3.810 \(\int \frac {(a-b x^2)^{3/4}}{x^6} \, dx\)

Optimal. Leaf size=128 \[ \frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3} \]

[Out]

-1/5*(-b*x^2+a)^(3/4)/x^5+1/10*b*(-b*x^2+a)^(3/4)/a/x^3+3/20*b^2*(-b*x^2+a)^(3/4)/a^2/x+3/20*b^(5/2)*(1-b*x^2/
a)^(1/4)*(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)))*EllipticE(sin(1/2*arc
sin(x*b^(1/2)/a^(1/2))),2^(1/2))/a^(3/2)/(-b*x^2+a)^(1/4)

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Rubi [A]  time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {277, 325, 229, 228} \[ \frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(3/4)/x^6,x]

[Out]

-(a - b*x^2)^(3/4)/(5*x^5) + (b*(a - b*x^2)^(3/4))/(10*a*x^3) + (3*b^2*(a - b*x^2)^(3/4))/(20*a^2*x) + (3*b^(5
/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(20*a^(3/2)*(a - b*x^2)^(1/4))

Rule 228

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

Rule 229

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

Rule 277

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

Rule 325

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 {\left (a-b x^2\right )^{3/4}}{x^6} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}-\frac {1}{10} (3 b) \int \frac {1}{x^4 \sqrt [4]{a-b x^2}} \, dx\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac {\left (3 b^2\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {\left (3 b^3\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^2}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {\left (3 b^3 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{40 a^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 52, normalized size = 0.41 \[ -\frac {\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac {5}{2},-\frac {3}{4};-\frac {3}{2};\frac {b x^2}{a}\right )}{5 x^5 \left (1-\frac {b x^2}{a}\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(3/4)/x^6,x]

[Out]

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

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fricas [F]  time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4)/x^6,x, algorithm="fricas")

[Out]

integral((-b*x^2 + a)^(3/4)/x^6, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4)/x^6,x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(3/4)/x^6, x)

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(3/4)/x^6,x)

[Out]

int((-b*x^2+a)^(3/4)/x^6,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4)/x^6,x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(3/4)/x^6, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-b\,x^2\right )}^{3/4}}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^(3/4)/x^6,x)

[Out]

int((a - b*x^2)^(3/4)/x^6, x)

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sympy [C]  time = 1.28, size = 36, normalized size = 0.28 \[ - \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(3/4)/x**6,x)

[Out]

-a**(3/4)*hyper((-5/2, -3/4), (-3/2,), b*x**2*exp_polar(2*I*pi)/a)/(5*x**5)

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