3.25.66 \(\int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx\) [2466]

Optimal. Leaf size=200 \[ \frac {\left (-9 b+8 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{12 a b x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {19 b-8 a^2 x}{12 b}+\frac {3 \sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{4 \sqrt {2} a^{3/2} x}\right ) \]

[Out]

1/12*(8*a^2*x-9*b)*(a^2*x^2-b*x)^(1/2)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/a/b/x+(x*(a*x+(a^2*x^2-b*x)^(1/2)))
^(1/2)*(1/12*(-8*a^2*x+19*b)/b+3/8*b^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)*arctan(2^(1/2)*a^(1/2)*(-a*x+(a^2*
x^2-b*x)^(1/2))^(1/2)/b^(1/2))*2^(1/2)/a^(3/2)/x)

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Rubi [F]
time = 2.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[-(b*x) + a^2*x^2]/Sqrt[a*x^2 + x*Sqrt[-(b*x) + a^2*x^2]],x]

[Out]

(2*Sqrt[-(b*x) + a^2*x^2]*Defer[Subst][Defer[Int][(x^2*Sqrt[-b + a^2*x^2])/Sqrt[a*x^4 + x^2*Sqrt[-(b*x^2) + a^
2*x^4]], x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[-b + a^2*x])

Rubi steps

\begin {align*} \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx &=\frac {\sqrt {-b x+a^2 x^2} \int \frac {\sqrt {x} \sqrt {-b+a^2 x}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx}{\sqrt {x} \sqrt {-b+a^2 x}}\\ &=\frac {\left (2 \sqrt {-b x+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-b+a^2 x^2}}{\sqrt {a x^4+x^2 \sqrt {-b x^2+a^2 x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a^2 x}}\\ \end {align*}

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Mathematica [A]
time = 3.99, size = 213, normalized size = 1.06 \begin {gather*} \frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (-9 b^2+a b \left (17 a x-19 \sqrt {x \left (-b+a^2 x\right )}\right )+8 a^3 x \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )\right )+9 \sqrt {2} b^{3/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{24 a^{3/2} b x \sqrt {x \left (-b+a^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-(b*x) + a^2*x^2]/Sqrt[a*x^2 + x*Sqrt[-(b*x) + a^2*x^2]],x]

[Out]

(Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(-2*Sqrt[a]*x*(-9*b^2 + a*b*(17*a*x - 19*Sqrt[x*(-b + a^2*x)]) + 8*a^3*x
*(-(a*x) + Sqrt[x*(-b + a^2*x)])) + 9*Sqrt[2]*b^(3/2)*Sqrt[x*(-b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]
*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]])/Sqrt[b]]))/(24*a^(3/2)*b*x*Sqrt[x*(-b + a^2*x)])

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b x}}{\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(sqrt(a^2*x^2 - b*x)/sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x), x)

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Fricas [A]
time = 0.41, size = 323, normalized size = 1.62 \begin {gather*} \left [\frac {9 \, \sqrt {2} \sqrt {a} b^{2} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{48 \, a^{2} b x}, \frac {9 \, \sqrt {2} \sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{24 \, a^{2} b x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(9*sqrt(2)*sqrt(a)*b^2*x*log(-(4*a^2*x^2 + 4*sqrt(a^2*x^2 - b*x)*a*x - b*x - 2*(sqrt(2)*a^(3/2)*x + sqrt
(2)*sqrt(a^2*x^2 - b*x)*sqrt(a))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/x) - 4*(8*a^4*x^2 - 19*a^2*b*x - (8*a^3*
x - 9*a*b)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b*x), 1/24*(9*sqrt(2)*sqrt(-a)*b^2*x
*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a*x)) - 2*(8*a^4*x^2 - 19*a^2*b*x - (8*a^3*x
- 9*a*b)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a^2*b*x)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (a^{2} x - b\right )}}{\sqrt {x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(a**2*x - b))/sqrt(x*(a*x + sqrt(a**2*x**2 - b*x))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a^2*x^2 - b*x)/sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a^2\,x^2-b\,x}}{\sqrt {a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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