\(\int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {a+b x}} \, dx\) [443]

Optimal result

Integrand size = 26, antiderivative size = 221 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {c+d x}}{e \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {b} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {a} d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

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

Mathematica [A] (verified)

Time = 4.54 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {1+\frac {a}{b x}} x^{3/2} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )}{\sqrt {-\frac {a}{b}} \sqrt {1+\frac {c}{d x}} (e x)^{3/2} \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {110, 27, 122, 120}

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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

Input:

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

Output:

(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*e*Sqrt[e*x]) + (2*Sqrt[-c]*Sqrt[d]*Sqr 
t[a + b*x]*Sqrt[1 + (d*x)/c]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[e*x])/(Sqrt[-c 
]*Sqrt[e])], (b*c)/(a*d)])/(a*e^(3/2)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x])
 

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])
 

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {2 \sqrt {x d +c}\, \sqrt {b x +a}\, \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}-b \,d^{2} x^{2}-x a \,d^{2}-b c d x -a c d \right )}{a d e \sqrt {e x}\, \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(223\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}{e^{2} a \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (\frac {d}{e}-\frac {a d +b c}{a e}+\frac {a d e +b c e}{e^{2} a}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 b c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{a e \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(429\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {c+d x}}{(e x)^{3/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} b d x {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b d + \sqrt {b d e} {\left (b c - 2 \, a d\right )} x {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, a b d e^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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