3.27.37 \(\int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [2637]

Optimal. Leaf size=58 \[ \frac {2 \sqrt {a} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {120} \begin {gather*} \frac {2 \sqrt {a} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(2*Sqrt[a]*EllipticF[ArcSin[(Sqrt[1 - c]*Sqrt[a + b*x])/Sqrt[a]], (1 - e)/(1 - c)])/(b*Sqrt[1 - c])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx &=\frac {2 \sqrt {a} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(58)=116\).
time = 10.97, size = 129, normalized size = 2.22 \begin {gather*} -\frac {2 (a+b x) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {a}{-1+c}}}{\sqrt {a+b x}}\right )|\frac {-1+c}{-1+e}\right )}{b \sqrt {-\frac {a}{-1+c}} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(-2*(a + b*x)*Sqrt[(-1 + c + a/(a + b*x))/(-1 + c)]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[ArcSin[Sqr
t[-(a/(-1 + c))]/Sqrt[a + b*x]], (-1 + c)/(-1 + e)])/(b*Sqrt[-(a/(-1 + c))]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e
+ (b*(-1 + e)*x)/a])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(49)=98\).
time = 0.13, size = 179, normalized size = 3.09

method	result	size
default	\(\frac {2 a \sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{a \left (c -e \right )}}\, \EllipticF \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) \left (c -e \right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, b \left (-1+e \right ) \left (-1+c \right )}\)	\(179\)
elliptic	\(\frac {2 \sqrt {\frac {\left (b x +a \right ) \left (b c x +a c -b x \right ) \left (b e x +a e -b x \right )}{a^{2}}}\, \left (-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}\)	\(425\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/2)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*x*(e - 1)/a + e)), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.17, size = 420, normalized size = 7.24 \begin {gather*} -\frac {2 \, a^{2} \sqrt {-\frac {b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e}{a^{2}}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} c^{2} - a^{2} c + a^{2} e^{2} + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2} + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} e^{3} + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3} - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, \frac {2 \, a c + 3 \, {\left (b c - b\right )} x - {\left (3 \, a c + 3 \, {\left (b c - b\right )} x - 2 \, a\right )} e - a}{3 \, {\left (b c - {\left (b c - b\right )} e - b\right )}}\right )}{b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^2*sqrt(-(b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*weierstrassPInverse(4/3*(a^2*c^2 - a^2*c + a^2*e^2 + a^2 - (
a^2*c + a^2)*e)/(b^2*c^2 - 2*b^2*c + b^2 + (b^2*c^2 - 2*b^2*c + b^2)*e^2 - 2*(b^2*c^2 - 2*b^2*c + b^2)*e), 4/2
7*(2*a^3*c^3 - 3*a^3*c^2 - 3*a^3*c + 2*a^3*e^3 + 2*a^3 - 3*(a^3*c + a^3)*e^2 - 3*(a^3*c^2 - 4*a^3*c + a^3)*e)/
(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3 - (b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^3 + 3*(b^3*c^3 - 3*b^3*c^2 + 3*
b^3*c - b^3)*e^2 - 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e), 1/3*(2*a*c + 3*(b*c - b)*x - (3*a*c + 3*(b*c -
b)*x - 2*a)*e - a)/(b*c - (b*c - b)*e - b))/(b^3*c - b^3 - (b^3*c - b^3)*e)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x} \sqrt {c + \frac {b c x}{a} - \frac {b x}{a}} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a + b*x)*sqrt(c + b*c*x/a - b*x/a)*sqrt(e + b*e*x/a - b*x/a)), x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, choosing
root of [1,0,%%%{2,[3,1,0,0]%%%}+%%%{-2,[3,0,0,0]%%%}+%%%{-2,[2,2,0,0]%%%}+%%%{2,[2,1,0,1]%%%}+%%%{2,[2,1,0,0
]%%%}+%%%{-2,[

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {c+\frac {b\,x\,\left (c-1\right )}{a}}\,\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________