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

Optimal. Leaf size=200 \[ \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (8 a^2 x-9 b\right )}{12 a b x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {19 b-8 a^2 x}{12 b}+\frac {3 \sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{4 \sqrt {2} a^{3/2} x}\right ) \]

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Rubi [F]  time = 3.07, 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 ) \operatorname {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 [C]  time = 1.81, size = 205, normalized size = 1.02 \begin {gather*} \frac {b \sqrt {x \left (a^2 x-b\right )} \left (\sqrt {x \left (a^2 x-b\right )}+a x\right )^2 \left (4 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {b}{2 a \left (a x+\sqrt {x \left (a^2 x-b\right )}\right )}\right )-\, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};1-\frac {b}{2 a \left (a x+\sqrt {x \left (a^2 x-b\right )}\right )}\right )-4\right )}{3 \sqrt {x \left (\sqrt {x \left (a^2 x-b\right )}+a x\right )} \left (a \left (\sqrt {x \left (a^2 x-b\right )}+a x\right )-b\right ) \left (2 a \left (\sqrt {x \left (a^2 x-b\right )}+a x\right )-b\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]

(b*Sqrt[x*(-b + a^2*x)]*(a*x + Sqrt[x*(-b + a^2*x)])^2*(-4 + 4*Hypergeometric2F1[-3/2, 1, -1/2, 1 - b/(2*a*(a*
x + Sqrt[x*(-b + a^2*x)]))] - Hypergeometric2F1[-3/2, 2, -1/2, 1 - b/(2*a*(a*x + Sqrt[x*(-b + a^2*x)]))]))/(3*
Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(-b + a*(a*x + Sqrt[x*(-b + a^2*x)]))*(-b + 2*a*(a*x + Sqrt[x*(-b + a^2*x
)])))

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IntegrateAlgebraic [A]  time = 5.40, size = 200, normalized size = 1.00 \begin {gather*} \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}} \tan ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*b + 8*a^2*x)*Sqrt[-(b*x) + a^2*x^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(12*a*b*x) + Sqrt[x*(a*x + Sq
rt[-(b*x) + a^2*x^2])]*((19*b - 8*a^2*x)/(12*b) + (3*Sqrt[b]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqr
t[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(4*Sqrt[2]*a^(3/2)*x))

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fricas [A]  time = 0.51, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}\,{d x} \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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maple [F]  time = 0.05, 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*} \int \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}\,{d x} \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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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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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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