c^2 e^2 \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}+\frac {4 \sqrt {b} (b c+a d) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 \sqrt {b} d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 e^{5/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

4/3*(a*d+b*c)*(d*x+c)^(1/2)/a/c^2/e^2/(e*x)^(1/2)/(b*x+a)^(1/2)-2/3*(b*x+a 
)^(1/2)*(d*x+c)^(1/2)/a/c/e/(e*x)^(3/2)+4/3*b^(1/2)*(a*d+b*c)*(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))/a^(3/2)/c^2/e^(5/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)-2/3 
*b^(1/2)*d*(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))/a^(1/2)/c^2/e^(5/2)/(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 = 4.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {x \left (-2 a c (a+b x) (c+d x)-4 i \sqrt {\frac {a}{b}} b d (b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d (b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{3 a^2 c^2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
 

Output:

(x*(-2*a*c*(a + b*x)*(c + d*x) - (4*I)*Sqrt[a/b]*b*d*(b*c + a*d)*Sqrt[1 + 
a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], 
 (b*c)/(a*d)] + (2*I)*Sqrt[a/b]*b*d*(b*c + 2*a*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 
 + c/(d*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])) 
/(3*a^2*c^2*(e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {115, 27, 169, 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.

\(\displaystyle \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

\(\displaystyle -\frac {-\frac {2 \int -\frac {b d e (a c+2 (b c+a d) x)}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e^2}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b d \int \frac {a c+2 (b c+a d) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle -\frac {\frac {b d \left (\frac {2 (a d+b c) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 122

\(\displaystyle -\frac {\frac {b d \left (\frac {2 \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 120

\(\displaystyle -\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle -\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (a d+2 b c) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

\(\Big \downarrow \) 126

\(\displaystyle -\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (a d+2 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c e (e x)^{3/2}}\)

Input:

Int[1/((e*x)^(5/2)*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
 

Output:

(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)) - ((-4*(b*c + a*d)* 
Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]) + (b*d*((4*Sqrt[-a]*(b*c + 
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*(2*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])))/(a*c*e))/(3*a*c*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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 3.18 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {2 \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x +\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d x -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x +2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3} x +2 a b \,d^{3} x^{3}+2 b^{2} c \,d^{2} x^{3}+2 a^{2} d^{3} x^{2}+3 a b c \,d^{2} x^{2}+2 b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2} x +a b \,c^{2} d x -a^{2} c^{2} d \right ) \sqrt {x d +c}\, \sqrt {b x +a}}{3 x \,c^{2} a^{2} d \left (b d \,x^{2}+a d x +b c x +a c \right ) e^{2} \sqrt {e x}}\)	\(445\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 e^{3} a c \,x^{2}}+\frac {4 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (a d +b c \right )}{3 e^{3} a^{2} c^{2} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (-\frac {d b}{3 a c \,e^{2}}+\frac {2 \left (a d +b c \right )^{2}}{3 a^{2} c^{2} e^{2}}-\frac {2 \left (a d e +b c e \right ) \left (a d +b c \right )}{3 e^{3} a^{2} c^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}-\frac {4 b \left (a d +b c \right ) \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{3 a^{2} c \,e^{2} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(515\)

Input:

int(1/(e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \, {\left ({\left (2 \, b^{2} c^{2} + a b c d + 2 \, a^{2} d^{2}\right )} \sqrt {b d e} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {b d e} x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (a b c d - 2 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{9 \, a^{2} b c^{2} d e^{3} x^{2}} \] Input:

integrate(1/(e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

2/9*((2*b^2*c^2 + a*b*c*d + 2*a^2*d^2)*sqrt(b*d*e)*x^2*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)) + 6*(b^2*c*d + a*b*d^2)*sqrt(b*d*e)*x^2*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))) - 3*(a*b*c*d - 2*(b^ 
2*c*d + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x))/(a^2*b*c^2*d*e^ 
3*x^2)
 

Sympy [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \sqrt {a + b x} \sqrt {c + d x}}\, dx \] Input:

integrate(1/(e*x)**(5/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
 

Output:

Integral(1/((e*x)**(5/2)*sqrt(a + b*x)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*(e*x)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*(e*x)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/((e*x)^(5/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((e*x)^(5/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{5}+a d \,x^{4}+b c \,x^{4}+a c \,x^{3}}d x \right )}{e^{3}} \] Input:

int(1/(e*x)^(5/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
 

Output:

(sqrt(e)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*x**3 + a*d*x**4 + 
b*c*x**4 + b*d*x**5),x))/e**3