Integrand size = 13, antiderivative size = 70 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=4 b \sqrt {a+\frac {b}{x^{2/3}}} \sqrt [3]{x}+a \sqrt {a+\frac {b}{x^{2/3}}} x-3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^{2/3}}} \sqrt [3]{x}}\right ) \] Output:
4*b*(a+b/x^(2/3))^(1/2)*x^(1/3)+a*(a+b/x^(2/3))^(1/2)*x-3*b^(3/2)*arctanh( b^(1/2)/(a+b/x^(2/3))^(1/2)/x^(1/3))
Time = 5.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=\frac {\sqrt {a+\frac {b}{x^{2/3}}} \sqrt [3]{x} \left (\sqrt {b+a x^{2/3}} \left (4 b+a x^{2/3}\right )-3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x^{2/3}}}{\sqrt {b}}\right )\right )}{\sqrt {b+a x^{2/3}}} \] Input:
Integrate[(a + b/x^(2/3))^(3/2),x]
Output:
(Sqrt[a + b/x^(2/3)]*x^(1/3)*(Sqrt[b + a*x^(2/3)]*(4*b + a*x^(2/3)) - 3*b^ (3/2)*ArcTanh[Sqrt[b + a*x^(2/3)]/Sqrt[b]]))/Sqrt[b + a*x^(2/3)]
Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {774, 858, 247, 247, 224, 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.
\(\displaystyle \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} x^{2/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -3 \int \frac {\left (a+b x^{2/3}\right )^{3/2}}{x^{4/3}}d\frac {1}{\sqrt [3]{x}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -3 \left (b \int \frac {\sqrt {a+b x^{2/3}}}{x^{2/3}}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+b x^{2/3}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -3 \left (b \left (b \int \frac {1}{\sqrt {a+b x^{2/3}}}d\frac {1}{\sqrt [3]{x}}-\frac {\sqrt {a+b x^{2/3}}}{\sqrt [3]{x}}\right )-\frac {\left (a+b x^{2/3}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -3 \left (b \left (b \int \frac {1}{1-b x^{2/3}}d\frac {1}{\sqrt {a+b x^{2/3}} \sqrt [3]{x}}-\frac {\sqrt {a+b x^{2/3}}}{\sqrt [3]{x}}\right )-\frac {\left (a+b x^{2/3}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -3 \left (b \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt [3]{x} \sqrt {a+b x^{2/3}}}\right )-\frac {\sqrt {a+b x^{2/3}}}{\sqrt [3]{x}}\right )-\frac {\left (a+b x^{2/3}\right )^{3/2}}{3 x}\right )\) |
Input:
Int[(a + b/x^(2/3))^(3/2),x]
Output:
-3*(-1/3*(a + b*x^(2/3))^(3/2)/x + b*(-(Sqrt[a + b*x^(2/3)]/x^(1/3)) + Sqr t[b]*ArcTanh[Sqrt[b]/(Sqrt[a + b*x^(2/3)]*x^(1/3))]))
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\left (\frac {b +a \,x^{\frac {2}{3}}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (\left (b +a \,x^{\frac {2}{3}}\right )^{\frac {3}{2}}-3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {b +a \,x^{\frac {2}{3}}}}{x^{\frac {1}{3}}}\right )+3 \sqrt {b +a \,x^{\frac {2}{3}}}\, b \right )}{\left (b +a \,x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\) | \(73\) |
default | \(-\frac {\left (\frac {b +a \,x^{\frac {2}{3}}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {b +a \,x^{\frac {2}{3}}}}{x^{\frac {1}{3}}}\right )-\left (b +a \,x^{\frac {2}{3}}\right )^{\frac {3}{2}}-3 \sqrt {b +a \,x^{\frac {2}{3}}}\, b \right )}{\left (b +a \,x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\) | \(76\) |
Input:
int((a+b/x^(2/3))^(3/2),x,method=_RETURNVERBOSE)
Output:
((b+a*x^(2/3))/x^(2/3))^(3/2)*x*((b+a*x^(2/3))^(3/2)-3*b^(3/2)*ln(2*(b^(1/ 2)*(b+a*x^(2/3))^(1/2)+b)/x^(1/3))+3*(b+a*x^(2/3))^(1/2)*b)/(b+a*x^(2/3))^ (3/2)
Timed out. \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((a+b/x^(2/3))^(3/2),x, algorithm="fricas")
