\(\int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c+d x}} \, dx\) [209]

Optimal result

Integrand size = 32, antiderivative size = 208 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {a} B \sqrt {1-\frac {b x}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a-b x} \sqrt {1+\frac {d x}{c}}}-\frac {2 \sqrt {a} (B c-A d) \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a-b x} \sqrt {c+d x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c+d x}} \, dx=\frac {-2 \sqrt {-\frac {a}{b}} B (a-b x) (c+d x)-2 i a B d \sqrt {1-\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right )|-\frac {b c}{a d}\right )+2 i (A b+a B) d \sqrt {1-\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right ),-\frac {b c}{a d}\right )}{\sqrt {-\frac {a}{b}} b d \sqrt {e x} \sqrt {a-b x} \sqrt {c+d x}} \] 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)]*Sqr 
t[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)* 
EllipticF[I*ArcSinh[Sqrt[-(a/b)]/Sqrt[x]], -((b*c)/(a*d))])/(Sqrt[-(a/b)]* 
b*d*Sqrt[e*x]*Sqrt[a - b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, 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

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*B*Sqrt[-c]*Sqrt[a - b*x]*Sqrt[1 + (d*x)/c]*EllipticE[ArcSin[(Sqrt[d]*S 
qrt[e*x])/(Sqrt[-c]*Sqrt[e])], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[e]*Sqrt[1 
- (b*x)/a]*Sqrt[c + d*x]) + (2*Sqrt[a]*(A*b + a*B)*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))])/(b^(3/2)*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 = 1.29 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {\frac {b c}{a d +b c}}\right ) b d +B \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {\frac {b c}{a d +b c}}\right ) a d -B \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {\frac {b c}{a d +b c}}\right ) a d -B \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {\frac {b c}{a d +b c}}\right ) b c \right ) c \sqrt {-\frac {x d}{c}}\, \sqrt {\frac {\left (-b x +a \right ) d}{a d +b c}}\, \sqrt {\frac {x d +c}{c}}\, \sqrt {x d +c}\, \sqrt {-b x +a}}{b \,d^{2} \left (-b d \,x^{2}+a d x -b c x +a c \right ) \sqrt {e x}}\)	\(214\)
elliptic	\(\frac {\sqrt {\left (x d +c \right ) \left (-b x +a \right ) e x}\, \left (\frac {2 A c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x -\frac {a}{b}}{-\frac {a}{b}-\frac {c}{d}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {a}{b}-\frac {c}{d}\right )}}\right )}{d \sqrt {-b d e \,x^{3}+a d e \,x^{2}-b c e \,x^{2}+a c e x}}+\frac {2 B c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x -\frac {a}{b}}{-\frac {a}{b}-\frac {c}{d}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {a}{b}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {a}{b}-\frac {c}{d}\right )}}\right )+\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {a}{b}-\frac {c}{d}\right )}}\right )}{b}\right )}{d \sqrt {-b d e \,x^{3}+a d e \,x^{2}-b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {-b x +a}\, \sqrt {x d +c}}\)	\(347\)

Input:

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

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 = 0.11 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a-b x} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-b d e} B b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c - a d}{3 \, b d}\right )\right ) + {\left (B b c - {\left (B a + 3 \, A b\right )} d\right )} \sqrt {-b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c - a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d^{2} e} \] Input:

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

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="m 
axima")
 

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="g 
iac")
 

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