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

Optimal. Leaf size=167 \[ \frac {2 \left (2 a x^2-1\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{3 b x^2}-\frac {4 \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{3 x}-\frac {\sqrt {2} a \tanh ^{-1}\left (\sqrt {2} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{b} \]

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Rubi [F]  time = 1.78, 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 {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx &=\int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\\ \end {align*}

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Mathematica [C]  time = 5.41, size = 341, normalized size = 2.04 \begin {gather*} \frac {\frac {3 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right ) \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};2 x \left (\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} b+a x\right )\right )+2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2+3\right )}{b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1}-\frac {4 a \left (3 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+3 a x^2-1\right )}{b x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}-\frac {3 \sqrt {2} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (\sqrt {2} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}+\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt {a}}\right )\right )}{b}}{6 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*a*(-1 + 3*a*x^2 + 3*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))/(b*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])) - (3*Sqrt
[2]*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(Sqrt[2]*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])] + S
qrt[a]*ArcTanh[Sqrt[a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2]/Sqrt[a]]))/b + (3*Sqrt[(a*(-1 + a*x^2))/b^2]*(
a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])*(3 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2] + Hypergeometric2F1[-1/2,
1, 1/2, 2*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]))/(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))/(6*Sqrt[x*(
a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])])

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IntegrateAlgebraic [A]  time = 3.30, size = 223, normalized size = 1.34 \begin {gather*} \frac {2 \left (-1+2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 b x^2}-\frac {4 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 x}+\frac {a \log \left (-1+\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b}-\frac {a \log \left (b+\sqrt {2} b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 37.49, size = 175, normalized size = 1.05 \begin {gather*} \frac {3 \, \sqrt {2} a x^{2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right ) + 4 \, {\left (2 \, a x^{2} - 2 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 1\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{6 \, b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*sqrt(2)*a*x^2*log(4*a*x^2 - 4*b*x*sqrt((a^2*x^2 - a)/b^2) + 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))
*(2*sqrt(2)*b*x*sqrt((a^2*x^2 - a)/b^2) - sqrt(2)*(2*a*x^2 - 1)) - 1) + 4*(2*a*x^2 - 2*b*x*sqrt((a^2*x^2 - a)/
b^2) - 1)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)))/(b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-a/b^2+a^2*x^2/b^2)^(1/2)/x^2/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(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 {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x^2\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((a^2*x^2)/b^2 - a/b^2)^(1/2)/(x^2*(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(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 {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x^{2} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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