Output:
Timed out
Time = 2.61 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=a \sqrt {b} x^{\frac {2}{3}} \sqrt {\frac {a x^{\frac {2}{3}}}{b} + 1} + 4 b^{\frac {3}{2}} \sqrt {\frac {a x^{\frac {2}{3}}}{b} + 1} + \frac {3 b^{\frac {3}{2}} \log {\left (\frac {a x^{\frac {2}{3}}}{b} \right )}}{2} - 3 b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x^{\frac {2}{3}}}{b} + 1} + 1 \right )} \] Input:
integrate((a+b/x**(2/3))**(3/2),x)
Output:
a*sqrt(b)*x**(2/3)*sqrt(a*x**(2/3)/b + 1) + 4*b**(3/2)*sqrt(a*x**(2/3)/b + 1) + 3*b**(3/2)*log(a*x**(2/3)/b)/2 - 3*b**(3/2)*log(sqrt(a*x**(2/3)/b + 1) + 1)
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx={\left (a + \frac {b}{x^{\frac {2}{3}}}\right )}^{\frac {3}{2}} x + \frac {3}{2} \, b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x^{\frac {2}{3}}}} x^{\frac {1}{3}} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{\frac {2}{3}}}} x^{\frac {1}{3}} + \sqrt {b}}\right ) + 3 \, \sqrt {a + \frac {b}{x^{\frac {2}{3}}}} b x^{\frac {1}{3}} \] Input:
integrate((a+b/x^(2/3))^(3/2),x, algorithm="maxima")
Output:
(a + b/x^(2/3))^(3/2)*x + 3/2*b^(3/2)*log((sqrt(a + b/x^(2/3))*x^(1/3) - s qrt(b))/(sqrt(a + b/x^(2/3))*x^(1/3) + sqrt(b))) + 3*sqrt(a + b/x^(2/3))*b *x^(1/3)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.80 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=\frac {3 \, b^{2} \arctan \left (\frac {\sqrt {a x^{\frac {2}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + {\left (a x^{\frac {2}{3}} + b\right )}^{\frac {3}{2}} + 3 \, \sqrt {a x^{\frac {2}{3}} + b} b + \frac {3 i \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 i \, \sqrt {3} \sqrt {-b} b^{\frac {3}{2}} + 4 \, \sqrt {-b} b^{\frac {3}{2}}}{-2 i \, \sqrt {3} \sqrt {-b} + 2 \, \sqrt {-b}} \] Input:
integrate((a+b/x^(2/3))^(3/2),x, algorithm="giac")
Output:
3*b^2*arctan(sqrt(a*x^(2/3) + b)/sqrt(-b))/sqrt(-b) + (a*x^(2/3) + b)^(3/2 ) + 3*sqrt(a*x^(2/3) + b)*b + (3*I*sqrt(3)*b^2*arctan(sqrt(b)/sqrt(-b)) + 3*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*I*sqrt(3)*sqrt(-b)*b^(3/2) + 4*sqrt(-b) *b^(3/2))/(-2*I*sqrt(3)*sqrt(-b) + 2*sqrt(-b))
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.53 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=\frac {x\,{\left (a+\frac {b}{x^{2/3}}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {3}{2};\ -\frac {1}{2};\ -\frac {b}{a\,x^{2/3}}\right )}{{\left (\frac {b}{a\,x^{2/3}}+1\right )}^{3/2}} \] Input:
int((a + b/x^(2/3))^(3/2),x)
Output:
(x*(a + b/x^(2/3))^(3/2)*hypergeom([-3/2, -3/2], -1/2, -b/(a*x^(2/3))))/(b /(a*x^(2/3)) + 1)^(3/2)
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19 \[ \int \left (a+\frac {b}{x^{2/3}}\right )^{3/2} \, dx=x^{\frac {2}{3}} \sqrt {x^{\frac {2}{3}} a +b}\, a +4 \sqrt {x^{\frac {2}{3}} a +b}\, b +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {x^{\frac {2}{3}} a +b}+x^{\frac {1}{3}} \sqrt {a}-\sqrt {b}}{\sqrt {b}}\right ) b -3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {x^{\frac {2}{3}} a +b}+x^{\frac {1}{3}} \sqrt {a}+\sqrt {b}}{\sqrt {b}}\right ) b \] Input:
int((a+b/x^(2/3))^(3/2),x)
Output:
x**(2/3)*sqrt(x**(2/3)*a + b)*a + 4*sqrt(x**(2/3)*a + b)*b + 3*sqrt(b)*log ((sqrt(x**(2/3)*a + b) + x**(1/3)*sqrt(a) - sqrt(b))/sqrt(b))*b - 3*sqrt(b )*log((sqrt(x**(2/3)*a + b) + x**(1/3)*sqrt(a) + sqrt(b))/sqrt(b))*b