3.1241 \(\int \frac{1}{x^9 (a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{21 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{21 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}-\frac{\sqrt [4]{a-b x^4}}{8 a x^8} \]

[Out]

-(a - b*x^4)^(1/4)/(8*a*x^8) - (7*b*(a - b*x^4)^(1/4))/(32*a^2*x^4) - (21*b^2*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)
])/(64*a^(11/4)) - (21*b^2*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(64*a^(11/4))

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Rubi [A]  time = 0.0590348, 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, 212, 206, 203} \[ -\frac{21 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{21 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}-\frac{\sqrt [4]{a-b x^4}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(8*a*x^8) - (7*b*(a - b*x^4)^(1/4))/(32*a^2*x^4) - (21*b^2*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)
])/(64*a^(11/4)) - (21*b^2*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(64*a^(11/4))

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 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{1}{x^9 \left (a-b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^3 (a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a-b x^4}}{8 a x^8}+\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^2 (a-b x)^{3/4}} \, dx,x,x^4\right )}{32 a}\\ &=-\frac{\sqrt [4]{a-b x^4}}{8 a x^8}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}+\frac{\left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a-b x)^{3/4}} \, dx,x,x^4\right )}{128 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{8 a x^8}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}-\frac{(21 b) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}-\frac{x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right )}{32 a^2}\\ &=-\frac{\sqrt [4]{a-b x^4}}{8 a x^8}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}-\frac{\left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{64 a^{5/2}}-\frac{\left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{64 a^{5/2}}\\ &=-\frac{\sqrt [4]{a-b x^4}}{8 a x^8}-\frac{7 b \sqrt [4]{a-b x^4}}{32 a^2 x^4}-\frac{21 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{21 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}\\ \end{align*}

Mathematica [C]  time = 0.0075023, size = 39, normalized size = 0.36 \[ -\frac{b^2 \sqrt [4]{a-b x^4} \, _2F_1\left (\frac{1}{4},3;\frac{5}{4};1-\frac{b x^4}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a - b*x^4)^(3/4)),x]

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^9/(-b*x^4+a)^(3/4),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)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89168, size = 528, normalized size = 4.89 \begin{align*} \frac{84 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{8} b^{2} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{3}{4}} - \sqrt{a^{6} \sqrt{\frac{b^{8}}{a^{11}}} + \sqrt{-b x^{4} + a} b^{4}} a^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{3}{4}}}{b^{8}}\right ) - 21 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \log \left (21 \, a^{3} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} + 21 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2}\right ) + 21 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-21 \, a^{3} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} + 21 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2}\right ) - 4 \,{\left (7 \, b x^{4} + 4 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{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)^(3/4),x, algorithm="fricas")

[Out]

1/128*(84*a^2*x^8*(b^8/a^11)^(1/4)*arctan(-((-b*x^4 + a)^(1/4)*a^8*b^2*(b^8/a^11)^(3/4) - sqrt(a^6*sqrt(b^8/a^
11) + sqrt(-b*x^4 + a)*b^4)*a^8*(b^8/a^11)^(3/4))/b^8) - 21*a^2*x^8*(b^8/a^11)^(1/4)*log(21*a^3*(b^8/a^11)^(1/
4) + 21*(-b*x^4 + a)^(1/4)*b^2) + 21*a^2*x^8*(b^8/a^11)^(1/4)*log(-21*a^3*(b^8/a^11)^(1/4) + 21*(-b*x^4 + a)^(
1/4)*b^2) - 4*(7*b*x^4 + 4*a)*(-b*x^4 + a)^(1/4))/(a^2*x^8)

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Sympy [C]  time = 2.70953, size = 42, normalized size = 0.39 \begin{align*} - \frac{e^{- \frac{3 i \pi }{4}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{11} \Gamma \left (\frac{15}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(-b*x**4+a)**(3/4),x)

[Out]

-exp(-3*I*pi/4)*gamma(11/4)*hyper((3/4, 11/4), (15/4,), a/(b*x**4))/(4*b**(3/4)*x**11*gamma(15/4))

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Giac [B]  time = 1.24519, size = 316, normalized size = 2.93 \begin{align*} -\frac{1}{256} \, b^{2}{\left (\frac{42 \, \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^{3}} + \frac{42 \, \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^{3}} + \frac{21 \, \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^{3}} - \frac{21 \, \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^{3}} - \frac{8 \,{\left (7 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} - 11 \,{\left (-b x^{4} + a\right )}^{\frac{1}{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)^(3/4),x, algorithm="giac")

[Out]

-1/256*b^2*(42*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
^3 + 42*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^3 + 2
1*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^3 - 21*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^3 - 8*(7*(-b*x^4 + a)^
(5/4) - 11*(-b*x^4 + a)^(1/4)*a)/(a^2*b^2*x^8))