3.1181 \(\int \frac{\sqrt [4]{a-b x^4}}{x^5} \, dx\)

Optimal. Leaf size=78 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{4 x^4} \]

[Out]

-(a - b*x^4)^(1/4)/(4*x^4) + (b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(3/4)) + (b*ArcTanh[(a - b*x^4)^(1/4)/
a^(1/4)])/(8*a^(3/4))

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Rubi [A]  time = 0.0431871, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {266, 47, 63, 212, 206, 203} \[ \frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a-b x^4}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(1/4)/x^5,x]

[Out]

-(a - b*x^4)^(1/4)/(4*x^4) + (b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(3/4)) + (b*ArcTanh[(a - b*x^4)^(1/4)/
a^(1/4)])/(8*a^(3/4))

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 212

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

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])

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{a-b x^4}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt [4]{a-b x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a-b x^4}}{4 x^4}-\frac{1}{16} b \operatorname{Subst}\left (\int \frac{1}{x (a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a-b x^4}}{4 x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}-\frac{x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right )\\ &=-\frac{\sqrt [4]{a-b x^4}}{4 x^4}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 \sqrt{a}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 \sqrt{a}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{4 x^4}+\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0084938, size = 39, normalized size = 0.5 \[ -\frac{b \left (a-b x^4\right )^{5/4} \, _2F_1\left (\frac{5}{4},2;\frac{9}{4};1-\frac{b x^4}{a}\right )}{5 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^4)^(1/4)/x^5,x]

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}\sqrt [4]{-b{x}^{4}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^4+a)^(1/4)/x^5,x)

[Out]

int((-b*x^4+a)^(1/4)/x^5,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((-b*x^4+a)^(1/4)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58971, size = 431, normalized size = 5.53 \begin{align*} -\frac{4 \, \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b \left (\frac{b^{4}}{a^{3}}\right )^{\frac{3}{4}} - \sqrt{\sqrt{-b x^{4} + a} b^{2} + a^{2} \sqrt{\frac{b^{4}}{a^{3}}}} a^{2} \left (\frac{b^{4}}{a^{3}}\right )^{\frac{3}{4}}}{b^{4}}\right ) - \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} b + a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} b - a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^5,x, algorithm="fricas")

[Out]

-1/16*(4*(b^4/a^3)^(1/4)*x^4*arctan(-((-b*x^4 + a)^(1/4)*a^2*b*(b^4/a^3)^(3/4) - sqrt(sqrt(-b*x^4 + a)*b^2 + a
^2*sqrt(b^4/a^3))*a^2*(b^4/a^3)^(3/4))/b^4) - (b^4/a^3)^(1/4)*x^4*log((-b*x^4 + a)^(1/4)*b + a*(b^4/a^3)^(1/4)
) + (b^4/a^3)^(1/4)*x^4*log((-b*x^4 + a)^(1/4)*b - a*(b^4/a^3)^(1/4)) + 4*(-b*x^4 + a)^(1/4))/x^4

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Sympy [C]  time = 1.49708, size = 42, normalized size = 0.54 \begin{align*} \frac{\sqrt [4]{b} e^{- \frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**4+a)**(1/4)/x**5,x)

[Out]

b**(1/4)*exp(-3*I*pi/4)*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), a/(b*x**4))/(4*x**3*gamma(7/4))

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Giac [B]  time = 1.18555, size = 286, normalized size = 3.67 \begin{align*} \frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{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} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{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} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{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} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{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} - \frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{b x^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^4+a)^(1/4)/x^5,x, algorithm="giac")

[Out]

1/32*b*(2*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a + 2*
sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a + sqrt(2)*(-a
)^(1/4)*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a - sqrt(2)*(-a)^(1/4)*log(-s
qrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a - 8*(-b*x^4 + a)^(1/4)/(b*x^4))