Integrand size = 16, antiderivative size = 105 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=-\frac {\sqrt [4]{a-b x^4}}{8 x^8}+\frac {b \sqrt [4]{a-b x^4}}{32 a x^4}+\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}} \] Output:
-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)
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=\frac {\sqrt [4]{a-b x^4} \left (-4 a+b x^4\right )}{32 a x^8}+\frac {3 b^2 \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}} \] Input:
Integrate[(a - b*x^4)^(1/4)/x^9,x]
Output:
((a - b*x^4)^(1/4)*(-4*a + b*x^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))
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {798, 51, 52, 73, 756, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt [4]{a-b x^4}}{x^{12}}dx^4\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}}dx^4-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \left (\frac {3 b \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}}dx^4}{4 a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \left (-\frac {3 \int \frac {1}{\frac {a}{b}-\frac {x^{16}}{b}}d\sqrt [4]{a-b x^4}}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \left (-\frac {3 \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \int \frac {1}{x^8+\sqrt {a}}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \left (-\frac {3 \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{8} b \left (-\frac {3 \left (\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{a}-\frac {\sqrt [4]{a-b x^4}}{a x^4}\right )-\frac {\sqrt [4]{a-b x^4}}{2 x^8}\right )\) |
Input:
Int[(a - b*x^4)^(1/4)/x^9,x]
Output:
(-1/2*(a - b*x^4)^(1/4)/x^8 - (b*(-((a - b*x^4)^(1/4)/(a*x^4)) - (3*((b*Ar cTan[(a - b*x^4)^(1/4)/a^(1/4)])/(2*a^(3/4)) + (b*ArcTanh[(a - b*x^4)^(1/4 )/a^(1/4)])/(2*a^(3/4))))/a))/8)/4
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] - Simp[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 ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.57 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {3 \ln \left (\frac {-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b^{2} x^{8}+6 \arctan \left (\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b^{2} x^{8}+4 b \,x^{4} a^{\frac {3}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}-16 a^{\frac {7}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{128 a^{\frac {7}{4}} x^{8}}\) | \(113\) |
Input:
int((-b*x^4+a)^(1/4)/x^9,x,method=_RETURNVERBOSE)
Output:
1/128*(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*a^(3/4)*(-b*x^4+a)^( 1/4)-16*a^(7/4)*(-b*x^4+a)^(1/4))/a^(7/4)/x^8
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=\frac {3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 3 \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) + 3 i \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + 3 i \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 i \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 3 i \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) - 3 \, a \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}} x^{8} \log \left (3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{2} - 3 \, a^{2} \left (\frac {b^{8}}{a^{7}}\right )^{\frac {1}{4}}\right ) + 4 \, {\left (b x^{4} - 4 \, a\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, a x^{8}} \] Input:
integrate((-b*x^4+a)^(1/4)/x^9,x, algorithm="fricas")
Output:
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)
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=- \frac {\sqrt [4]{b} e^{\frac {i \pi }{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((-b*x**4+a)**(1/4)/x**9,x)
Output:
-b**(1/4)*exp(I*pi/4)*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), a/(b*x**4))/( 4*x**7*gamma(11/4))
Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{2} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a b^{2}}{32 \, {\left ({\left (b x^{4} - a\right )}^{2} a + 2 \, {\left (b x^{4} - a\right )} a^{2} + a^{3}\right )}} + \frac {3 \, {\left (\frac {2 \, b^{2} \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b^{2} \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{128 \, a} \] Input:
integrate((-b*x^4+a)^(1/4)/x^9,x, algorithm="maxima")
Output:
-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
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (81) = 162\).
Time = 0.13 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.39 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=\frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{a^{2}} + \frac {3 \, \sqrt {2} b^{3} \log \left (-\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}} a} - \frac {8 \, {\left ({\left (-b x^{4} + a\right )}^{\frac {5}{4}} b^{3} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a b^{3}\right )}}{a b^{2} x^{8}}}{256 \, b} \] Input:
integrate((-b*x^4+a)^(1/4)/x^9,x, algorithm="giac")
Output:
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
Time = 0.64 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=\frac {3\,b^2\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{64\,a^{7/4}}-\frac {3\,{\left (a-b\,x^4\right )}^{1/4}}{32\,x^8}-\frac {{\left (a-b\,x^4\right )}^{5/4}}{32\,a\,x^8}-\frac {b^2\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,3{}\mathrm {i}}{64\,a^{7/4}} \] Input:
int((a - b*x^4)^(1/4)/x^9,x)
Output:
(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)
\[ \int \frac {\sqrt [4]{a-b x^4}}{x^9} \, dx=\frac {-4 \left (-b \,x^{4}+a \right )^{\frac {1}{4}} a +\left (-b \,x^{4}+a \right )^{\frac {1}{4}} b \,x^{4}-3 \left (\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{-b \,x^{5}+a x}d x \right ) b^{2} x^{8}}{32 a \,x^{8}} \] Input:
int((-b*x^4+a)^(1/4)/x^9,x)
Output:
( - 4*(a - b*x**4)**(1/4)*a + (a - b*x**4)**(1/4)*b*x**4 - 3*int((a - b*x* *4)**(1/4)/(a*x - b*x**5),x)*b**2*x**8)/(32*a*x**8)