3.7.8 \(\int (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx\) [608]

Optimal. Leaf size=117 \[ -\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {6^{3/4} a c^{3/2} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{7 \sqrt {a \left (3-2 x^2\right )}} \]

[Out]

1/7*6^(3/4)*a*c^(3/2)*EllipticF(1/3*2^(1/4)*3^(3/4)*(c*x)^(1/2)/c^(1/2),I)*(-2*x^2+3)^(1/2)/(a*(-2*x^2+3))^(1/
2)+2/7*(c*x)^(5/2)*(-2*a*x^2+3*a)^(1/2)/c-2/7*c*(c*x)^(1/2)*(-2*a*x^2+3*a)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {285, 327, 335, 230, 227} \begin {gather*} \frac {6^{3/4} a c^{3/2} \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 )}{7 \sqrt {a \left (3-2 x^2\right )}}+\frac {2 \sqrt {3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac {2}{7} c \sqrt {3 a-2 a x^2} \sqrt {c x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2],x]

[Out]

(-2*c*Sqrt[c*x]*Sqrt[3*a - 2*a*x^2])/7 + (2*(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2])/(7*c) + (6^(3/4)*a*c^(3/2)*Sqrt[3
- 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/Sqrt[c]], -1])/(7*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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 (c x)^{3/2} \sqrt {3 a-2 a x^2} \, dx &=\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} (6 a) \int \frac {(c x)^{3/2}}{\sqrt {3 a-2 a x^2}} \, dx\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} \left (3 a c^2\right ) \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {1}{7} (6 a c) \text {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {\left (2 \sqrt {3} a c \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 )}{7 \sqrt {a \left (3-2 x^2\right )}}\\ &=-\frac {2}{7} c \sqrt {c x} \sqrt {3 a-2 a x^2}+\frac {2 (c x)^{5/2} \sqrt {3 a-2 a x^2}}{7 c}+\frac {6^{3/4} a c^{3/2} \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 )}{7 \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 = 8.37, size = 74, normalized size = 0.63 \begin {gather*} \frac {c \sqrt {c x} \sqrt {a \left (3-2 x^2\right )} \left (-\left (3-2 x^2\right )^{3/2}+3 \sqrt {3} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {2 x^2}{3}\right )\right )}{7 \sqrt {3-2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c*x]*Sqrt[a*(3 - 2*x^2)]*(-(3 - 2*x^2)^(3/2) + 3*Sqrt[3]*Hypergeometric2F1[-1/2, 1/4, 5/4, (2*x^2)/3])
)/(7*Sqrt[3 - 2*x^2])

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Maple [A]
time = 0.10, size = 133, normalized size = 1.14

method	result	size
default	\(-\frac {c \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \left (-8 x^{5}+\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )+20 x^{3}-12 x \right )}{14 x \left (2 x^{2}-3\right )}\)	\(133\)
risch	\(-\frac {2 \left (x^{2}-1\right ) x \left (2 x^{2}-3\right ) c^{2} a}{7 \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}+\frac {\sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right ) c^{2} a \sqrt {-c x a \left (2 x^{2}-3\right )}}{126 \sqrt {-2 a c \,x^{3}+3 a c x}\, \sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(155\)
elliptic	\(-\frac {\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}\, \sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 c \,x^{2} \sqrt {-2 a c \,x^{3}+3 a c x}}{7}-\frac {2 c \sqrt {-2 a c \,x^{3}+3 a c x}}{7}+\frac {c^{2} a \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{126 \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{c x a \left (2 x^{2}-3\right )}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*c*(c*x)^(1/2)*(-a*(2*x^2-3))^(1/2)*(-8*x^5+((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))+20*x^3-12*x)/x/(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((c*x)^(3/2)*(-2*a*x^2+3*a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 45, normalized size = 0.38 \begin {gather*} -\frac {3}{7} \, \sqrt {2} \sqrt {-a c} c {\rm weierstrassPInverse}\left (6, 0, x\right ) + \frac {2}{7} \, \sqrt {-2 \, a x^{2} + 3 \, a} {\left (c x^{2} - c\right )} \sqrt {c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/7*sqrt(2)*sqrt(-a*c)*c*weierstrassPInverse(6, 0, x) + 2/7*sqrt(-2*a*x^2 + 3*a)*(c*x^2 - c)*sqrt(c*x)

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Sympy [A]
time = 1.39, size = 53, normalized size = 0.45 \begin {gather*} \frac {\sqrt {3} \sqrt {a} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*sqrt(a)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), 2*x**2*exp_polar(2*I*pi)/3)/(2*gamma(9
/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((c*x)^(3/2)*(-2*a*x^2+3*a)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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