Integrand size = 13, antiderivative size = 128 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {105 b^3}{8 a^4 \sqrt {a+\frac {b}{\sqrt [3]{x}}}}+\frac {35 b^2 \sqrt [3]{x}}{8 a^3 \sqrt {a+\frac {b}{\sqrt [3]{x}}}}-\frac {7 b x^{2/3}}{4 a^2 \sqrt {a+\frac {b}{\sqrt [3]{x}}}}+\frac {x}{a \sqrt {a+\frac {b}{\sqrt [3]{x}}}}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Output:
105/8*b^3/a^4/(a+b/x^(1/3))^(1/2)+35/8*b^2*x^(1/3)/a^3/(a+b/x^(1/3))^(1/2) -7/4*b*x^(2/3)/a^2/(a+b/x^(1/3))^(1/2)+x/a/(a+b/x^(1/3))^(1/2)-105/8*b^3*a rctanh((a+b/x^(1/3))^(1/2)/a^(1/2))/a^(9/2)
Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}} \left (105 b^3 \sqrt [3]{x}+35 a b^2 x^{2/3}-14 a^2 b x+8 a^3 x^{4/3}\right )}{8 a^4 \left (b+a \sqrt [3]{x}\right )}-\frac {105 b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Input:
Integrate[(a + b/x^(1/3))^(-3/2),x]
Output:
(Sqrt[a + b/x^(1/3)]*(105*b^3*x^(1/3) + 35*a*b^2*x^(2/3) - 14*a^2*b*x + 8* a^3*x^(4/3)))/(8*a^4*(b + a*x^(1/3))) - (105*b^3*ArcTanh[Sqrt[a + b/x^(1/3 )]/Sqrt[a]])/(8*a^(9/2))
Time = 0.38 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {774, 798, 52, 52, 52, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -3 \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} x^{4/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -3 \left (-\frac {7 b \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} x}d\frac {1}{\sqrt [3]{x}}}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -3 \left (-\frac {7 b \left (-\frac {5 b \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} x^{2/3}}d\frac {1}{\sqrt [3]{x}}}{4 a}-\frac {1}{2 a x^{2/3} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -3 \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2} \sqrt [3]{x}}d\frac {1}{\sqrt [3]{x}}}{2 a}-\frac {1}{a \sqrt [3]{x} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{4 a}-\frac {1}{2 a x^{2/3} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -3 \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {\int \frac {1}{\sqrt {a+\frac {b}{\sqrt [3]{x}}} \sqrt [3]{x}}d\frac {1}{\sqrt [3]{x}}}{a}+\frac {2}{a \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{2 a}-\frac {1}{a \sqrt [3]{x} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{4 a}-\frac {1}{2 a x^{2/3} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -3 \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2 \int \frac {1}{\frac {x^{2/3}}{b}-\frac {a}{b}}d\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{2 a}-\frac {1}{a \sqrt [3]{x} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{4 a}-\frac {1}{2 a x^{2/3} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -3 \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {2}{a \sqrt {a+\frac {b}{\sqrt [3]{x}}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {1}{a \sqrt [3]{x} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{4 a}-\frac {1}{2 a x^{2/3} \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )}{6 a}-\frac {1}{3 a x \sqrt {a+\frac {b}{\sqrt [3]{x}}}}\right )\) |
Input:
Int[(a + b/x^(1/3))^(-3/2),x]
Output:
-3*(-1/3*1/(a*Sqrt[a + b/x^(1/3)]*x) - (7*b*(-1/2*1/(a*Sqrt[a + b/x^(1/3)] *x^(2/3)) - (5*b*(-(1/(a*Sqrt[a + b/x^(1/3)]*x^(1/3))) - (3*b*(2/(a*Sqrt[a + b/x^(1/3)]) - (2*ArcTanh[Sqrt[a + b/x^(1/3)]/Sqrt[a]])/a^(3/2)))/(2*a)) )/(4*a)))/(6*a))
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] :> 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[((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]]
Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(92)=184\).
