3.1257 \(\int \frac{x^8}{(a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]

[Out]

(-5*a*x*(a - b*x^4)^(1/4))/(12*b^2) - (x^5*(a - b*x^4)^(1/4))/(6*b) - (5*a^(3/2)*(1 - a/(b*x^4))^(3/4)*x^3*Ell
ipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*b^(3/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.0477997, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {321, 237, 335, 275, 232} \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a - b*x^4)^(3/4),x]

[Out]

(-5*a*x*(a - b*x^4)^(1/4))/(12*b^2) - (x^5*(a - b*x^4)^(1/4))/(6*b) - (5*a^(3/2)*(1 - a/(b*x^4))^(3/4)*x^3*Ell
ipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*b^(3/2)*(a - b*x^4)^(3/4))

Rule 321

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 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 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]

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 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]

Rubi steps

\begin{align*} \int \frac{x^8}{\left (a-b x^4\right )^{3/4}} \, dx &=-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac{(5 a) \int \frac{x^4}{\left (a-b x^4\right )^{3/4}} \, dx}{6 b}\\ &=-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\left (a-b x^4\right )^{3/4}} \, dx}{12 b^2}\\ &=-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}+\frac{\left (5 a^2 \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}{12 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac{\left (5 a^2 \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 )}{12 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac{\left (5 a^2 \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 )}{24 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b}-\frac{5 a^{3/2} \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 )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0202742, size = 80, normalized size = 0.73 \[ \frac{5 a^2 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-5 a^2 x+3 a b x^5+2 b^2 x^9}{12 b^2 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a - b*x^4)^(3/4),x]

[Out]

(-5*a^2*x + 3*a*b*x^5 + 2*b^2*x^9 + 5*a^2*x*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^4)/a])
/(12*b^2*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \begin{align*} \int{{x}^{8} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(-b*x^4+a)^(3/4),x)

[Out]

int(x^8/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^8/(-b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{8}}{b x^{4} - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-b*x^4 + a)^(1/4)*x^8/(b*x^4 - a), x)

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Sympy [C]  time = 1.02037, size = 39, normalized size = 0.36 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-b*x**4+a)**(3/4),x)

[Out]

x**9*gamma(9/4)*hyper((3/4, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/4)*gamma(13/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^8/(-b*x^4 + a)^(3/4), x)