\(\int \frac {1}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [472]

Optimal result

Integrand size = 26, antiderivative size = 259 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{a c e \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {b} (2 b c-a d) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} c (b c-a d) e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {b} d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {a} c (b c-a d) e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

-2*(d*x+c)^(1/2)/a/c/e/(e*x)^(1/2)/(b*x+a)^(1/2)-2*b^(1/2)*(-a*d+2*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/(-a*d+b*c)/e^(3/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c 
/(b*x+a))^(1/2)+2*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/(-a*d+b*c)/e^(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 = 5.91 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {x \left (-2 a b c (c+d x)+2 i \sqrt {\frac {a}{b}} b d (-2 b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b d (-b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{a^2 c (-b c+a d) (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

Integrate[1/((e*x)^(3/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.30, 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.

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[1/((e*x)^(3/2)*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

(-2*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]*Sqrt[a + b*x]) - (b*((2*(2*b*c - a*d)* 
Sqrt[e*x]*Sqrt[c + d*x])/(a*(b*c - a*d)*e*Sqrt[a + b*x]) - (d*((2*Sqrt[-a] 
*(2*b*c - a*d)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*S 
qrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[e]*Sqrt[a + b 
*x]*Sqrt[1 + (d*x)/c]) - (4*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])))/(a*(b*c - a*d 
))))/(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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(475\) vs. \(2(226)=452\).

Time = 3.44 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.84

method	result	size
default	\(-\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}-\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3}+a b \,d^{3} x^{2}-2 b^{2} c \,d^{2} x^{2}+a^{2} d^{3} x -2 b^{2} c^{2} d x +c \,a^{2} d^{2}-a b \,c^{2} d \right )}{e \sqrt {e x}\, \left (a d -b c \right ) d \,a^{2} c \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(476\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}{e^{2} a^{2} c \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (b d e \,x^{2}+b c e x \right ) b}{\left (a d -b c \right ) a^{2} e^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (-\frac {a d +b c}{a^{2} c e}+\frac {a d e +b c e}{e^{2} a^{2} c}-\frac {b}{a^{2} e}-\frac {b^{2} c}{e \left (a d -b c \right ) a^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (\frac {d b}{a^{2} c e}-\frac {d \,b^{2}}{\left (a d -b c \right ) a^{2} e}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(550\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (226) = 452\).

Time = 0.09 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(e x)^{3/2} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, {\left (a b^{2} c d - a^{2} b d^{2} + {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} + \sqrt {b d e} {\left ({\left (2 \, b^{3} c^{2} - 2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d - a^{3} d^{2}\right )} x\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, \sqrt {b d e} {\left ({\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x\right )} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left ({\left (a^{2} b^{3} c^{2} d - a^{3} b^{2} c d^{2}\right )} e^{2} x^{2} + {\left (a^{3} b^{2} c^{2} d - a^{4} b c d^{2}\right )} e^{2} x\right )}} \] Input:

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

Output:

-2/3*(3*(a*b^2*c*d - a^2*b*d^2 + (2*b^3*c*d - a*b^2*d^2)*x)*sqrt(b*x + a)* 
sqrt(d*x + c)*sqrt(e*x) + sqrt(b*d*e)*((2*b^3*c^2 - 2*a*b^2*c*d - a^2*b*d^ 
2)*x^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d - a^3*d^2)*x)*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*sqrt(b*d*e)*((2*b^3*c*d - a*b^2*d^2)*x^2 + (2*a*b^2*c*d - a^2*b*d^2)*x) 
*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), weierstrassPI 
nverse(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))))/((a^2*b^3*c^2*d - a^3*b^2*c*d^2)*e^2*x^2 + (a^3*b^2*c^2*d - a^ 
4*b*c*d^2)*e^2*x)
 

Sympy [F]

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

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

Output:

Integral(1/((e*x)**(3/2)*(a + b*x)**(3/2)*sqrt(c + d*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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