3.2639 \(\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx\)

Optimal. Leaf size=134 \[ \frac {2 \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {f (b c-a d)}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {121, 120} \[ \frac {2 \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 120

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/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

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

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Mathematica [A]  time = 0.57, size = 126, normalized size = 0.94 \[ -\frac {2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{d \sqrt {e+f x} \sqrt {a-\frac {b c}{d}} \sqrt {\frac {b (c+d x)}{d (a+b x)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b d f x^{3} + a c e + {\left (b d e + {\left (b c + a d\right )} f\right )} x^{2} + {\left (a c f + {\left (b c + a d\right )} e\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 192, normalized size = 1.43 \[ \frac {2 \left (a d -b c \right ) \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \EllipticF \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )}{\left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right ) b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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