\(\int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx\) [207]

Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 B \sqrt {e x} \sqrt {c+d x}}{d e \sqrt {a+b x}}-\frac {2 \sqrt {a} B \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {a} A \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*B*(e*x)^(1/2)*(d*x+c)^(1/2)/d/e/(b*x+a)^(1/2)-2*a^(1/2)*B*(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))/b^(1/2)/d/e^(1/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+2*a^(1/ 
2)*A*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1 
/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/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 = 12.16 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\frac {2 a B (a+b x) (c+d x)}{b}+2 i a \sqrt {\frac {a}{b}} B d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} (-A b+a B) d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{a d \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] 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)]*S 
qrt[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 = 0.31 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, 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

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

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 = 1.17 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b d -B \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d +B \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d -B \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c \right ) c \sqrt {-\frac {x d}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {\frac {x d +c}{c}}\, \sqrt {x d +c}\, \sqrt {b x +a}}{b \,d^{2} \left (b d \,x^{2}+a d x +b c x +a c \right ) \sqrt {e x}}\)	\(219\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 A c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 B c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(334\)

Input:

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

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)*(d*(b*x+a)/(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 = 310, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} B b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + {\left (B b c + {\left (B a - 3 \, A b\right )} d\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d^{2} e} \] Input:

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

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="ma 
xima")
 

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="gi 
ac")
 

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