\(\int \frac {\sqrt {e x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [470]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int \frac {\sqrt {e x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {a} \sqrt {e} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} (b c-a d) \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {a} \sqrt {e} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} (b c-a d) \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

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

Mathematica [A] (verified)

Time = 4.47 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {e x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 e \left (a \sqrt {-\frac {c}{d}} \sqrt {1+\frac {a}{b x}} (c+d x)-c \sqrt {1+\frac {c}{d x}} \sqrt {x} (a+b x) E\left (\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{\sqrt {x}}\right )|\frac {a d}{b c}\right )\right )}{b \sqrt {-\frac {c}{d}} (b c-a d) \sqrt {1+\frac {a}{b x}} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.70, 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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

Input:

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

Output:

(-2*Sqrt[e*x]*Sqrt[c + d*x])/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[-a]*Sqr 
t[e]*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]*(b*c - a*d)*Sqrt[a + b*x]*Sqrt 
[1 + (d*x)/c])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(180)=360\).

Time = 1.95 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}-\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d +\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}+b \,d^{2} x^{2}+b c d x \right ) \sqrt {x d +c}\, \sqrt {b x +a}\, \sqrt {e x}}{d b \left (a d -b c \right ) x \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(363\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 b d e \,x^{2}+2 b c e x}{\left (a d -b c \right ) b \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}-\frac {2 c^{2} e \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{\left (a d -b c \right ) d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}-\frac {2 e c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{e x \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(410\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (180) = 360\).

Time = 0.09 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {e x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b^{2} d - {\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (b^{2} d x + a b d\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (a b^{3} c d - a^{2} b^{2} d^{2} + {\left (b^{4} c d - a b^{3} d^{2}\right )} x\right )}} \] Input:

integrate((e*x)^(1/2)/(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)*b^2*d - (2*a*b*c - a^2*d + ( 
2*b^2*c - a*b*d)*x)*sqrt(b*d*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*(b^2*d*x + a*b* 
d)*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*b^3*c*d - a^2*b^2*d^2 + (b^4*c*d - a*b^3*d^2) 
*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((e*x)^(1/2)/(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))/(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)