3.9.41 \(\int \frac {1}{x^4 (a-b x^2)^{3/4}} \, dx\) [841]

Optimal. Leaf size=106 \[ -\frac {\sqrt [4]{a-b x^2}}{3 a x^3}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}+\frac {5 b^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a-b x^2\right )^{3/4}} \]

[Out]

-1/3*(-b*x^2+a)^(1/4)/a/x^3-5/6*b*(-b*x^2+a)^(1/4)/a^2/x+5/6*b^(3/2)*(1-b*x^2/a)^(3/4)*(cos(1/2*arcsin(x*b^(1/
2)/a^(1/2)))^2)^(1/2)/cos(1/2*arcsin(x*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2*arcsin(x*b^(1/2)/a^(1/2))),2^(1/2))
/a^(3/2)/(-b*x^2+a)^(3/4)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {331, 239, 238} \begin {gather*} \frac {5 b^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a-b x^2\right )^{3/4}}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}-\frac {\sqrt [4]{a-b x^2}}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a - b*x^2)^(3/4)),x]

[Out]

-1/3*(a - b*x^2)^(1/4)/(a*x^3) - (5*b*(a - b*x^2)^(1/4))/(6*a^2*x) + (5*b^(3/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic
F[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*a^(3/2)*(a - b*x^2)^(3/4))

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 239

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + b*(x^2/a))^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + b*(x^2
/a))^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

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]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a-b x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a-b x^2}}{3 a x^3}+\frac {(5 b) \int \frac {1}{x^2 \left (a-b x^2\right )^{3/4}} \, dx}{6 a}\\ &=-\frac {\sqrt [4]{a-b x^2}}{3 a x^3}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}+\frac {\left (5 b^2\right ) \int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx}{12 a^2}\\ &=-\frac {\sqrt [4]{a-b x^2}}{3 a x^3}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}+\frac {\left (5 b^2 \left (1-\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx}{12 a^2 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^2}}{3 a x^3}-\frac {5 b \sqrt [4]{a-b x^2}}{6 a^2 x}+\frac {5 b^{3/2} \left (1-\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a-b x^2\right )^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 52, normalized size = 0.49 \begin {gather*} -\frac {\left (1-\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};-\frac {1}{2};\frac {b x^2}{a}\right )}{3 x^3 \left (a-b x^2\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a - b*x^2)^(3/4)),x]

[Out]

-1/3*((1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-3/2, 3/4, -1/2, (b*x^2)/a])/(x^3*(a - b*x^2)^(3/4))

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-b*x^2+a)^(3/4),x)

[Out]

int(1/x^4/(-b*x^2+a)^(3/4),x)

________________________________________________________________________________________

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/x^4/(-b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*x^4), x)

________________________________________________________________________________________

Fricas [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/x^4/(-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^2 + a)^(1/4)/(b*x^6 - a*x^4), x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.56, size = 34, normalized size = 0.32 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {3}{4}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(-b*x**2+a)**(3/4),x)

[Out]

-hyper((-3/2, 3/4), (-1/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*a**(3/4)*x**3)

________________________________________________________________________________________

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/x^4/(-b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*x^4), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (a-b\,x^2\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a - b*x^2)^(3/4)),x)

[Out]

int(1/(x^4*(a - b*x^2)^(3/4)), x)

________________________________________________________________________________________