\(\int \frac {\sqrt {c+d x}}{(e x)^{3/2} (a+b x)^{3/2}} \, dx\) [463]

Optimal result

Integrand size = 26, antiderivative size = 224 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \sqrt {c+d x}}{a e \sqrt {e x} \sqrt {a+b x}}-\frac {4 \sqrt {b} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} c e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

-2*(d*x+c)^(1/2)/a/e/(e*x)^(1/2)/(b*x+a)^(1/2)-4*b^(1/2)*(d*x+c)^(1/2)*Ell 
ipticE(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)/e^(3/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+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)/b^(1/2)/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 = 8.52 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} (a+b x)^{3/2}} \, dx=\frac {x \left (2 \sqrt {\frac {a}{b}} (c+d x)+4 i d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{a \sqrt {\frac {a}{b}} (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.31, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {110, 27, 169, 27, 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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*Sqrt[c + d*x])/(a*e*Sqrt[e*x]*Sqrt[a + b*x]) - ((4*b*Sqrt[e*x]*Sqrt[c 
+ d*x])/(a*e*Sqrt[a + b*x]) - (d*((4*Sqrt[-a]*Sqrt[b]*Sqrt[1 + (b*x)/a]*Sq 
rt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d 
)/(b*c)])/(d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (2*Sqrt[-a]*(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)/(a*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 110

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 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 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 = 2.68 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.22

method	result	size
default	\(-\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d +2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}+2 b \,d^{2} x^{2}+x a \,d^{2}+2 b c d x +a c d \right )}{e \sqrt {e x}\, d \,a^{2} \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(273\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}{e^{2} a^{2} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}-\frac {2 \left (b d e \,x^{2}+b c e x \right )}{e^{2} a^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (-\frac {a d +b c}{a^{2} e}+\frac {a d e +b c e}{e^{2} a^{2}}+\frac {a d -b c}{a^{2} e}+\frac {b c}{e \,a^{2}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {4 b c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{a^{2} e \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(492\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (191) = 382\).

Time = 0.10 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (2 \, b^{2} d x + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} + \sqrt {b d e} {\left ({\left (2 \, b^{2} c - a b d\right )} x^{2} + {\left (2 \, a b c - a^{2} d\right )} x\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left (b^{2} d x^{2} + a b d x\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (a^{2} b^{2} d e^{2} x^{2} + a^{3} b d e^{2} x\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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