Time = 0.56 (sec) , antiderivative size = 527, normalized size of antiderivative = 4.12
| method | result | size |
| derivativedivides | \(\frac {16 a^{\frac {9}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} x^{\frac {2}{3}}-60 a^{\frac {9}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b x +32 a^{\frac {7}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b \,x^{\frac {1}{3}}-150 a^{\frac {7}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}+240 a^{\frac {7}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}-120 a^{3} \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3} x^{\frac {2}{3}}+16 a^{\frac {5}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{2}-120 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{3} x^{\frac {1}{3}}-96 a^{\frac {5}{2}} {\left (\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}\right )}^{\frac {3}{2}} b^{2}+480 a^{\frac {5}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{3} x^{\frac {1}{3}}-240 a^{2} \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{4} x^{\frac {1}{3}}+15 \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 ) a^{3} b^{3} x^{\frac {2}{3}}-30 a^{\frac {3}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{4}+240 a^{\frac {3}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{4}-120 a \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{5}+30 \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 ) a^{2} b^{4} x^{\frac {1}{3}}+15 \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 ) a \,b^{5}}{16 a^{\frac {11}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, x^{\frac {1}{3}} \left (\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right )^{\frac {3}{2}}}\) | \(527\) |
| default | \(\frac {\sqrt {\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}}\, x^{\frac {1}{3}} \left (16 a^{\frac {9}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} x^{\frac {2}{3}}-60 a^{\frac {9}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b x +32 a^{\frac {7}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b \,x^{\frac {1}{3}}-150 a^{\frac {7}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}+240 a^{\frac {7}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}-120 a^{3} \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3} x^{\frac {2}{3}}+16 a^{\frac {5}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{2}-120 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{3} x^{\frac {1}{3}}-96 a^{\frac {5}{2}} {\left (\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}\right )}^{\frac {3}{2}} b^{2}+480 a^{\frac {5}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{3} x^{\frac {1}{3}}-240 a^{2} \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{4} x^{\frac {1}{3}}+15 \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 ) a^{3} b^{3} x^{\frac {2}{3}}-30 a^{\frac {3}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{4}+240 a^{\frac {3}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, b^{4}-120 a \ln \left (\frac {2 \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{5}+30 \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 ) a^{2} b^{4} x^{\frac {1}{3}}+15 \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 ) a \,b^{5}\right )}{16 a^{\frac {11}{2}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, \left (b +a \,x^{\frac {1}{3}}\right )^{2}}\) | \(536\) |
Input:
int(1/(a+b/x^(1/3))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/16/a^(11/2)*(16*a^(9/2)*(a*x^(2/3)+b*x^(1/3))^(3/2)*x^(2/3)-60*a^(9/2)*( a*x^(2/3)+b*x^(1/3))^(1/2)*b*x+32*a^(7/2)*(a*x^(2/3)+b*x^(1/3))^(3/2)*b*x^ (1/3)-150*a^(7/2)*(a*x^(2/3)+b*x^(1/3))^(1/2)*b^2*x^(2/3)+240*a^(7/2)*((b+ a*x^(1/3))*x^(1/3))^(1/2)*b^2*x^(2/3)-120*a^3*ln(1/2*(2*((b+a*x^(1/3))*x^( 1/3))^(1/2)*a^(1/2)+2*a*x^(1/3)+b)/a^(1/2))*b^3*x^(2/3)+16*a^(5/2)*(a*x^(2 /3)+b*x^(1/3))^(3/2)*b^2-120*a^(5/2)*(a*x^(2/3)+b*x^(1/3))^(1/2)*b^3*x^(1/ 3)-96*a^(5/2)*((b+a*x^(1/3))*x^(1/3))^(3/2)*b^2+480*a^(5/2)*((b+a*x^(1/3)) *x^(1/3))^(1/2)*b^3*x^(1/3)-240*a^2*ln(1/2*(2*((b+a*x^(1/3))*x^(1/3))^(1/2 )*a^(1/2)+2*a*x^(1/3)+b)/a^(1/2))*b^4*x^(1/3)+15*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))*a^3*b^3*x^(2/3)-30*a^(3/2)*(a *x^(2/3)+b*x^(1/3))^(1/2)*b^4+240*a^(3/2)*((b+a*x^(1/3))*x^(1/3))^(1/2)*b^ 4-120*a*ln(1/2*(2*((b+a*x^(1/3))*x^(1/3))^(1/2)*a^(1/2)+2*a*x^(1/3)+b)/a^( 1/2))*b^5+30*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))*a^2*b^4*x^(1/3)+15*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))*a*b^5)/((b+a*x^(1/3))*x^(1/3))^(1/2)/x^(1/3)/((b+a *x^(1/3))/x^(1/3))^(3/2)
