\(\int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x)^{3/2}} \, dx\) [462]

Optimal result

Integrand size = 26, antiderivative size = 93 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x)^{3/2}} \, dx=\frac {2 \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

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))/a^(1/2)/b^(1/2)/e^(1/2)/(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 = 4.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x)^{3/2}} \, dx=\frac {2 \sqrt {\frac {b x}{a}} \left (\sqrt {\frac {b x}{a}} (c+d x)+i c \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right )|\frac {a d}{b c}\right )-i c \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right ),\frac {a d}{b c}\right )\right )}{b \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

(2*Sqrt[(b*x)/a]*(Sqrt[(b*x)/a]*(c + d*x) + I*c*Sqrt[1 + (b*x)/a]*Sqrt[1 + 
 (d*x)/c]*EllipticE[I*ArcSinh[Sqrt[(b*x)/a]], (a*d)/(b*c)] - I*c*Sqrt[1 + 
(b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[I*ArcSinh[Sqrt[(b*x)/a]], (a*d)/(b*c) 
]))/(b*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.62, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {110, 8, 27, 124, 123}

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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 8

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m   Int[u*(a* 
x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
 

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 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(279\) vs. \(2(84)=168\).

Time = 1.69 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.01

method	result	size
default	\(\frac {2 \sqrt {x d +c}\, \sqrt {b x +a}\, \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d +\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}+b \,d^{2} x^{2}+b c d x \right )}{b a d \sqrt {e x}\, \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(280\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 b d e \,x^{2}+2 b c e x}{a e b \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (\frac {d}{b}-\frac {a d -b c}{b a}-\frac {c}{a}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}-\frac {2 c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{a \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(407\)

Input:

int((d*x+c)^(1/2)/(e*x)^(1/2)/(b*x+a)^(3/2),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)*EllipticF(((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))*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)/b/a/d/(e*x)^(1/2)/(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. 361 vs. \(2 (84) = 168\).

Time = 0.07 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.88 \[ \int \frac {\sqrt {c+d x}}{\sqrt {e x} (a+b x)^{3/2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b^{2} d + {\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (b^{2} d x + a b d\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (a b^{3} d e x + a^{2} b^{2} d e\right )}} \] Input:

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

Output:

2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)*b^2*d + (a*b*c + a^2*d + (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)*sq 
rt(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), weie 
rstrassPInverse(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*d*e*x + a^2*b^2*d*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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