\(\int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx\) [995]

Optimal result

Integrand size = 15, antiderivative size = 101 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=-\frac {\sqrt [4]{a+b x^4}}{8 x^8}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}+\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}} \]

[Out]

-1/8*(b*x^4+a)^(1/4)/x^8-1/32*b*(b*x^4+a)^(1/4)/a/x^4+3/64*b^2*arctan((b*x^4+a)^(1/4)/a^(1/4))/a^(7/4)+3/64*b^
2*arctanh((b*x^4+a)^(1/4)/a^(1/4))/a^(7/4)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {272, 43, 44, 65, 218, 212, 209} \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac {\sqrt [4]{a+b x^4}}{8 x^8} \]

[In]

Int[(a + b*x^4)^(1/4)/x^9,x]

[Out]

-1/8*(a + b*x^4)^(1/4)/x^8 - (b*(a + b*x^4)^(1/4))/(32*a*x^4) + (3*b^2*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*
a^(7/4)) + (3*b^2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(7/4))

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 272

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x^3} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 x^8}+\frac {1}{32} b \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/4}} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 x^8}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )}{128 a} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 x^8}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{32 a} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 x^8}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{3/2}}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{64 a^{3/2}} \\ & = -\frac {\sqrt [4]{a+b x^4}}{8 x^8}-\frac {b \sqrt [4]{a+b x^4}}{32 a x^4}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=\frac {\left (-4 a-b x^4\right ) \sqrt [4]{a+b x^4}}{32 a x^8}+\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}} \]

[In]

Integrate[(a + b*x^4)^(1/4)/x^9,x]

[Out]

((-4*a - b*x^4)*(a + b*x^4)^(1/4))/(32*a*x^8) + (3*b^2*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(7/4)) + (3*b^
2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(7/4))

Maple [A] (verified)

Time = 4.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {3 \ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b^{2} x^{8}+6 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b^{2} x^{8}-4 b \,x^{4} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}-16 a^{\frac {7}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{128 a^{\frac {7}{4}} x^{8}}\)	\(108\)

[In]

int((b*x^4+a)^(1/4)/x^9,x,method=_RETURNVERBOSE)

[Out]

1/128/a^(7/4)*(3*ln((-(b*x^4+a)^(1/4)-a^(1/4))/(-(b*x^4+a)^(1/4)+a^(1/4)))*b^2*x^8+6*arctan((b*x^4+a)^(1/4)/a^
(1/4))*b^2*x^8-4*b*x^4*(b*x^4+a)^(1/4)*a^(3/4)-16*a^(7/4)*(b*x^4+a)^(1/4))/x^8

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=\frac {3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 3 \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) + 3 i \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 3 i \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 i \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 3 i \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 3 \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, a x^{8}} \]

[In]

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

[Out]

1/128*(3*a*(b^8/a^7)^(1/4)*x^8*log(3*(b*x^4 + a)^(1/4)*b^2 + 3*a^2*(b^8/a^7)^(1/4)) + 3*I*a*(b^8/a^7)^(1/4)*x^
8*log(3*(b*x^4 + a)^(1/4)*b^2 + 3*I*a^2*(b^8/a^7)^(1/4)) - 3*I*a*(b^8/a^7)^(1/4)*x^8*log(3*(b*x^4 + a)^(1/4)*b
^2 - 3*I*a^2*(b^8/a^7)^(1/4)) - 3*a*(b^8/a^7)^(1/4)*x^8*log(3*(b*x^4 + a)^(1/4)*b^2 - 3*a^2*(b^8/a^7)^(1/4)) -
4*(b*x^4 + 4*a)*(b*x^4 + a)^(1/4))/(a*x^8)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=- \frac {\sqrt [4]{b} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (\frac {11}{4}\right )} \]

[In]

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

[Out]

-b**(1/4)*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), a*exp_polar(I*pi)/(b*x**4))/(4*x**7*gamma(11/4))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=-\frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2} + 3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} + a\right )}^{2} a - 2 \, {\left (b x^{4} + a\right )} a^{2} + a^{3}\right )}} + \frac {3 \, {\left (\frac {2 \, b^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b^{2} \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{128 \, a} \]

[In]

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

[Out]

-1/32*((b*x^4 + a)^(5/4)*b^2 + 3*(b*x^4 + a)^(1/4)*a*b^2)/((b*x^4 + a)^2*a - 2*(b*x^4 + a)*a^2 + a^3) + 3/128*
(2*b^2*arctan((b*x^4 + a)^(1/4)/a^(1/4))/a^(3/4) - b^2*log(((b*x^4 + a)^(1/4) - a^(1/4))/((b*x^4 + a)^(1/4) +
a^(1/4)))/a^(3/4))/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (77) = 154\).

Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.41 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=\frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{a^{2}} + \frac {3 \, \sqrt {2} b^{3} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}} a} - \frac {8 \, {\left ({\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3} + 3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b^{3}\right )}}{a b^{2} x^{8}}}{256 \, b} \]

[In]

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

[Out]

1/256*(6*sqrt(2)*(-a)^(1/4)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a^2
+ 6*sqrt(2)*(-a)^(1/4)*b^3*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a^2 + 3*
sqrt(2)*(-a)^(1/4)*b^3*log(sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a^2 + 3*sqrt(2)*
b^3*log(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/((-a)^(3/4)*a) - 8*((b*x^4 + a)^(5
/4)*b^3 + 3*(b*x^4 + a)^(1/4)*a*b^3)/(a*b^2*x^8))/b

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt [4]{a+b x^4}}{x^9} \, dx=\frac {3\,b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{7/4}}-\frac {3\,{\left (b\,x^4+a\right )}^{1/4}}{32\,x^8}-\frac {{\left (b\,x^4+a\right )}^{5/4}}{32\,a\,x^8}-\frac {b^2\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,3{}\mathrm {i}}{64\,a^{7/4}} \]

[In]

int((a + b*x^4)^(1/4)/x^9,x)

[Out]

(3*b^2*atan((a + b*x^4)^(1/4)/a^(1/4)))/(64*a^(7/4)) - (3*(a + b*x^4)^(1/4))/(32*x^8) - (b^2*atan(((a + b*x^4)
^(1/4)*1i)/a^(1/4))*3i)/(64*a^(7/4)) - (a + b*x^4)^(5/4)/(32*a*x^8)