Integrand size = 15, antiderivative size = 77 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=-\frac {1}{2} a^2 \sqrt {a+\frac {b}{x^4}}-\frac {1}{6} a \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {1}{10} \left (a+\frac {b}{x^4}\right )^{5/2}+\frac {1}{2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \] Output:
-1/2*a^2*(a+b/x^4)^(1/2)-1/6*a*(a+b/x^4)^(3/2)-1/10*(a+b/x^4)^(5/2)+1/2*a^ (5/2)*arctanh((a+b/x^4)^(1/2)/a^(1/2))
Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=\frac {\sqrt {a+\frac {b}{x^4}} \left (-3 b^2-11 a b x^4-23 a^2 x^8+\frac {15 a^{5/2} x^{10} \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{\sqrt {b+a x^4}}\right )}{30 x^8} \] Input:
Integrate[(a + b/x^4)^(5/2)/x,x]
Output:
(Sqrt[a + b/x^4]*(-3*b^2 - 11*a*b*x^4 - 23*a^2*x^8 + (15*a^(5/2)*x^10*Log[ Sqrt[a]*x^2 + Sqrt[b + a*x^4]])/Sqrt[b + a*x^4]))/(30*x^8)
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 60, 60, 60, 73, 221}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{4} \int \left (a+\frac {b}{x^4}\right )^{5/2} x^4d\frac {1}{x^4}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-a \int \left (a+\frac {b}{x^4}\right )^{3/2} x^4d\frac {1}{x^4}-\frac {2}{5} \left (a+\frac {b}{x^4}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-a \left (a \int \sqrt {a+\frac {b}{x^4}} x^4d\frac {1}{x^4}+\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )-\frac {2}{5} \left (a+\frac {b}{x^4}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-a \left (a \left (a \int \frac {x^4}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x^4}+2 \sqrt {a+\frac {b}{x^4}}\right )+\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )-\frac {2}{5} \left (a+\frac {b}{x^4}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-a \left (a \left (\frac {2 a \int \frac {1}{\frac {1}{b x^8}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^4}}}{b}+2 \sqrt {a+\frac {b}{x^4}}\right )+\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )-\frac {2}{5} \left (a+\frac {b}{x^4}\right )^{5/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (-a \left (a \left (2 \sqrt {a+\frac {b}{x^4}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+\frac {b}{x^4}\right )^{3/2}\right )-\frac {2}{5} \left (a+\frac {b}{x^4}\right )^{5/2}\right )\) |
Input:
Int[(a + b/x^4)^(5/2)/x,x]
Output:
((-2*(a + b/x^4)^(5/2))/5 - a*((2*(a + b/x^4)^(3/2))/3 + a*(2*Sqrt[a + b/x ^4] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {\left (23 a^{2} x^{8}+11 a b \,x^{4}+3 b^{2}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{30 x^{8}}+\frac {a^{\frac {5}{2}} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{2 \sqrt {a \,x^{4}+b}}\) | \(89\) |
default | \(-\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} \left (-15 a^{\frac {5}{2}} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) x^{10}+23 a^{2} x^{8} \sqrt {a \,x^{4}+b}+11 a b \sqrt {a \,x^{4}+b}\, x^{4}+3 b^{2} \sqrt {a \,x^{4}+b}\right )}{30 \left (a \,x^{4}+b \right )^{\frac {5}{2}}}\) | \(99\) |
Input:
int((a+b/x^4)^(5/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/30*(23*a^2*x^8+11*a*b*x^4+3*b^2)/x^8*((a*x^4+b)/x^4)^(1/2)+1/2*a^(5/2)* ln(a^(1/2)*x^2+(a*x^4+b)^(1/2))*((a*x^4+b)/x^4)^(1/2)*x^2/(a*x^4+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=\left [\frac {15 \, a^{\frac {5}{2}} x^{8} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - 2 \, {\left (23 \, a^{2} x^{8} + 11 \, a b x^{4} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{60 \, x^{8}}, -\frac {15 \, \sqrt {-a} a^{2} x^{8} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a}\right ) + {\left (23 \, a^{2} x^{8} + 11 \, a b x^{4} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{30 \, x^{8}}\right ] \] Input:
integrate((a+b/x^4)^(5/2)/x,x, algorithm="fricas")
Output:
[1/60*(15*a^(5/2)*x^8*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 2*(23*a^2*x^8 + 11*a*b*x^4 + 3*b^2)*sqrt((a*x^4 + b)/x^4))/x^8, -1/3 0*(15*sqrt(-a)*a^2*x^8*arctan(sqrt(-a)*sqrt((a*x^4 + b)/x^4)/a) + (23*a^2* x^8 + 11*a*b*x^4 + 3*b^2)*sqrt((a*x^4 + b)/x^4))/x^8]
Time = 3.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=- \frac {23 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x^{4}}}}{30} - \frac {a^{\frac {5}{2}} \log {\left (\frac {b}{a x^{4}} \right )}}{4} + \frac {a^{\frac {5}{2}} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{2} - \frac {11 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x^{4}}}}{30 x^{4}} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x^{4}}}}{10 x^{8}} \] Input:
integrate((a+b/x**4)**(5/2)/x,x)
Output:
-23*a**(5/2)*sqrt(1 + b/(a*x**4))/30 - a**(5/2)*log(b/(a*x**4))/4 + a**(5/ 2)*log(sqrt(1 + b/(a*x**4)) + 1)/2 - 11*a**(3/2)*b*sqrt(1 + b/(a*x**4))/(3 0*x**4) - sqrt(a)*b**2*sqrt(1 + b/(a*x**4))/(10*x**8)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=-\frac {1}{4} \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{10} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} a^{2} \] Input:
integrate((a+b/x^4)^(5/2)/x,x, algorithm="maxima")
Output:
-1/4*a^(5/2)*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a))) - 1/10*(a + b/x^4)^(5/2) - 1/6*(a + b/x^4)^(3/2)*a - 1/2*sqrt(a + b/x^4)*a ^2
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (57) = 114\).
Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=-\frac {1}{4} \, a^{\frac {5}{2}} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {45 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{8} a^{\frac {5}{2}} b - 90 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{6} a^{\frac {5}{2}} b^{2} + 140 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{4} a^{\frac {5}{2}} b^{3} - 70 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} a^{\frac {5}{2}} b^{4} + 23 \, a^{\frac {5}{2}} b^{5}}{15 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b\right )}^{5}} \] Input:
integrate((a+b/x^4)^(5/2)/x,x, algorithm="giac")
Output:
-1/4*a^(5/2)*log((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2) + 1/15*(45*(sqrt(a)*x^ 2 - sqrt(a*x^4 + b))^8*a^(5/2)*b - 90*(sqrt(a)*x^2 - sqrt(a*x^4 + b))^6*a^ (5/2)*b^2 + 140*(sqrt(a)*x^2 - sqrt(a*x^4 + b))^4*a^(5/2)*b^3 - 70*(sqrt(a )*x^2 - sqrt(a*x^4 + b))^2*a^(5/2)*b^4 + 23*a^(5/2)*b^5)/((sqrt(a)*x^2 - s qrt(a*x^4 + b))^2 - b)^5
Time = 1.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=-\frac {a\,{\left (a+\frac {b}{x^4}\right )}^{3/2}}{6}-\frac {{\left (a+\frac {b}{x^4}\right )}^{5/2}}{10}-\frac {a^2\,\sqrt {a+\frac {b}{x^4}}}{2}-\frac {a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x^4}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2} \] Input:
int((a + b/x^4)^(5/2)/x,x)
Output:
- (a^(5/2)*atan(((a + b/x^4)^(1/2)*1i)/a^(1/2))*1i)/2 - (a*(a + b/x^4)^(3/ 2))/6 - (a + b/x^4)^(5/2)/10 - (a^2*(a + b/x^4)^(1/2))/2
Time = 0.25 (sec) , antiderivative size = 441, normalized size of antiderivative = 5.73 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{5/2}}{x} \, dx=\frac {240 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{4} x^{18}+180 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{3} b \,x^{14}+15 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} b^{2} x^{10}-288 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{4} x^{18}-576 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{3} b \,x^{14}-378 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{2} b^{2} x^{10}-115 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a \,b^{3} x^{6}-15 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, b^{4} x^{2}+240 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{5} x^{20}+300 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{4} b \,x^{16}+75 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{3} b^{2} x^{12}-288 a^{5} x^{20}-720 a^{4} b \,x^{16}-630 a^{3} b^{2} x^{12}-250 a^{2} b^{3} x^{8}-50 a \,b^{4} x^{4}-3 b^{5}}{30 x^{10} \left (16 \sqrt {a \,x^{4}+b}\, a^{2} x^{8}+12 \sqrt {a \,x^{4}+b}\, a b \,x^{4}+\sqrt {a \,x^{4}+b}\, b^{2}+16 \sqrt {a}\, a^{2} x^{10}+20 \sqrt {a}\, a b \,x^{6}+5 \sqrt {a}\, b^{2} x^{2}\right )} \] Input:
int((a+b/x^4)^(5/2)/x,x)
Output:
(240*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b ))*a**4*x**18 + 180*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt( a)*x**2)/sqrt(b))*a**3*b*x**14 + 15*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x **4 + b) + sqrt(a)*x**2)/sqrt(b))*a**2*b**2*x**10 - 288*sqrt(a)*sqrt(a*x** 4 + b)*a**4*x**18 - 576*sqrt(a)*sqrt(a*x**4 + b)*a**3*b*x**14 - 378*sqrt(a )*sqrt(a*x**4 + b)*a**2*b**2*x**10 - 115*sqrt(a)*sqrt(a*x**4 + b)*a*b**3*x **6 - 15*sqrt(a)*sqrt(a*x**4 + b)*b**4*x**2 + 240*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**5*x**20 + 300*log((sqrt(a*x**4 + b) + sqrt(a)*x* *2)/sqrt(b))*a**4*b*x**16 + 75*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt( b))*a**3*b**2*x**12 - 288*a**5*x**20 - 720*a**4*b*x**16 - 630*a**3*b**2*x* *12 - 250*a**2*b**3*x**8 - 50*a*b**4*x**4 - 3*b**5)/(30*x**10*(16*sqrt(a*x **4 + b)*a**2*x**8 + 12*sqrt(a*x**4 + b)*a*b*x**4 + sqrt(a*x**4 + b)*b**2 + 16*sqrt(a)*a**2*x**10 + 20*sqrt(a)*a*b*x**6 + 5*sqrt(a)*b**2*x**2))