Integrand size = 15, antiderivative size = 74 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {b \sqrt {a+\frac {b}{x^3}} x^3}{4 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^6}{6 a}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:
-1/4*b*(a+b/x^3)^(1/2)*x^3/a^2+1/6*(a+b/x^3)^(1/2)*x^6/a+1/4*b^2*arctanh(( a+b/x^3)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {\sqrt {a} x^{3/2} \left (-3 b^2-a b x^3+2 a^2 x^6\right )+3 b^2 \sqrt {b+a x^3} \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )}{12 a^{5/2} \sqrt {a+\frac {b}{x^3}} x^{3/2}} \] Input:
Integrate[x^5/Sqrt[a + b/x^3],x]
Output:
(Sqrt[a]*x^(3/2)*(-3*b^2 - a*b*x^3 + 2*a^2*x^6) + 3*b^2*Sqrt[b + a*x^3]*Lo g[Sqrt[a]*x^(3/2) + Sqrt[b + a*x^3]])/(12*a^(5/2)*Sqrt[a + b/x^3]*x^(3/2))
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 52, 52, 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.
\(\displaystyle \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \frac {x^9}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3 b \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x^3}}{4 a}+\frac {x^6 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {3 b \left (-\frac {b \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x^3}}{2 a}-\frac {x^3 \sqrt {a+\frac {b}{x^3}}}{a}\right )}{4 a}+\frac {x^6 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3 b \left (-\frac {\int \frac {1}{\frac {1}{b x^6}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^3}}}{a}-\frac {x^3 \sqrt {a+\frac {b}{x^3}}}{a}\right )}{4 a}+\frac {x^6 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^3 \sqrt {a+\frac {b}{x^3}}}{a}\right )}{4 a}+\frac {x^6 \sqrt {a+\frac {b}{x^3}}}{2 a}\right )\) |
Input:
Int[x^5/Sqrt[a + b/x^3],x]
Output:
((Sqrt[a + b/x^3]*x^6)/(2*a) + (3*b*(-((Sqrt[a + b/x^3]*x^3)/a) + (b*ArcTa nh[Sqrt[a + b/x^3]/Sqrt[a]])/a^(3/2)))/(4*a))/3
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[(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 = 0.82 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\frac {\left (2 a \,x^{3}-3 b \right ) \left (a \,x^{3}+b \right )}{12 a^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) \sqrt {x \left (a \,x^{3}+b \right )}}{4 a^{\frac {5}{2}} x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) | \(91\) |
default | \(\frac {\left (a \,x^{3}+b \right ) \left (2 \sqrt {x \left (a \,x^{3}+b \right )}\, a^{\frac {7}{2}} x^{4}-3 \sqrt {x \left (a \,x^{3}+b \right )}\, a^{\frac {5}{2}} b x +3 a^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) b^{2}\right )}{12 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x \sqrt {x \left (a \,x^{3}+b \right )}\, a^{\frac {9}{2}}}\) | \(105\) |
Input:
int(x^5/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/12*(2*a*x^3-3*b)/a^2*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)+1/4*b^2/a^(5/2)*arc tanh((x*(a*x^3+b))^(1/2)/x^2/a^(1/2))/x^2/((a*x^3+b)/x^3)^(1/2)*(x*(a*x^3+ b))^(1/2)
Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.53 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ) + 4 \, {\left (2 \, a^{2} x^{6} - 3 \, a b x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{48 \, a^{3}}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {{\left (2 \, a x^{3} + b\right )} \sqrt {-a} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, {\left (a^{2} x^{3} + a b\right )}}\right ) - 2 \, {\left (2 \, a^{2} x^{6} - 3 \, a b x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{24 \, a^{3}}\right ] \] Input:
