3.1211 \(\int \frac{1}{x^9 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=108 \[ \frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b \left (a-b x^4\right )^{3/4}}{32 a^2 x^4}-\frac{\left (a-b x^4\right )^{3/4}}{8 a x^8} \]

[Out]

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

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Rubi [A]  time = 0.0612817, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {266, 51, 63, 298, 203, 206} \[ \frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b \left (a-b x^4\right )^{3/4}}{32 a^2 x^4}-\frac{\left (a-b x^4\right )^{3/4}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a - b*x^4)^(1/4)),x]

[Out]

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

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 298

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

Rule 203

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

Rule 206

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

Rubi steps

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

Mathematica [C]  time = 0.0074485, size = 41, normalized size = 0.38 \[ -\frac{b^2 \left (a-b x^4\right )^{3/4} \, _2F_1\left (\frac{3}{4},3;\frac{7}{4};1-\frac{b x^4}{a}\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a - b*x^4)^(1/4)),x]

[Out]

-(b^2*(a - b*x^4)^(3/4)*Hypergeometric2F1[3/4, 3, 7/4, 1 - (b*x^4)/a])/(3*a^3)

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^9/(-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(1/x^9/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62164, size = 528, normalized size = 4.89 \begin{align*} -\frac{20 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b^{6} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} - \sqrt{a^{5} b^{8} \sqrt{\frac{b^{8}}{a^{9}}} + \sqrt{-b x^{4} + a} b^{12}} a^{2} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}}}{b^{8}}\right ) + 5 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) - 5 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) + 4 \,{\left (5 \, b x^{4} + 4 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, a^{2} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 3.02537, size = 41, normalized size = 0.38 \begin{align*} - \frac{e^{- \frac{i \pi }{4}} \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{a}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x^{9} \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(-b*x**4+a)**(1/4),x)

[Out]

-exp(-I*pi/4)*gamma(9/4)*hyper((1/4, 9/4), (13/4,), a/(b*x**4))/(4*b**(1/4)*x**9*gamma(13/4))

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Giac [B]  time = 1.16492, size = 316, normalized size = 2.93 \begin{align*} -\frac{1}{256} \, b^{2}{\left (\frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} - \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} + \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{8 \,{\left (5 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} - 9 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a\right )}}{a^{2} b^{2} x^{8}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

-1/256*b^2*(10*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a
^3 + 10*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a^3 - 5
*sqrt(2)*(-a)^(3/4)*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a^3 + 5*sqrt(2)*(
-a)^(3/4)*log(-sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a^3 - 8*(5*(-b*x^4 + a)^(7
/4) - 9*(-b*x^4 + a)^(3/4)*a)/(a^2*b^2*x^8))