Timed out. \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b/x^(1/3))^(3/2),x, algorithm="fricas")
Output:
Timed out
Time = 13.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {x^{\frac {7}{6}}}{a \sqrt {b} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} - \frac {7 \sqrt {b} x^{\frac {5}{6}}}{4 a^{2} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {35 b^{\frac {3}{2}} \sqrt {x}}{8 a^{3} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {105 b^{\frac {5}{2}} \sqrt [6]{x}}{8 a^{4} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} - \frac {105 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt [6]{x}}{\sqrt {b}} \right )}}{8 a^{\frac {9}{2}}} \] Input:
integrate(1/(a+b/x**(1/3))**(3/2),x)
Output:
x**(7/6)/(a*sqrt(b)*sqrt(a*x**(1/3)/b + 1)) - 7*sqrt(b)*x**(5/6)/(4*a**2*s qrt(a*x**(1/3)/b + 1)) + 35*b**(3/2)*sqrt(x)/(8*a**3*sqrt(a*x**(1/3)/b + 1 )) + 105*b**(5/2)*x**(1/6)/(8*a**4*sqrt(a*x**(1/3)/b + 1)) - 105*b**3*asin h(sqrt(a)*x**(1/6)/sqrt(b))/(8*a**(9/2))
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {105 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} b^{3} - 280 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a b^{3} + 231 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{2} b^{3} - 48 \, a^{3} b^{3}}{8 \, {\left ({\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {7}{2}} a^{4} - 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {5}{2}} a^{5} + 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {3}{2}} a^{6} - \sqrt {a + \frac {b}{x^{\frac {1}{3}}}} a^{7}\right )}} + \frac {105 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} + \sqrt {a}}\right )}{16 \, a^{\frac {9}{2}}} \] Input:
integrate(1/(a+b/x^(1/3))^(3/2),x, algorithm="maxima")
Output:
1/8*(105*(a + b/x^(1/3))^3*b^3 - 280*(a + b/x^(1/3))^2*a*b^3 + 231*(a + b/ x^(1/3))*a^2*b^3 - 48*a^3*b^3)/((a + b/x^(1/3))^(7/2)*a^4 - 3*(a + b/x^(1/ 3))^(5/2)*a^5 + 3*(a + b/x^(1/3))^(3/2)*a^6 - sqrt(a + b/x^(1/3))*a^7) + 1 05/16*b^3*log((sqrt(a + b/x^(1/3)) - sqrt(a))/(sqrt(a + b/x^(1/3)) + sqrt( a)))/a^(9/2)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, x^{\frac {1}{3}} {\left (\frac {4 \, x^{\frac {1}{3}} \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}{a} - \frac {7 \, b \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}{a^{2}}\right )} + \frac {35 \, b^{2} \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}{a^{3}}\right )} x^{\frac {1}{3}} + \frac {105 \, b^{3} \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}{a^{4}}\right )} x^{\frac {1}{6}}}{8 \, \sqrt {a x^{\frac {1}{3}} + b}} + \frac {105 \, b^{3} \log \left ({\left | -\sqrt {a} x^{\frac {1}{6}} + \sqrt {a x^{\frac {1}{3}} + b} \right |}\right ) \mathrm {sgn}\left (x\right )^{\frac {1}{3}}}{8 \, a^{\frac {9}{2}}} - \frac {105 \, {\left (-i \, \sqrt {3} b^{3} \log \left ({\left | b \right |}\right ) + b^{3} \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{64 \, a^{\frac {9}{2}}} \] Input:
integrate(1/(a+b/x^(1/3))^(3/2),x, algorithm="giac")
Output:
1/8*((2*x^(1/3)*(4*x^(1/3)*sgn(x)^(1/3)/a - 7*b*sgn(x)^(1/3)/a^2) + 35*b^2 *sgn(x)^(1/3)/a^3)*x^(1/3) + 105*b^3*sgn(x)^(1/3)/a^4)*x^(1/6)/sqrt(a*x^(1 /3) + b) + 105/8*b^3*log(abs(-sqrt(a)*x^(1/6) + sqrt(a*x^(1/3) + b)))*sgn( x)^(1/3)/a^(9/2) - 105/64*(-I*sqrt(3)*b^3*log(abs(b)) + b^3*log(abs(b)))*s gn(x)/a^(9/2)
Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {2\,x\,{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {9}{2};\ \frac {11}{2};\ -\frac {a\,x^{1/3}}{b}\right )}{3\,{\left (a+\frac {b}{x^{1/3}}\right )}^{3/2}} \] Input:
int(1/(a + b/x^(1/3))^(3/2),x)
Output:
(2*x*((a*x^(1/3))/b + 1)^(3/2)*hypergeom([3/2, 9/2], 11/2, -(a*x^(1/3))/b) )/(3*(a + b/x^(1/3))^(3/2))
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}} \, dx=\frac {-840 \sqrt {a}\, \sqrt {x^{\frac {1}{3}} a +b}\, \mathrm {log}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}+x^{\frac {1}{6}} \sqrt {a}}{\sqrt {b}}\right ) b^{3}+525 \sqrt {a}\, \sqrt {x^{\frac {1}{3}} a +b}\, b^{3}-112 x^{\frac {5}{6}} a^{3} b +280 \sqrt {x}\, a^{2} b^{2}+64 x^{\frac {7}{6}} a^{4}+840 x^{\frac {1}{6}} a \,b^{3}}{64 \sqrt {x^{\frac {1}{3}} a +b}\, a^{5}} \] Input:
int(1/(a+b/x^(1/3))^(3/2),x)
Output:
( - 840*sqrt(a)*sqrt(x**(1/3)*a + b)*log((sqrt(x**(1/3)*a + b) + x**(1/6)* sqrt(a))/sqrt(b))*b**3 + 525*sqrt(a)*sqrt(x**(1/3)*a + b)*b**3 - 112*x**(5 /6)*a**3*b + 280*sqrt(x)*a**2*b**2 + 64*x**(1/6)*a**4*x + 840*x**(1/6)*a*b **3)/(64*sqrt(x**(1/3)*a + b)*a**5)