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

Optimal. Leaf size=114 \[ \frac {2 \sqrt {a x} \sqrt {d f-e^2} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {d f-e^2}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {114, 113} \[ \frac {2 \sqrt {a x} \sqrt {d f-e^2} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {d f-e^2}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-e^2 + d*f]*Sqrt[a*x]*Sqrt[(e*(e + f*x))/(e^2 - d*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[d + e*x])/Sqrt[-e
^2 + d*f]], 1 - e^2/(d*f)])/(e*Sqrt[f]*Sqrt[-((e*x)/d)]*Sqrt[e + f*x])

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

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Mathematica [C]  time = 0.21, size = 106, normalized size = 0.93 \[ -\frac {2 i e \sqrt {a x} \sqrt {\frac {f x}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )\right )}{f \sqrt {\frac {e x}{d+e x}} \sqrt {d+e x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*e*Sqrt[a*x]*Sqrt[1 + (f*x)/e]*(EllipticE[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)/e^2] - EllipticF[I*ArcSinh[Sq
rt[(e*x)/d]], (d*f)/e^2]))/(f*Sqrt[(e*x)/(d + e*x)]*Sqrt[d + e*x]*Sqrt[e + f*x])

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x} \sqrt {e x + d} \sqrt {f x + e}}{e f x^{2} + d e + {\left (e^{2} + d f\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 191, normalized size = 1.68 \[ -\frac {2 \left (d f \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d f}{d f -e^{2}}}\right )-e^{2} \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d f}{d f -e^{2}}}\right )+e^{2} \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d f}{d f -e^{2}}}\right )\right ) \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (f x +e \right ) e}{d f -e^{2}}}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {f x +e}\, \sqrt {e x +d}\, \sqrt {a x}\, d}{\left (e f \,x^{2}+d f x +e^{2} x +d e \right ) e^{2} f x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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