integrate(x^5/(a+b/x^3)^(1/2),x, algorithm="fricas")
Output:
[1/48*(3*sqrt(a)*b^2*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3 )*sqrt(a)*sqrt((a*x^3 + b)/x^3)) + 4*(2*a^2*x^6 - 3*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a^3, -1/24*(3*sqrt(-a)*b^2*arctan(1/2*(2*a*x^3 + b)*sqrt(-a)*sqr t((a*x^3 + b)/x^3)/(a^2*x^3 + a*b)) - 2*(2*a^2*x^6 - 3*a*b*x^3)*sqrt((a*x^ 3 + b)/x^3))/a^3]
Time = 2.89 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {x^{\frac {15}{2}}}{6 \sqrt {b} \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {\sqrt {b} x^{\frac {9}{2}}}{12 a \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {b^{\frac {3}{2}} x^{\frac {3}{2}}}{4 a^{2} \sqrt {\frac {a x^{3}}{b} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{4 a^{\frac {5}{2}}} \] Input:
integrate(x**5/(a+b/x**3)**(1/2),x)
Output:
x**(15/2)/(6*sqrt(b)*sqrt(a*x**3/b + 1)) - sqrt(b)*x**(9/2)/(12*a*sqrt(a*x **3/b + 1)) - b**(3/2)*x**(3/2)/(4*a**2*sqrt(a*x**3/b + 1)) + b**2*asinh(s qrt(a)*x**(3/2)/sqrt(b))/(4*a**(5/2))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=-\frac {b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {a + \frac {b}{x^{3}}} a b^{2}}{12 \, {\left ({\left (a + \frac {b}{x^{3}}\right )}^{2} a^{2} - 2 \, {\left (a + \frac {b}{x^{3}}\right )} a^{3} + a^{4}\right )}} \] Input:
integrate(x^5/(a+b/x^3)^(1/2),x, algorithm="maxima")
Output:
-1/8*b^2*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a)))/a^(5 /2) - 1/12*(3*(a + b/x^3)^(3/2)*b^2 - 5*sqrt(a + b/x^3)*a*b^2)/((a + b/x^3 )^2*a^2 - 2*(a + b/x^3)*a^3 + a^4)
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {1}{12} \, \sqrt {a x^{4} + b x} {\left (\frac {2 \, x^{3}}{a} - \frac {3 \, b}{a^{2}}\right )} x - \frac {b^{2} \arctan \left (\frac {\sqrt {a + \frac {b}{x^{3}}}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} \] Input:
integrate(x^5/(a+b/x^3)^(1/2),x, algorithm="giac")
Output:
1/12*sqrt(a*x^4 + b*x)*(2*x^3/a - 3*b/a^2)*x - 1/4*b^2*arctan(sqrt(a + b/x ^3)/sqrt(-a))/(sqrt(-a)*a^2)
Time = 0.98 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {x^6\,\sqrt {a+\frac {b}{x^3}}}{6\,a}+\frac {b^2\,\ln \left (x^6\,\left (\sqrt {a+\frac {b}{x^3}}-\sqrt {a}\right )\,{\left (\sqrt {a+\frac {b}{x^3}}+\sqrt {a}\right )}^3\right )}{8\,a^{5/2}}-\frac {b\,x^3\,\sqrt {a+\frac {b}{x^3}}}{4\,a^2} \] Input:
int(x^5/(a + b/x^3)^(1/2),x)
Output:
(x^6*(a + b/x^3)^(1/2))/(6*a) + (b^2*log(x^6*((a + b/x^3)^(1/2) - a^(1/2)) *((a + b/x^3)^(1/2) + a^(1/2))^3))/(8*a^(5/2)) - (b*x^3*(a + b/x^3)^(1/2)) /(4*a^2)
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {4 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a^{2} x^{4}-6 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a b x -3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}-\sqrt {x}\, \sqrt {a}\, x \right ) b^{2}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}+\sqrt {x}\, \sqrt {a}\, x \right ) b^{2}}{24 a^{3}} \] Input:
int(x^5/(a+b/x^3)^(1/2),x)
Output:
(4*sqrt(x)*sqrt(a*x**3 + b)*a**2*x**4 - 6*sqrt(x)*sqrt(a*x**3 + b)*a*b*x - 3*sqrt(a)*log(sqrt(a*x**3 + b) - sqrt(x)*sqrt(a)*x)*b**2 + 3*sqrt(a)*log( sqrt(a*x**3 + b) + sqrt(x)*sqrt(a)*x)*b**2)/(24*a**3)