3.7.48 \(\int \frac {1}{(c x)^{5/2} (3 a-2 a x^2)^{3/2}} \, dx\) [648]

Optimal. Leaf size=132 \[ \frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}-\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {5\ 2^{3/4} \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{27 \sqrt [4]{3} a c^{5/2} \sqrt {a \left (3-2 x^2\right )}} \]

[Out]

5/81*2^(3/4)*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)/a/c^(5/2)/(a*(-2*x^
2+3))^(1/2)+1/3/a/c/(c*x)^(3/2)/(-2*a*x^2+3*a)^(1/2)-5/27*(-2*a*x^2+3*a)^(1/2)/a^2/c/(c*x)^(3/2)

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Rubi [A]
time = 0.05, antiderivative size = 132, 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 = {296, 331, 335, 230, 227} \begin {gather*} -\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {5\ 2^{3/4} \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 )}{27 \sqrt [4]{3} a c^{5/2} \sqrt {a \left (3-2 x^2\right )}}+\frac {1}{3 a c \sqrt {3 a-2 a x^2} (c x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x]

[Out]

1/(3*a*c*(c*x)^(3/2)*Sqrt[3*a - 2*a*x^2]) - (5*Sqrt[3*a - 2*a*x^2])/(27*a^2*c*(c*x)^(3/2)) + (5*2^(3/4)*Sqrt[3
- 2*x^2]*EllipticF[ArcSin[((2/3)^(1/4)*Sqrt[c*x])/Sqrt[c]], -1])/(27*3^(1/4)*a*c^(5/2)*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 296

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 {1}{(c x)^{5/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}+\frac {5 \int \frac {1}{(c x)^{5/2} \sqrt {3 a-2 a x^2}} \, dx}{6 a}\\ &=\frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}-\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {5 \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx}{27 a c^2}\\ &=\frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}-\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {10 \text {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{27 a c^3}\\ &=\frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}-\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {\left (10 \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 )}{27 \sqrt {3} a c^3 \sqrt {a \left (3-2 x^2\right )}}\\ &=\frac {1}{3 a c (c x)^{3/2} \sqrt {3 a-2 a x^2}}-\frac {5 \sqrt {3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac {5\ 2^{3/4} \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 )}{27 \sqrt [4]{3} a c^{5/2} \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 = 10.02, size = 58, normalized size = 0.44 \begin {gather*} -\frac {2 x \left (3-2 x^2\right )^{3/2} \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};\frac {2 x^2}{3}\right )}{9 \sqrt {3} (c x)^{5/2} \left (a \left (3-2 x^2\right )\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x]

[Out]

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

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Maple [A]
time = 0.07, size = 133, normalized size = 1.01

method	result	size
default	\(-\frac {\sqrt {-a \left (2 x^{2}-3\right )}\, \left (5 \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-x \sqrt {2}\, \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right ) x +60 x^{2}-36\right )}{162 x \,a^{2} c^{2} \sqrt {c x}\, \left (2 x^{2}-3\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-c x a \left (2 x^{2}-3\right )}\, \left (\frac {2 x}{9 a \,c^{2} \sqrt {-2 \left (x^{2}-\frac {3}{2}\right ) a c x}}-\frac {2 \sqrt {-2 a c \,x^{3}+3 a c x}}{27 a^{2} c^{3} x^{2}}+\frac {5 \sqrt {6}\, \sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-6 \left (x -\frac {\sqrt {6}}{2}\right ) \sqrt {6}}\, \sqrt {-3 x \sqrt {6}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {6}}{2}\right ) \sqrt {6}}}{3}, \frac {\sqrt {2}}{2}\right )}{1458 c^{2} a \sqrt {-2 a c \,x^{3}+3 a c x}}\right )}{\sqrt {c x}\, \sqrt {-a \left (2 x^{2}-3\right )}}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/162*(-a*(2*x^2-3))^(1/2)*(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))*x+60*x^2-36)/x/a^2/c^2/(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(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 80, normalized size = 0.61 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (2 \, x^{4} - 3 \, x^{2}\right )} \sqrt {-a c} {\rm weierstrassPInverse}\left (6, 0, x\right ) + 2 \, \sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x} {\left (5 \, x^{2} - 3\right )}}{27 \, {\left (2 \, a^{2} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="fricas")

[Out]

-1/27*(5*sqrt(2)*(2*x^4 - 3*x^2)*sqrt(-a*c)*weierstrassPInverse(6, 0, x) + 2*sqrt(-2*a*x^2 + 3*a)*sqrt(c*x)*(5
*x^2 - 3))/(2*a^2*c^3*x^4 - 3*a^2*c^3*x^2)

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Sympy [A]
time = 4.69, size = 54, normalized size = 0.41 \begin {gather*} \frac {\sqrt {3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac {3}{2}} c^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/2)/(-2*a*x**2+3*a)**(3/2),x)

[Out]

sqrt(3)*gamma(-3/4)*hyper((-3/4, 3/2), (1/4,), 2*x**2*exp_polar(2*I*pi)/3)/(18*a**(3/2)*c**(5/2)*x**(3/2)*gamm
a(1/4))

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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(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{5/2}\,{\left (3\,a-2\,a\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x)

[Out]

int(1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)), x)

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