3.7.10 \(\int \frac {\sqrt {3 a-2 a x^2}}{\sqrt {c x}} \, dx\) [610]

Optimal. Leaf size=94 \[ \frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {2\ 2^{3/4} a \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}} \]

[Out]

2/3*2^(3/4)*a*EllipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*(-2*x^2+3)^(1/2)*3^(3/4)/c^(1/2)/(a*(-2*x^2
+3))^(1/2)+2/3*(c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2)/c

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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {285, 335, 230, 227} \begin {gather*} \frac {2\ 2^{3/4} a \sqrt {3-2 x^2} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}}+\frac {2 \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3*a - 2*a*x^2]/Sqrt[c*x],x]

[Out]

(2*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/(3*c) + (2*2^(3/4)*a*Sqrt[3 - 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x]
)/Sqrt[c]], -1])/(3^(1/4)*Sqrt[c]*Sqrt[a*(3 - 2*x^2)])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {3 a-2 a x^2}}{\sqrt {c x}} \, dx &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+(2 a) \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {(4 a) \text {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{c}\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {\left (4 a \sqrt {3-2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {2 x^4}{3 c^2}}} \, dx,x,\sqrt {c x}\right )}{\sqrt {3} c \sqrt {a \left (3-2 x^2\right )}}\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {2\ 2^{3/4} a \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.81, size = 51, normalized size = 0.54 \begin {gather*} \frac {2 x \sqrt {a \left (9-6 x^2\right )} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {2 x^2}{3}\right )}{\sqrt {c x} \sqrt {3-2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3*a - 2*a*x^2]/Sqrt[c*x],x]

[Out]

(2*x*Sqrt[a*(9 - 6*x^2)]*Hypergeometric2F1[-1/2, 1/4, 5/4, (2*x^2)/3])/(Sqrt[c*x]*Sqrt[3 - 2*x^2])

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Maple [A]
time = 0.08, size = 124, normalized size = 1.32

method	result	size
default	\(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \left (\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-4 x^{3}+6 x \right )}{3 \sqrt {c x}\, \left (2 x^{2}-3\right )}\)	\(124\)
elliptic	\(\frac {\sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 \sqrt {-2 a c \,x^{3}+3 a c x}}{3 c}+\frac {a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{27 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(137\)
risch	\(-\frac {2 x \left (2 x^{2}-3\right ) a}{3 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-c x a \left (2 x^{2}-3\right )}}{27 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*a*x^2+3*a)^(1/2)/(c*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-a*(2*x^2-3))^(1/2)*(((2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/2)*((-2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/
2))^(1/2)*(-x*2^(1/2)*3^(1/2))^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*((2*x+2^(1/2)*3^(1/2))*2^(1/2)*3^(1/2))^(1/
2),1/2*2^(1/2))-4*x^3+6*x)/(c*x)^(1/2)/(2*x^2-3)

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

[Out]

integrate(sqrt(-2*a*x^2 + 3*a)/sqrt(c*x), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 40, normalized size = 0.43 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {-a c} {\rm weierstrassPInverse}\left (6, 0, x\right ) - \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}\right )}}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*sqrt(2)*sqrt(-a*c)*weierstrassPInverse(6, 0, x) - sqrt(-2*a*x^2 + 3*a)*sqrt(c*x))/c

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Sympy [A]
time = 0.44, size = 53, normalized size = 0.56 \begin {gather*} \frac {\sqrt {3} \sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*sqrt(a)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), 2*x**2*exp_polar(2*I*pi)/3)/(2*sqrt(c)*gamma(5/4
))

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

[Out]

integrate(sqrt(-2*a*x^2 + 3*a)/sqrt(c*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {3\,a-2\,a\,x^2}}{\sqrt {c\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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