3.3.91 \(\int \sqrt {\frac {-a+b x^3}{x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {2}{3} x \sqrt {b x-\frac {a}{x^2}}+\frac {2}{3} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {b x-\frac {a}{x^2}}}\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1979, 2007, 2029, 203} \begin {gather*} \frac {2}{3} x \sqrt {b x-\frac {a}{x^2}}+\frac {2}{3} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {b x-\frac {a}{x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-a + b*x^3)/x^2],x]

[Out]

(2*x*Sqrt[-(a/x^2) + b*x])/3 + (2*Sqrt[a]*ArcTan[Sqrt[a]/(x*Sqrt[-(a/x^2) + b*x])])/3

Rule 203

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

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2007

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(p*(n - j)), x] + Dist
[a, Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[
Simplify[j*p + 1], 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

\begin {align*} \int \sqrt {\frac {-a+b x^3}{x^2}} \, dx &=\int \sqrt {-\frac {a}{x^2}+b x} \, dx\\ &=\frac {2}{3} x \sqrt {-\frac {a}{x^2}+b x}-a \int \frac {1}{x^2 \sqrt {-\frac {a}{x^2}+b x}} \, dx\\ &=\frac {2}{3} x \sqrt {-\frac {a}{x^2}+b x}+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {1}{x \sqrt {-\frac {a}{x^2}+b x}}\right )\\ &=\frac {2}{3} x \sqrt {-\frac {a}{x^2}+b x}+\frac {2}{3} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {-\frac {a}{x^2}+b x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 73, normalized size = 1.38 \begin {gather*} \frac {2 x \sqrt {b x-\frac {a}{x^2}} \left (\sqrt {b x^3-a}-\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x^3-a}}{\sqrt {a}}\right )\right )}{3 \sqrt {b x^3-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-a + b*x^3)/x^2],x]

[Out]

(2*x*Sqrt[-(a/x^2) + b*x]*(Sqrt[-a + b*x^3] - Sqrt[a]*ArcTan[Sqrt[-a + b*x^3]/Sqrt[a]]))/(3*Sqrt[-a + b*x^3])

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IntegrateAlgebraic [A]  time = 4.13, size = 76, normalized size = 1.43 \begin {gather*} \frac {x \sqrt {b x-\frac {a}{x^2}} \left (\frac {2}{3} \sqrt {b x^3-a}-\frac {2}{3} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x^3-a}}{\sqrt {a}}\right )\right )}{\sqrt {b x^3-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[-(a/x^2) + b*x]*((2*Sqrt[-a + b*x^3])/3 - (2*Sqrt[a]*ArcTan[Sqrt[-a + b*x^3]/Sqrt[a]])/3))/Sqrt[-a + b
*x^3]

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fricas [A]  time = 0.40, size = 109, normalized size = 2.06 \begin {gather*} \left [\frac {2}{3} \, x \sqrt {\frac {b x^{3} - a}{x^{2}}} + \frac {1}{3} \, \sqrt {-a} \log \left (\frac {b x^{3} - 2 \, \sqrt {-a} x \sqrt {\frac {b x^{3} - a}{x^{2}}} - 2 \, a}{x^{3}}\right ), \frac {2}{3} \, x \sqrt {\frac {b x^{3} - a}{x^{2}}} - \frac {2}{3} \, \sqrt {a} \arctan \left (\frac {x \sqrt {\frac {b x^{3} - a}{x^{2}}}}{\sqrt {a}}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3-a)/x^2)^(1/2),x, algorithm="fricas")

[Out]

[2/3*x*sqrt((b*x^3 - a)/x^2) + 1/3*sqrt(-a)*log((b*x^3 - 2*sqrt(-a)*x*sqrt((b*x^3 - a)/x^2) - 2*a)/x^3), 2/3*x
*sqrt((b*x^3 - a)/x^2) - 2/3*sqrt(a)*arctan(x*sqrt((b*x^3 - a)/x^2)/sqrt(a))]

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giac [A]  time = 0.19, size = 65, normalized size = 1.23 \begin {gather*} -\frac {2}{3} \, \sqrt {a} \arctan \left (\frac {\sqrt {b x^{3} - a}}{\sqrt {a}}\right ) \mathrm {sgn}\relax (x) + \frac {2}{3} \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {a}}\right ) - \sqrt {-a}\right )} \mathrm {sgn}\relax (x) + \frac {2}{3} \, \sqrt {b x^{3} - a} \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3-a)/x^2)^(1/2),x, algorithm="giac")

[Out]

-2/3*sqrt(a)*arctan(sqrt(b*x^3 - a)/sqrt(a))*sgn(x) + 2/3*(sqrt(a)*arctan(sqrt(-a)/sqrt(a)) - sqrt(-a))*sgn(x)
+ 2/3*sqrt(b*x^3 - a)*sgn(x)

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maple [A]  time = 0.11, size = 73, normalized size = 1.38 \begin {gather*} \frac {2 \sqrt {\frac {b \,x^{3}-a}{x^{2}}}\, \left (a \arctanh \left (\frac {\sqrt {b \,x^{3}-a}}{\sqrt {-a}}\right )+\sqrt {b \,x^{3}-a}\, \sqrt {-a}\right ) x}{3 \sqrt {b \,x^{3}-a}\, \sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^3-a)/x^2)^(1/2),x)

[Out]

2/3*((b*x^3-a)/x^2)^(1/2)*x*((b*x^3-a)^(1/2)*(-a)^(1/2)+a*arctanh((b*x^3-a)^(1/2)/(-a)^(1/2)))/(b*x^3-a)^(1/2)
/(-a)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {b x^{3} - a}{x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3-a)/x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt((b*x^3 - a)/x^2), x)

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mupad [B]  time = 5.35, size = 63, normalized size = 1.19 \begin {gather*} \frac {2\,x\,\sqrt {b\,x-\frac {a}{x^2}}}{3}+\frac {2\,\sqrt {a}\,\mathrm {asin}\left (\frac {\sqrt {a}}{\sqrt {b}\,x^{3/2}}\right )\,\sqrt {b\,x-\frac {a}{x^2}}}{3\,\sqrt {b}\,\sqrt {x}\,\sqrt {1-\frac {a}{b\,x^3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(a - b*x^3)/x^2)^(1/2),x)

[Out]

(2*x*(b*x - a/x^2)^(1/2))/3 + (2*a^(1/2)*asin(a^(1/2)/(b^(1/2)*x^(3/2)))*(b*x - a/x^2)^(1/2))/(3*b^(1/2)*x^(1/
2)*(1 - a/(b*x^3))^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**3-a)/x**2)**(1/2),x)

[Out]

Timed out

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