Integrand size = 15, antiderivative size = 69 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {5}{12} b \sqrt {a+\frac {b}{x^3}} x^3+\frac {1}{6} a \sqrt {a+\frac {b}{x^3}} x^6+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \] Output:
5/12*b*(a+b/x^3)^(1/2)*x^3+1/6*a*(a+b/x^3)^(1/2)*x^6+1/4*b^2*arctanh((a+b/ x^3)^(1/2)/a^(1/2))/a^(1/2)
Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {1}{12} \sqrt {a+\frac {b}{x^3}} x^{3/2} \left (x^{3/2} \left (5 b+2 a x^3\right )+\frac {3 b^2 \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )}{\sqrt {a} \sqrt {b+a x^3}}\right ) \] Input:
Integrate[(a + b/x^3)^(3/2)*x^5,x]
Output:
(Sqrt[a + b/x^3]*x^(3/2)*(x^(3/2)*(5*b + 2*a*x^3) + (3*b^2*Log[Sqrt[a]*x^( 3/2) + Sqrt[b + a*x^3]])/(Sqrt[a]*Sqrt[b + a*x^3])))/12
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 51, 51, 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.
\(\displaystyle \int x^5 \left (a+\frac {b}{x^3}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \left (a+\frac {b}{x^3}\right )^{3/2} x^9d\frac {1}{x^3}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+\frac {b}{x^3}\right )^{3/2}-\frac {3}{4} b \int \sqrt {a+\frac {b}{x^3}} x^6d\frac {1}{x^3}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+\frac {b}{x^3}\right )^{3/2}-\frac {3}{4} b \left (\frac {1}{2} b \int \frac {x^3}{\sqrt {a+\frac {b}{x^3}}}d\frac {1}{x^3}-x^3 \sqrt {a+\frac {b}{x^3}}\right )\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+\frac {b}{x^3}\right )^{3/2}-\frac {3}{4} b \left (\int \frac {1}{\frac {1}{b x^6}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^3}}-x^3 \sqrt {a+\frac {b}{x^3}}\right )\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+\frac {b}{x^3}\right )^{3/2}-\frac {3}{4} b \left (x^3 \left (-\sqrt {a+\frac {b}{x^3}}\right )-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{\sqrt {a}}\right )\right )\) |
Input:
Int[(a + b/x^3)^(3/2)*x^5,x]
Output:
(((a + b/x^3)^(3/2)*x^6)/2 - (3*b*(-(Sqrt[a + b/x^3]*x^3) - (b*ArcTanh[Sqr t[a + b/x^3]/Sqrt[a]])/Sqrt[a]))/4)/3
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] :> 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 = 0.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {x^{3} \left (2 a \,x^{3}+5 b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{12}+\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x \sqrt {x \left (a \,x^{3}+b \right )}}{4 \sqrt {a}\, \left (a \,x^{3}+b \right )}\) | \(91\) |
default | \(\frac {\left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} x^{5} \left (2 \sqrt {x \left (a \,x^{3}+b \right )}\, a^{\frac {3}{2}} x^{4}+5 b x \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a \,x^{3}+b \right )}}{x^{2} \sqrt {a}}\right ) b^{2}\right )}{12 \left (a \,x^{3}+b \right ) \sqrt {x \left (a \,x^{3}+b \right )}\, \sqrt {a}}\) | \(104\) |
Input:
int((a+b/x^3)^(3/2)*x^5,x,method=_RETURNVERBOSE)
Output:
1/12*x^3*(2*a*x^3+5*b)*((a*x^3+b)/x^3)^(1/2)+1/4*b^2/a^(1/2)*arctanh((x*(a *x^3+b))^(1/2)/x^2/a^(1/2))*((a*x^3+b)/x^3)^(1/2)*x*(x*(a*x^3+b))^(1/2)/(a *x^3+b)
Time = 0.15 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.71 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ) + 4 \, {\left (2 \, a^{2} x^{6} + 5 \, a b x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{48 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {{\left (2 \, a x^{3} + b\right )} \sqrt {-a} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, {\left (a^{2} x^{3} + a b\right )}}\right ) - 2 \, {\left (2 \, a^{2} x^{6} + 5 \, a b x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{24 \, a}\right ] \] Input:
integrate((a+b/x^3)^(3/2)*x^5,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 + 5*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a, -1/24*(3*sqrt(-a)*b^2*arctan(1/2*(2*a*x^3 + b)*sqrt(-a)*sqrt( (a*x^3 + b)/x^3)/(a^2*x^3 + a*b)) - 2*(2*a^2*x^6 + 5*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a]
Time = 2.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {a \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{6} + \frac {5 b^{\frac {3}{2}} x^{\frac {3}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{12} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \] Input:
integrate((a+b/x**3)**(3/2)*x**5,x)
Output:
a*sqrt(b)*x**(9/2)*sqrt(a*x**3/b + 1)/6 + 5*b**(3/2)*x**(3/2)*sqrt(a*x**3/ b + 1)/12 + b**2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(4*sqrt(a))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.42 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=-\frac {b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x^{3}}} a b^{2}}{12 \, {\left ({\left (a + \frac {b}{x^{3}}\right )}^{2} - 2 \, {\left (a + \frac {b}{x^{3}}\right )} a + a^{2}\right )}} \] Input:
integrate((a+b/x^3)^(3/2)*x^5,x, algorithm="maxima")
Output:
-1/8*b^2*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a)))/sqrt (a) + 1/12*(5*(a + b/x^3)^(3/2)*b^2 - 3*sqrt(a + b/x^3)*a*b^2)/((a + b/x^3 )^2 - 2*(a + b/x^3)*a + a^2)
Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {1}{12} \, \sqrt {a x^{4} + b x} {\left (2 \, a x^{3} + 5 \, b\right )} x - \frac {b^{2} \arctan \left (\frac {\sqrt {a + \frac {b}{x^{3}}}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} \] Input:
integrate((a+b/x^3)^(3/2)*x^5,x, algorithm="giac")
Output:
1/12*sqrt(a*x^4 + b*x)*(2*a*x^3 + 5*b)*x - 1/4*b^2*arctan(sqrt(a + b/x^3)/ sqrt(-a))/sqrt(-a)
Time = 1.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {a\,x^6\,\sqrt {a+\frac {b}{x^3}}}{6}+\frac {5\,b\,x^3\,\sqrt {a+\frac {b}{x^3}}}{12}+\frac {b^2\,\ln \left (x^6\,\left (\sqrt {a+\frac {b}{x^3}}-\sqrt {a}\right )\,{\left (\sqrt {a+\frac {b}{x^3}}+\sqrt {a}\right )}^3\right )}{8\,\sqrt {a}} \] Input:
int(x^5*(a + b/x^3)^(3/2),x)
Output:
(a*x^6*(a + b/x^3)^(1/2))/6 + (5*b*x^3*(a + b/x^3)^(1/2))/12 + (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^(1/ 2))
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25 \[ \int \left (a+\frac {b}{x^3}\right )^{3/2} x^5 \, dx=\frac {4 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a^{2} x^{4}+10 \sqrt {x}\, \sqrt {a \,x^{3}+b}\, a b x -3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}-\sqrt {x}\, \sqrt {a}\, x \right ) b^{2}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{3}+b}+\sqrt {x}\, \sqrt {a}\, x \right ) b^{2}}{24 a} \] Input:
int((a+b/x^3)^(3/2)*x^5,x)
Output:
(4*sqrt(x)*sqrt(a*x**3 + b)*a**2*x**4 + 10*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)