Integrand size = 13, antiderivative size = 100 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {3 b^2 \sqrt {a+\frac {b}{\sqrt [3]{x}}} \sqrt [3]{x}}{8 a}+\frac {7}{4} b \sqrt {a+\frac {b}{\sqrt [3]{x}}} x^{2/3}+a \sqrt {a+\frac {b}{\sqrt [3]{x}}} x-\frac {3 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Output:
3/8*b^2*(a+b/x^(1/3))^(1/2)*x^(1/3)/a+7/4*b*(a+b/x^(1/3))^(1/2)*x^(2/3)+a* (a+b/x^(1/3))^(1/2)*x-3/8*b^3*arctanh((a+b/x^(1/3))^(1/2)/a^(1/2))/a^(3/2)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {\sqrt {a} \sqrt {a+\frac {b}{\sqrt [3]{x}}} \left (3 b^2+14 a b \sqrt [3]{x}+8 a^2 x^{2/3}\right ) \sqrt [3]{x}-3 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Input:
Integrate[(a + b/x^(1/3))^(3/2),x]
Output:
(Sqrt[a]*Sqrt[a + b/x^(1/3)]*(3*b^2 + 14*a*b*x^(1/3) + 8*a^2*x^(2/3))*x^(1 /3) - 3*b^3*ArcTanh[Sqrt[a + b/x^(1/3)]/Sqrt[a]])/(8*a^(3/2))
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {774, 798, 51, 51, 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.
\(\displaystyle \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} x^{2/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -3 \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{x^{4/3}}d\frac {1}{\sqrt [3]{x}}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle -3 \left (\frac {1}{2} b \int \frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{x}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle -3 \left (\frac {1}{2} b \left (\frac {1}{4} b \int \frac {1}{\sqrt {a+\frac {b}{\sqrt [3]{x}}} x^{2/3}}d\frac {1}{\sqrt [3]{x}}-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -3 \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{\sqrt {a+\frac {b}{\sqrt [3]{x}}} \sqrt [3]{x}}d\frac {1}{\sqrt [3]{x}}}{2 a}-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{a \sqrt [3]{x}}\right )-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -3 \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {x^{2/3}}{b}-\frac {a}{b}}d\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{a}-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{a \sqrt [3]{x}}\right )-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -3 \left (\frac {1}{2} b \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{a \sqrt [3]{x}}\right )-\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{3 x}\right )\) |
Input:
Int[(a + b/x^(1/3))^(3/2),x]
Output:
-3*(-1/3*(a + b/x^(1/3))^(3/2)/x + (b*(-1/2*Sqrt[a + b/x^(1/3)]/x^(2/3) + (b*(-(Sqrt[a + b/x^(1/3)]/(a*x^(1/3))) + (b*ArcTanh[Sqrt[a + b/x^(1/3)]/Sq rt[a]])/a^(3/2)))/4))/2)
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] :> 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[((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] :> 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 = 0.36 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\sqrt {\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}}\, x^{\frac {1}{3}} \left (16 \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} a^{\frac {5}{2}}+12 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b \,x^{\frac {1}{3}}+6 a^{\frac {3}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2}-3 a \ln \left (\frac {2 \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3}\right )}{16 a^{\frac {5}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}}\) | \(135\) |
derivativedivides | \(\frac {\left (\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right )^{\frac {3}{2}} x^{\frac {2}{3}} \left (16 \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} a^{\frac {5}{2}}+12 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b \,x^{\frac {1}{3}}+6 a^{\frac {3}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2}-3 a \ln \left (\frac {2 \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3}\right )}{16 a^{\frac {5}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}}\) | \(144\) |
Input:
int((a+b/x^(1/3))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/16*((b+a*x^(1/3))/x^(1/3))^(1/2)*x^(1/3)/a^(5/2)*(16*(a*x^(2/3)+b*x^(1/3 ))^(3/2)*a^(5/2)+12*a^(5/2)*(a*x^(2/3)+b*x^(1/3))^(1/2)*b*x^(1/3)+6*a^(3/2 )*(a*x^(2/3)+b*x^(1/3))^(1/2)*b^2-3*a*ln(1/2*(2*(a*x^(2/3)+b*x^(1/3))^(1/2 )*a^(1/2)+2*a*x^(1/3)+b)/a^(1/2))*b^3)/((b+a*x^(1/3))*x^(1/3))^(1/2)
