\(\int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx\) [210]

Optimal result

Integrand size = 33, antiderivative size = 206 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=-\frac {2 B \sqrt {c} \sqrt {a-b x} \sqrt {1-\frac {d x}{c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )|\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {e} \sqrt {1-\frac {b x}{a}} \sqrt {c-d x}}+\frac {2 \sqrt {a} (A b+a B) \sqrt {1-\frac {b x}{a}} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{b^{3/2} \sqrt {e} \sqrt {a-b x} \sqrt {c-d x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=\frac {2 \sqrt {-\frac {a}{b}} B (a-b x) (c-d x)-2 i a B d \sqrt {1-\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i (A b+a B) d \sqrt {1-\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{\sqrt {-\frac {a}{b}} b d \sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \] Input:

Integrate[(A + B*x)/(Sqrt[e*x]*Sqrt[a - b*x]*Sqrt[c - d*x]),x]
 

Output:

(2*Sqrt[-(a/b)]*B*(a - b*x)*(c - d*x) - (2*I)*a*B*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)*(A*b + a*B)*d*Sqrt[1 - a/(b*x)]*Sqrt[1 - c/(d*x)]*x^(3/2)*Elli 
pticF[I*ArcSinh[Sqrt[-(a/b)]/Sqrt[x]], (b*c)/(a*d)])/(Sqrt[-(a/b)]*b*d*Sqr 
t[e*x]*Sqrt[a - b*x]*Sqrt[c - d*x])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {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.

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[(A + B*x)/(Sqrt[e*x]*Sqrt[a - b*x]*Sqrt[c - d*x]),x]
 

Output:

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

Defintions of rubi rules used

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 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 = 1.34 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.10

method	result	size
default	\(-\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {-x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b d +B \operatorname {EllipticF}\left (\sqrt {\frac {-x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d -B \operatorname {EllipticE}\left (\sqrt {\frac {-x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d +B \operatorname {EllipticE}\left (\sqrt {\frac {-x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c \right ) c \sqrt {\frac {x d}{c}}\, \sqrt {\frac {\left (-b x +a \right ) d}{a d -b c}}\, \sqrt {\frac {-x d +c}{c}}\, \sqrt {-x d +c}\, \sqrt {-b x +a}}{b \,d^{2} \left (b d \,x^{2}-a d x -b c x +a c \right ) \sqrt {e x}}\)	\(227\)
elliptic	\(\frac {\sqrt {\left (-x d +c \right ) \left (-b x +a \right ) e x}\, \left (-\frac {2 A c \sqrt {-\frac {\left (x -\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x -\frac {a}{b}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (x -\frac {c}{d}\right ) d}{c}}, \sqrt {\frac {c}{d \left (\frac {c}{d}-\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}-a d e \,x^{2}-b c e \,x^{2}+a c e x}}-\frac {2 B c \sqrt {-\frac {\left (x -\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x -\frac {a}{b}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x d}{c}}\, \left (\left (\frac {c}{d}-\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {-\frac {\left (x -\frac {c}{d}\right ) d}{c}}, \sqrt {\frac {c}{d \left (\frac {c}{d}-\frac {a}{b}\right )}}\right )+\frac {a \operatorname {EllipticF}\left (\sqrt {-\frac {\left (x -\frac {c}{d}\right ) d}{c}}, \sqrt {\frac {c}{d \left (\frac {c}{d}-\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}-a d e \,x^{2}-b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {-b x +a}\, \sqrt {-x d +c}}\)	\(348\)

Input:

int((B*x+A)/(e*x)^(1/2)/(-b*x+a)^(1/2)/(-d*x+c)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

-2*(A*EllipticF(((-d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*b*d+B*EllipticF 
(((-d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*a*d-B*EllipticE(((-d*x+c)/c)^( 
1/2),(-b*c/(a*d-b*c))^(1/2))*a*d+B*EllipticE(((-d*x+c)/c)^(1/2),(-b*c/(a*d 
-b*c))^(1/2))*b*c)*c*(1/c*x*d)^(1/2)*((-b*x+a)*d/(a*d-b*c))^(1/2)*((-d*x+c 
)/c)^(1/2)*(-d*x+c)^(1/2)*(-b*x+a)^(1/2)/b/d^2/(b*d*x^2-a*d*x-b*c*x+a*c)/( 
e*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.53 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} B b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x - b c - a d}{3 \, b d}\right )\right ) - {\left (B b c + {\left (B a + 3 \, A b\right )} d\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x - b c - a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d^{2} e} \] Input:

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

Output:

-2/3*(3*sqrt(b*d*e)*B*b*d*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))) - (B*b*c + (B*a + 3*A*b)*d)*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)))/(b^2*d^2*e)
 

Sympy [F]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=\int \frac {A + B x}{\sqrt {e x} \sqrt {a - b x} \sqrt {c - d x}}\, dx \] Input:

integrate((B*x+A)/(e*x)**(1/2)/(-b*x+a)**(1/2)/(-d*x+c)**(1/2),x)
 

Output:

Integral((A + B*x)/(sqrt(e*x)*sqrt(a - b*x)*sqrt(c - d*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=\int { \frac {B x + A}{\sqrt {-b x + a} \sqrt {-d x + c} \sqrt {e x}} \,d x } \] Input:

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

Output:

integrate((B*x + A)/(sqrt(-b*x + a)*sqrt(-d*x + c)*sqrt(e*x)), x)
 

Giac [F]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=\int { \frac {B x + A}{\sqrt {-b x + a} \sqrt {-d x + c} \sqrt {e x}} \,d x } \] Input:

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

Output:

integrate((B*x + A)/(sqrt(-b*x + a)*sqrt(-d*x + c)*sqrt(e*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c-d x}} \, dx=\int \frac {A+B\,x}{\sqrt {e\,x}\,\sqrt {a-b\,x}\,\sqrt {c-d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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