\(\int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx\) [454]

Optimal result

Integrand size = 26, antiderivative size = 170 \[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 b \sqrt {e x} \sqrt {c+d x}}{a c e^2 \sqrt {a+b x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c e \sqrt {e x}}-\frac {2 \sqrt {b} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} c e^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*b*(e*x)^(1/2)*(d*x+c)^(1/2)/a/c/e^2/(b*x+a)^(1/2)-2*(b*x+a)^(1/2)*(d*x+c 
)^(1/2)/a/c/e/(e*x)^(1/2)-2*b^(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))/a^(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 = 2.82 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 x \left ((a+b x) (c+d x)+i a c \sqrt {\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right )|\frac {a d}{b c}\right )-i a c \sqrt {\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right ),\frac {a d}{b c}\right )\right )}{a c (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {115, 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.

\(\displaystyle \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*e*Sqrt[e*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*c*e^2*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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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 [A] (verified)

Time = 2.15 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.72

method	result	size
default	\(-\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d +\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}+b \,d^{2} x^{2}+x a \,d^{2}+b c d x +a c d \right ) \sqrt {x d +c}\, \sqrt {b x +a}}{c a d \left (b d \,x^{2}+a d x +b c x +a c \right ) e \sqrt {e x}}\)	\(293\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}{e^{2} a c \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (-\frac {a d +b c}{a c e}+\frac {a d e +b c e}{e^{2} a c}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 b \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{a e \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(432\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (149) = 298\).

Time = 0.08 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} b d x {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + 3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b d + \sqrt {b d e} {\left (b c + a d\right )} x {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, a b c d e^{2} x} \] Input:

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

Sympy [F]

\[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \sqrt {a + b x} \sqrt {c + d x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}+\left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{2}+a d x +b c x +a c}d x \right ) b d x \right )}{a c \,e^{2} x} \] Input:

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

Output:

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