3.2644 \(\int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx\)

Optimal. Leaf size=101 \[ \frac {2 \sqrt {c} \sqrt {\frac {c-d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{d \sqrt {d x-c} \sqrt {e+f x}} \]

[Out]

2*EllipticF(1/2*(d*x+c)^(1/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))*c^(1/2)*((-d*x+c)/c)^(1/2)*(d*(f*x+e)
/(-c*f+d*e))^(1/2)/d/(d*x-c)^(1/2)/(f*x+e)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {121, 119} \[ \frac {2 \sqrt {c} \sqrt {\frac {c-d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right )|-\frac {2 c f}{d e-c f}\right )}{d \sqrt {d x-c} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*Sqrt[c]*Sqrt[(c - d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSin[Sqrt[c + d*x]/(Sqrt[2]*Sqrt[c])]
, (-2*c*f)/(d*e - c*f)])/(d*Sqrt[-c + d*x]*Sqrt[e + f*x])

Rule 119

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

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rubi steps

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

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Mathematica [A]  time = 0.37, size = 123, normalized size = 1.22 \[ \frac {\sqrt {2} (c-d x) \sqrt {\frac {c+d x}{d x-c}} \sqrt {\frac {d (e+f x)}{f (d x-c)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {-c}}{\sqrt {d x-c}}\right ),\frac {1}{2} \left (\frac {d e}{c f}+1\right )\right )}{\sqrt {-c} d \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

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

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fricas [F]  time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x + c} \sqrt {d x - c} \sqrt {f x + e}}{d^{2} f x^{3} + d^{2} e x^{2} - c^{2} f x - c^{2} e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)/(d^2*f*x^3 + d^2*e*x^2 - c^2*f*x - c^2*e), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x + c} \sqrt {d x - c} \sqrt {f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)), x)

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maple [A]  time = 0.09, size = 174, normalized size = 1.72 \[ -\frac {2 \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {d x -c}\, \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {-\frac {\left (d x -c \right ) f}{c f +d e}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \left (c f -d e \right ) \EllipticF \left (\sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right )}{\left (d^{2} f \,x^{3}+d^{2} e \,x^{2}-c^{2} f x -c^{2} e \right ) d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x-c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)

[Out]

-2*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(d*x-c)^(1/2)/d/f*(-(f*x+e)*d/(c*f-d*e))^(1/2)*(-(d*x-c)*f/(c*f+d*e))^(1/2)*((d
*x+c)*f/(c*f-d*e))^(1/2)*EllipticF((-(f*x+e)*d/(c*f-d*e))^(1/2),(-(c*f-d*e)/(c*f+d*e))^(1/2))*(c*f-d*e)/(d^2*f
*x^3+d^2*e*x^2-c^2*f*x-c^2*e)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x + c} \sqrt {d x - c} \sqrt {f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\sqrt {d\,x-c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^(1/2)*(c + d*x)^(1/2)*(d*x - c)^(1/2)),x)

[Out]

int(1/((e + f*x)^(1/2)*(c + d*x)^(1/2)*(d*x - c)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- c + d x} \sqrt {c + d x} \sqrt {e + f x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x-c)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral(1/(sqrt(-c + d*x)*sqrt(c + d*x)*sqrt(e + f*x)), x)

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