3.11.13 \(\int \frac {\sqrt {x (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}})}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [1013]

Optimal. Leaf size=46 \[ \frac {\sqrt {2} b \sinh ^{-1}\left (\frac {a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

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

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Rubi [A]
time = 0.74, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2156, 2155, 221} \begin {gather*} \frac {\sqrt {2} b \sinh ^{-1}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2155

Int[Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)*Sqrt[(c_) + (d_.)*(x_)^2]]/((x_)*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> D
ist[Sqrt[2]*(b/a), Subst[Int[1/Sqrt[1 + x^2/a], x], x, a*x + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]

Rule 2156

Int[Sqrt[(e_.)*(x_)*((a_.)*(x_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2])]/((x_)*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol
] :> Int[Sqrt[a*e*x^2 + b*e*x*Sqrt[c + d*x^2]]/(x*Sqrt[c + d*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2
- b^2*d, 0] && EqQ[b^2*c*e + a, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {x \left (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ &=\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a}\\ &=\frac {\sqrt {2} b \sinh ^{-1}\left (\frac {a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(46)=92\).
time = 0.03, size = 107, normalized size = 2.33 \begin {gather*} -\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \tan ^{-1}\left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2]*b*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*ArcT
an[Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]])/(a*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 5.76, size = 161, normalized size = 3.50 \begin {gather*} \left [\frac {\sqrt {2} b \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 (\sqrt {2} \sqrt {a} x + \frac {\sqrt {2} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}{\sqrt {a}}\right )} + 1\right )}{2 \, \sqrt {a}}, -\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*b*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))*(sqr
t(2)*sqrt(a)*x + sqrt(2)*b*sqrt((a^2*x^2 - a)/b^2)/sqrt(a)) + 1)/sqrt(a), -sqrt(2)*b*sqrt(-1/a)*arctan(1/2*sqr
t(2)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*sqrt(-1/a)/x)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}{x \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x\,\left (a\,x+b\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}}{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((x*(a*x + b*((a^2*x^2)/b^2 - a/b^2)^(1/2)))^(1/2)/(x*((a^2*x^2)/b^2 - a/b^2)^(1/2)),x)

[Out]

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

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