\(\int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c-d x}} \, dx\) [208]

Optimal result

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c-d x}} \, dx=\frac {\frac {2 a B (a+b x) (-c+d x)}{b}+2 i a \sqrt {\frac {a}{b}} B d \sqrt {1+\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|-\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} (-A b+a B) d \sqrt {1+\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),-\frac {b c}{a d}\right )}{a d \sqrt {e x} \sqrt {a+b x} \sqrt {c-d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (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

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

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]*S 
qrt[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[c]*(B*c + A*d)*Sqrt[1 + (b*x)/a]*Sqrt[ 
1 - (d*x)/c]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])], -((b 
*c)/(a*d))])/(d^(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 = 1.30 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03

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

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(((b*x+a)/a)^(1/2),(a*d/(a*d+b*c))^(1/2))*b*d+B*EllipticF((( 
b*x+a)/a)^(1/2),(a*d/(a*d+b*c))^(1/2))*b*c-B*EllipticE(((b*x+a)/a)^(1/2),( 
a*d/(a*d+b*c))^(1/2))*a*d-B*EllipticE(((b*x+a)/a)^(1/2),(a*d/(a*d+b*c))^(1 
/2))*b*c)*a*(-b*x/a)^(1/2)*((-d*x+c)*b/(a*d+b*c))^(1/2)*((b*x+a)/a)^(1/2)* 
(-d*x+c)^(1/2)*(b*x+a)^(1/2)/b^2/d/(-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 = 313, 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]

\[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c-d x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, c -\sqrt {x}\, d x}d x \right )}{e} \] 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(c - d*x)*sqrt(a + b*x))/(sqrt(x)*c - sqrt(x)*d*x),x))/e