Integrand size = 15, antiderivative size = 64 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=-\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {1}{2 a^2 \sqrt {a+\frac {b}{x^4}}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Output:
-1/6/a/(a+b/x^4)^(3/2)-1/2/a^2/(a+b/x^4)^(1/2)+1/2*arctanh((a+b/x^4)^(1/2) /a^(1/2))/a^(5/2)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=\frac {-\sqrt {a} x^2 \left (3 b+4 a x^4\right )+3 \left (b+a x^4\right )^{3/2} \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{6 a^{5/2} \sqrt {a+\frac {b}{x^4}} x^2 \left (b+a x^4\right )} \] Input:
Integrate[1/((a + b/x^4)^(5/2)*x),x]
Output:
(-(Sqrt[a]*x^2*(3*b + 4*a*x^4)) + 3*(b + a*x^4)^(3/2)*Log[Sqrt[a]*x^2 + Sq rt[b + a*x^4]])/(6*a^(5/2)*Sqrt[a + b/x^4]*x^2*(b + a*x^4))
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 61, 61, 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.
\(\displaystyle \int \frac {1}{x \left (a+\frac {b}{x^4}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{4} \int \frac {x^4}{\left (a+\frac {b}{x^4}\right )^{5/2}}d\frac {1}{x^4}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {x^4}{\left (a+\frac {b}{x^4}\right )^{3/2}}d\frac {1}{x^4}}{a}-\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{4} \left (-\frac {\frac {\int \frac {x^4}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x^4}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x^4}}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {\frac {2 \int \frac {1}{\frac {1}{b x^8}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^4}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x^4}}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (-\frac {\frac {2}{a \sqrt {a+\frac {b}{x^4}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
Input:
Int[1/((a + b/x^4)^(5/2)*x),x]
Output:
(-2/(3*a*(a + b/x^4)^(3/2)) - (2/(a*Sqrt[a + b/x^4]) - (2*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/a^(3/2))/a)/4
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] && 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]
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs. \(2(48)=96\).
Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.45
method | result | size |
default | \(-\frac {\left (a \,x^{4}+b \right )^{\frac {5}{2}} \left (4 a^{\frac {9}{2}} \sqrt {-\frac {\left (a \,x^{2}+\sqrt {-a b}\right ) \left (-a \,x^{2}+\sqrt {-a b}\right )}{a}}\, x^{6}-3 \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) a^{5} x^{8}+3 a^{\frac {7}{2}} b \sqrt {-\frac {\left (a \,x^{2}+\sqrt {-a b}\right ) \left (-a \,x^{2}+\sqrt {-a b}\right )}{a}}\, x^{2}-6 a^{4} b \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) x^{4}-3 a^{3} b^{2} \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )\right )}{6 a^{\frac {7}{2}} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} x^{10} \left (-a \,x^{2}+\sqrt {-a b}\right )^{2} \left (a \,x^{2}+\sqrt {-a b}\right )^{2}}\) | \(221\) |
Input:
int(1/(a+b/x^4)^(5/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/6*(a*x^4+b)^(5/2)*(4*a^(9/2)*(-1/a*(a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^ (1/2)))^(1/2)*x^6-3*ln(a^(1/2)*x^2+(a*x^4+b)^(1/2))*a^5*x^8+3*a^(7/2)*b*(- 1/a*(a*x^2+(-a*b)^(1/2))*(-a*x^2+(-a*b)^(1/2)))^(1/2)*x^2-6*a^4*b*ln(a^(1/ 2)*x^2+(a*x^4+b)^(1/2))*x^4-3*a^3*b^2*ln(a^(1/2)*x^2+(a*x^4+b)^(1/2)))/a^( 7/2)/((a*x^4+b)/x^4)^(5/2)/x^10/(-a*x^2+(-a*b)^(1/2))^2/(a*x^2+(-a*b)^(1/2 ))^2
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (48) = 96\).
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.48 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=\left [\frac {3 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {a} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - 2 \, {\left (4 \, a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}}, -\frac {3 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a}\right ) + {\left (4 \, a^{2} x^{8} + 3 \, a b x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{6 \, {\left (a^{5} x^{8} + 2 \, a^{4} b x^{4} + a^{3} b^{2}\right )}}\right ] \] Input:
integrate(1/(a+b/x^4)^(5/2)/x,x, algorithm="fricas")
Output:
[1/12*(3*(a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt(a)*log(-2*a*x^4 - 2*sqrt(a)*x^4* sqrt((a*x^4 + b)/x^4) - b) - 2*(4*a^2*x^8 + 3*a*b*x^4)*sqrt((a*x^4 + b)/x^ 4))/(a^5*x^8 + 2*a^4*b*x^4 + a^3*b^2), -1/6*(3*(a^2*x^8 + 2*a*b*x^4 + b^2) *sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x^4 + b)/x^4)/a) + (4*a^2*x^8 + 3*a*b*x^ 4)*sqrt((a*x^4 + b)/x^4))/(a^5*x^8 + 2*a^4*b*x^4 + a^3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (54) = 108\).