Timed out. \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((a+b/x^(1/3))^(3/2),x, algorithm="fricas")
Output:
Timed out
Time = 4.42 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.39 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {a^{2} x^{\frac {7}{6}}}{\sqrt {b} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {11 a \sqrt {b} x^{\frac {5}{6}}}{4 \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {17 b^{\frac {3}{2}} \sqrt {x}}{8 \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {3 b^{\frac {5}{2}} \sqrt [6]{x}}{8 a \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} - \frac {3 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt [6]{x}}{\sqrt {b}} \right )}}{8 a^{\frac {3}{2}}} \] Input:
integrate((a+b/x**(1/3))**(3/2),x)
Output:
a**2*x**(7/6)/(sqrt(b)*sqrt(a*x**(1/3)/b + 1)) + 11*a*sqrt(b)*x**(5/6)/(4* sqrt(a*x**(1/3)/b + 1)) + 17*b**(3/2)*sqrt(x)/(8*sqrt(a*x**(1/3)/b + 1)) + 3*b**(5/2)*x**(1/6)/(8*a*sqrt(a*x**(1/3)/b + 1)) - 3*b**3*asinh(sqrt(a)*x **(1/6)/sqrt(b))/(8*a**(3/2))
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.35 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {3 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} + \sqrt {a}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {5}{2}} b^{3} + 8 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {a + \frac {b}{x^{\frac {1}{3}}}} a^{2} b^{3}}{8 \, {\left ({\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a - 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{2} + 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{3} - a^{4}\right )}} \] Input:
integrate((a+b/x^(1/3))^(3/2),x, algorithm="maxima")
Output:
3/16*b^3*log((sqrt(a + b/x^(1/3)) - sqrt(a))/(sqrt(a + b/x^(1/3)) + sqrt(a )))/a^(3/2) + 1/8*(3*(a + b/x^(1/3))^(5/2)*b^3 + 8*(a + b/x^(1/3))^(3/2)*a *b^3 - 3*sqrt(a + b/x^(1/3))*a^2*b^3)/((a + b/x^(1/3))^3*a - 3*(a + b/x^(1 /3))^2*a^2 + 3*(a + b/x^(1/3))*a^3 - a^4)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {3 \, b^{3} \log \left ({\left | -\sqrt {a} x^{\frac {1}{6}} + \sqrt {a x^{\frac {1}{3}} + b} \right |}\right )}{8 \, a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + \frac {1}{8} \, \sqrt {a x^{\frac {1}{3}} + b} {\left (2 \, x^{\frac {1}{3}} {\left (\frac {4 \, a x^{\frac {1}{3}}}{\mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + \frac {7 \, b}{\mathrm {sgn}\left (x\right )^{\frac {1}{3}}}\right )} + \frac {3 \, b^{2}}{a \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}\right )} x^{\frac {1}{6}} - \frac {3 \, {\left (i \, \sqrt {3} b^{3} \log \left ({\left | b \right |}\right ) + b^{3} \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{64 \, a^{\frac {3}{2}}} \] Input:
integrate((a+b/x^(1/3))^(3/2),x, algorithm="giac")
Output:
3/8*b^3*log(abs(-sqrt(a)*x^(1/6) + sqrt(a*x^(1/3) + b)))/(a^(3/2)*sgn(x)^( 1/3)) + 1/8*sqrt(a*x^(1/3) + b)*(2*x^(1/3)*(4*a*x^(1/3)/sgn(x)^(1/3) + 7*b /sgn(x)^(1/3)) + 3*b^2/(a*sgn(x)^(1/3)))*x^(1/6) - 3/64*(I*sqrt(3)*b^3*log (abs(b)) + b^3*log(abs(b)))*sgn(x)/a^(3/2)
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.38 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {2\,x\,{\left (a+\frac {b}{x^{1/3}}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {3}{2};\ \frac {5}{2};\ -\frac {a\,x^{1/3}}{b}\right )}{{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}} \] Input:
int((a + b/x^(1/3))^(3/2),x)
Output:
(2*x*(a + b/x^(1/3))^(3/2)*hypergeom([-3/2, 3/2], 5/2, -(a*x^(1/3))/b))/(( a*x^(1/3))/b + 1)^(3/2)
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.83 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \, dx=\frac {8 x^{\frac {5}{6}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{3}+14 \sqrt {x}\, \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b +3 x^{\frac {1}{6}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}+x^{\frac {1}{6}} \sqrt {a}}{\sqrt {b}}\right ) b^{3}}{8 a^{2}} \] Input:
int((a+b/x^(1/3))^(3/2),x)
Output:
(8*x**(5/6)*sqrt(x**(1/3)*a + b)*a**3 + 14*sqrt(x)*sqrt(x**(1/3)*a + b)*a* *2*b + 3*x**(1/6)*sqrt(x**(1/3)*a + b)*a*b**2 - 3*sqrt(a)*log((sqrt(x**(1/ 3)*a + b) + x**(1/6)*sqrt(a))/sqrt(b))*b**3)/(8*a**2)