3.1219 \(\int \frac{x^8}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]

[Out]

(-5*a*x*(a - b*x^4)^(3/4))/(32*b^2) - (x^5*(a - b*x^4)^(3/4))/(8*b) - (5*a^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a
- b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) + (5*a^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]
*b^(9/4)) - (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2
]*b^(9/4)) + (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[
2]*b^(9/4))

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Rubi [A]  time = 0.135438, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {321, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a - b*x^4)^(1/4),x]

[Out]

(-5*a*x*(a - b*x^4)^(3/4))/(32*b^2) - (x^5*(a - b*x^4)^(3/4))/(8*b) - (5*a^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a
- b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) + (5*a^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]
*b^(9/4)) - (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2
]*b^(9/4)) + (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[
2]*b^(9/4))

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 240

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

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{\sqrt [4]{a-b x^4}} \, dx &=-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac{(5 a) \int \frac{x^4}{\sqrt [4]{a-b x^4}} \, dx}{8 b}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt [4]{a-b x^4}} \, dx}{32 b^2}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{32 b^2}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{64 b^2}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{64 b^2}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}+2 x}{-\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}-2 x}{-\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}\\ &=-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{9/4}}\\ \end{align*}

Mathematica [A]  time = 0.129546, size = 245, normalized size = 0.93 \[ -\frac{5 \sqrt{2} a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-5 \sqrt{2} a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )+10 \sqrt{2} a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )-10 \sqrt{2} a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )+32 b^{5/4} x^5 \left (a-b x^4\right )^{3/4}+40 a \sqrt [4]{b} x \left (a-b x^4\right )^{3/4}}{256 b^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a - b*x^4)^(1/4),x]

[Out]

-(40*a*b^(1/4)*x*(a - b*x^4)^(3/4) + 32*b^(5/4)*x^5*(a - b*x^4)^(3/4) + 10*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*b^(
1/4)*x)/(a - b*x^4)^(1/4)] - 10*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)] + 5*Sqrt[2]*a^2*
Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)] - 5*Sqrt[2]*a^2*Log[1 + (Sqrt[b
]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(256*b^(9/4))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{{x}^{8}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.95223, size = 544, normalized size = 2.06 \begin{align*} \frac{20 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6} b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} - b^{2} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \sqrt{-\frac{a^{8} b^{5} x^{2} \sqrt{-\frac{a^{8}}{b^{9}}} - \sqrt{-b x^{4} + a} a^{12}}{x^{2}}}}{a^{8} x}\right ) - 5 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{125 \,{\left (b^{7} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 5 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-\frac{125 \,{\left (b^{7} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 4 \,{\left (4 \, b x^{5} + 5 \, a x\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

1/128*(20*b^2*(-a^8/b^9)^(1/4)*arctan(-((-b*x^4 + a)^(1/4)*a^6*b^2*(-a^8/b^9)^(1/4) - b^2*x*(-a^8/b^9)^(1/4)*s
qrt(-(a^8*b^5*x^2*sqrt(-a^8/b^9) - sqrt(-b*x^4 + a)*a^12)/x^2))/(a^8*x)) - 5*b^2*(-a^8/b^9)^(1/4)*log(125*(b^7
*x*(-a^8/b^9)^(3/4) + (-b*x^4 + a)^(1/4)*a^6)/x) + 5*b^2*(-a^8/b^9)^(1/4)*log(-125*(b^7*x*(-a^8/b^9)^(3/4) - (
-b*x^4 + a)^(1/4)*a^6)/x) - 4*(4*b*x^5 + 5*a*x)*(-b*x^4 + a)^(3/4))/b^2

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Sympy [C]  time = 1.79394, size = 39, normalized size = 0.15 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \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)**(1/4),x)

[Out]

x**9*gamma(9/4)*hyper((1/4, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(1/4)*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{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^8/(-b*x^4 + a)^(1/4), x)