\(\int \frac {(e x)^{3/2}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [469]

Optimal result

Integrand size = 26, antiderivative size = 252 \[ \int \frac {(e x)^{3/2}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 e \sqrt {e x} \sqrt {c+d x}}{b d \sqrt {a+b x}}-\frac {2 \sqrt {a} (b c-2 a d) e^{3/2} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{b^{3/2} d (b c-a d) \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 a^{3/2} e^{3/2} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} (b c-a d) \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*e*(e*x)^(1/2)*(d*x+c)^(1/2)/b/d/(b*x+a)^(1/2)-2*a^(1/2)*(-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/(-a*d+b*c)/(b*x+a)^(1/2)/(a*(d*x+c)/c/ 
(b*x+a))^(1/2)-2*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)/(-a*d+b*c)/(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 = 6.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.98 \[ \int \frac {(e x)^{3/2}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 e^2 \left ((c+d x) \left (-2 a^2 d+b^2 c x+a b (c-d x)\right )-i \sqrt {\frac {a}{b}} b d (-b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d (-b c+a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{b^2 d (b c-a d) \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

(2*e^2*((c + d*x)*(-2*a^2*d + b^2*c*x + a*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*Ar 
cSinh[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)]))/(b^2*d*(b*c - a*d)*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt 
[c + d*x])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(2*a*e*Sqrt[e*x]*Sqrt[c + d*x])/(b*(b*c - a*d)*Sqrt[a + b*x]) - (e^2*((-2* 
Sqrt[-a]*(b*c - 2*a*d)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(S 
qrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[e]*Sq 
rt[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])))/(b*(b 
*c - a*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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 2.25 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.73

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+b c e x \right ) a e}{\left (a d -b c \right ) b^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 c^{2} e^{2} a \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b \left (a d -b c \right ) d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 \left (\frac {e^{2}}{b}+\frac {a d \,e^{2}}{b \left (a d -b c \right )}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(437\)
default	\(-\frac {2 \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3}+a b \,d^{3} x^{2}+a b c \,d^{2} x \right ) \sqrt {x d +c}\, \sqrt {b x +a}\, e \sqrt {e x}}{x \,d^{2} b^{2} \left (a d -b c \right ) \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(457\)

Input:

int((e*x)^(3/2)/(b*x+a)^(3/2)/(d*x+c)^(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*(b*d*e*x^2+b*c*e*x)/(a*d-b*c)/b^2*a*e/((x+a/b)*(b*d*e*x^2+b*c*e*x))^(1 
/2)+2/b*c^2*e^2/(a*d-b*c)*a/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)*Elliptic 
F(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(1/b*e^2+a*d/b*e^2/(a*d-b 
*c))*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (219) = 438\).

Time = 0.15 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.79 \[ \int \frac {(e x)^{3/2}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} a b^{2} d^{2} e - \sqrt {b d e} {\left ({\left (b^{3} c^{2} + 2 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e x + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 2 \, a^{3} d^{2}\right )} e\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) - 3 \, \sqrt {b d e} {\left ({\left (b^{3} c d - 2 \, a b^{2} d^{2}\right )} e x + {\left (a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e\right )} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3} + {\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} x\right )}} \] Input:

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

Output:

2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)*a*b^2*d^2*e - sqrt(b*d*e)*((b 
^3*c^2 + 2*a*b^2*c*d - 2*a^2*b*d^2)*e*x + (a*b^2*c^2 + 2*a^2*b*c*d - 2*a^3 
*d^2)*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*sqrt(b*d*e)*((b^3*c*d - 2*a*b^2*d^2)*e 
*x + (a*b^2*c*d - 2*a^2*b*d^2)*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))))/(a*b^4*c*d^2 - a^2*b^3*d^3 + 
(b^5*c*d^2 - a*b^4*d^3)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(e)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x)*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)*e