Time = 2.27 (sec) , antiderivative size = 743, normalized size of antiderivative = 11.61 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b/x**4)**(5/2)/x,x)
Output:
-8*a**7*x**12*sqrt(1 + b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x* *8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 3*a**7*x**12*log(b/(a*x **4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) + 6*a**7*x**12*log(sqrt(1 + b/(a*x**4)) + 1)/(12*a**(1 9/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b **3) - 14*a**6*b*x**8*sqrt(1 + b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17 /2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 9*a**6*b*x**8*l og(b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b* *2*x**4 + 12*a**(13/2)*b**3) + 18*a**6*b*x**8*log(sqrt(1 + b/(a*x**4)) + 1 )/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12* a**(13/2)*b**3) - 6*a**5*b**2*x**4*sqrt(1 + b/(a*x**4))/(12*a**(19/2)*x**1 2 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) - 9* a**5*b**2*x**4*log(b/(a*x**4))/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3) + 18*a**5*b**2*x**4*log(sqrt( 1 + b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15 /2)*b**2*x**4 + 12*a**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x**4))/(12*a**(1 9/2)*x**12 + 36*a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b **3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x**4)) + 1)/(12*a**(19/2)*x**12 + 36* a**(17/2)*b*x**8 + 36*a**(15/2)*b**2*x**4 + 12*a**(13/2)*b**3)
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} - \frac {4 \, a + \frac {3 \, b}{x^{4}}}{6 \, {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{2}} \] Input:
integrate(1/(a+b/x^4)^(5/2)/x,x, algorithm="maxima")
Output:
-1/4*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a)))/a^(5/2) - 1/6*(4*a + 3*b/x^4)/((a + b/x^4)^(3/2)*a^2)
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=-\frac {{\left (\frac {4 \, x^{4}}{a} + \frac {3 \, b}{a^{2}}\right )} x^{2}}{6 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, a^{\frac {5}{2}}} \] Input:
integrate(1/(a+b/x^4)^(5/2)/x,x, algorithm="giac")
Output:
-1/6*(4*x^4/a + 3*b/a^2)*x^2/(a*x^4 + b)^(3/2) - 1/2*log(abs(-sqrt(a)*x^2 + sqrt(a*x^4 + b)))/a^(5/2)
Time = 0.64 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {\frac {a+\frac {b}{x^4}}{a^2}+\frac {1}{3\,a}}{2\,{\left (a+\frac {b}{x^4}\right )}^{3/2}} \] Input:
int(1/(x*(a + b/x^4)^(5/2)),x)
Output:
atanh((a + b/x^4)^(1/2)/a^(1/2))/(2*a^(5/2)) - ((a + b/x^4)/a^2 + 1/(3*a)) /(2*(a + b/x^4)^(3/2))
Time = 0.23 (sec) , antiderivative size = 407, normalized size of antiderivative = 6.36 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x} \, dx=\frac {12 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} x^{10}+21 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a b \,x^{6}+9 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) b^{2} x^{2}-16 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{2} x^{10}-16 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a b \,x^{6}-3 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, b^{2} x^{2}+12 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{3} x^{12}+27 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} b \,x^{8}+18 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a \,b^{2} x^{4}+3 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) b^{3}-16 a^{3} x^{12}-24 a^{2} b \,x^{8}-9 a \,b^{2} x^{4}}{6 a^{2} \left (4 \sqrt {a \,x^{4}+b}\, a^{3} x^{10}+7 \sqrt {a \,x^{4}+b}\, a^{2} b \,x^{6}+3 \sqrt {a \,x^{4}+b}\, a \,b^{2} x^{2}+4 \sqrt {a}\, a^{3} x^{12}+9 \sqrt {a}\, a^{2} b \,x^{8}+6 \sqrt {a}\, a \,b^{2} x^{4}+\sqrt {a}\, b^{3}\right )} \] Input:
int(1/(a+b/x^4)^(5/2)/x,x)
Output:
(12*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b) )*a**2*x**10 + 21*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a) *x**2)/sqrt(b))*a*b*x**6 + 9*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b ) + sqrt(a)*x**2)/sqrt(b))*b**2*x**2 - 16*sqrt(a)*sqrt(a*x**4 + b)*a**2*x* *10 - 16*sqrt(a)*sqrt(a*x**4 + b)*a*b*x**6 - 3*sqrt(a)*sqrt(a*x**4 + b)*b* *2*x**2 + 12*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**3*x**12 + 2 7*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**2*b*x**8 + 18*log((sqr t(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a*b**2*x**4 + 3*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*b**3 - 16*a**3*x**12 - 24*a**2*b*x**8 - 9*a*b* *2*x**4)/(6*a**2*(4*sqrt(a*x**4 + b)*a**3*x**10 + 7*sqrt(a*x**4 + b)*a**2* b*x**6 + 3*sqrt(a*x**4 + b)*a*b**2*x**2 + 4*sqrt(a)*a**3*x**12 + 9*sqrt(a) *a**2*b*x**8 + 6*sqrt(a)*a*b**2*x**4 + sqrt(a)*b**3))