3.2672 \(\int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {114, 113} \[ \frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(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] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 114

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

Rubi steps

\begin {align*} \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx &=\frac {\left (\sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=\frac {2 \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\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 {\frac {b (e+f x)}{b e-a f}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.86, size = 154, normalized size = 1.15 \[ \frac {2 \sqrt {c+d x} \left (\frac {(a f-b e) \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {a-\frac {b e}{f}}}{\sqrt {a+b x}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )}{b \sqrt {a-\frac {b e}{f}} \sqrt {\frac {b (c+d x)}{d (a+b x)}}}+\frac {e+f x}{\sqrt {a+b x}}\right )}{d \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 209, normalized size = 1.56 \[ -\frac {2 \left (a^{2} d f -a b c f -b e a d +b^{2} c e \right ) \sqrt {-\frac {\left (d x +c \right ) b}{a d -b c}}\, \sqrt {-\frac {\left (f x +e \right ) b}{a f -b e}}\, \sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}\, \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {f x +e}\, \EllipticE \left (\sqrt {\frac {\left (b x +a \right ) d}{a d -b c}}, \sqrt {\frac {\left (a d -b c \right ) f}{\left (a f -b e \right ) d}}\right )}{\left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right ) b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________