3.1262 \(\int \frac {1}{x^{12} (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=136 \[ -\frac {40 b^{7/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}} \]

[Out]

-1/11*(-b*x^4+a)^(1/4)/a/x^11-10/77*b*(-b*x^4+a)^(1/4)/a^2/x^7-20/77*b^2*(-b*x^4+a)^(1/4)/a^3/x^3-40/77*b^(7/2
)*(1-a/b/x^4)^(3/4)*x^3*(cos(1/2*arccsc(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccsc(x^2*b^(1/2)/a^(1/2)))*El
lipticF(sin(1/2*arccsc(x^2*b^(1/2)/a^(1/2))),2^(1/2))/a^(7/2)/(-b*x^4+a)^(3/4)

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Rubi [A]  time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {325, 237, 335, 275, 232} \[ -\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {40 b^{7/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(11*a*x^11) - (10*b*(a - b*x^4)^(1/4))/(77*a^2*x^7) - (20*b^2*(a - b*x^4)^(1/4))/(77*a^3*x^
3) - (40*b^(7/2)*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*a^(7/2)*(a - b*x
^4)^(3/4))

Rule 232

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

Rule 237

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

Rule 275

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

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]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}+\frac {(10 b) \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}} \, dx}{11 a}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}+\frac {\left (60 b^2\right ) \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}} \, dx}{77 a^2}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}+\frac {\left (40 b^3\right ) \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{77 a^3}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}+\frac {\left (40 b^3 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{77 a^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {\left (40 b^3 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{77 a^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {\left (20 b^3 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{77 a^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {40 b^{7/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 52, normalized size = 0.38 \[ -\frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac {11}{4},\frac {3}{4};-\frac {7}{4};\frac {b x^4}{a}\right )}{11 x^{11} \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^12*(a - b*x^4)^(3/4)),x]

[Out]

-1/11*((1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[-11/4, 3/4, -7/4, (b*x^4)/a])/(x^11*(a - b*x^4)^(3/4))

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{b x^{16} - a x^{12}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^4 + a)^(1/4)/(b*x^16 - a*x^12), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^12), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}} x^{12}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^12/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^12/(-b*x^4+a)^(3/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^12), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{12}\,{\left (a-b\,x^4\right )}^{3/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^12*(a - b*x^4)^(3/4)),x)

[Out]

int(1/(x^12*(a - b*x^4)^(3/4)), x)

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sympy [C]  time = 2.94, size = 34, normalized size = 0.25 \[ - \frac {i e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{14 b^{\frac {3}{4}} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**12/(-b*x**4+a)**(3/4),x)

[Out]

-I*exp(3*I*pi/4)*hyper((3/4, 7/2), (9/2,), a/(b*x**4))/(14*b**(3/4)*x**14)

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