\(\int \frac {(e x)^{3/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\) [451]

Optimal result

Integrand size = 26, antiderivative size = 283 \[ \int \frac {(e x)^{3/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {4 (b c+a d) e \sqrt {e x} \sqrt {c+d x}}{3 b d^2 \sqrt {a+b x}}+\frac {2 e \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}{3 b d}+\frac {4 \sqrt {a} (b c+a d) e^{3/2} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 a^{3/2} e^{3/2} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} d \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

-4/3*(a*d+b*c)*e*(e*x)^(1/2)*(d*x+c)^(1/2)/b/d^2/(b*x+a)^(1/2)+2/3*e*(e*x) 
^(1/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d+4/3*a^(1/2)*(a*d+b*c)*e^(3/2)*(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^2/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)-2/3* 
a^(3/2)*e^(3/2)*(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)/d/(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 = 8.60 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^{3/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 e^2 \left ((a+b x) (c+d x) (2 b c+2 a d-b d x)+2 i \sqrt {\frac {a}{b}} b d (b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-i \sqrt {\frac {a}{b}} b d (b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{3 b^2 d^2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 267, 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 = {113, 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.

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

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

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 113

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 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 = 1.14 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {2 \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3}+b^{2} d^{3} x^{3}+a b \,d^{3} x^{2}+b^{2} c \,d^{2} x^{2}+a b c \,d^{2} x \right ) \sqrt {x d +c}\, \sqrt {b x +a}\, e \sqrt {e x}}{3 x \,d^{3} b^{2} \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(392\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 e \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{3 b d}-\frac {2 e^{2} a \,c^{2} \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{3 b \,d^{2} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}-\frac {4 e \left (a d e +b c e \right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{3 b \,d^{2} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(398\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.25 \[ \int \frac {(e x)^{3/2}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b^{2} d^{2} e + {\left (2 \, b^{2} c^{2} + a b c d + 2 \, a^{2} d^{2}\right )} \sqrt {b d e} e {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left (b^{2} c d + a b d^{2}\right )} \sqrt {b d e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{9 \, b^{3} d^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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