Integrand size = 11, antiderivative size = 174 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {x}{a^3}-\frac {b^2 x}{3 a^3 \left (b+a x^{3/2}\right )^2}+\frac {14 b x}{9 a^3 \left (b+a x^{3/2}\right )}+\frac {40 b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} a^{11/3}}+\frac {40 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )}{27 a^{11/3}}-\frac {20 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{27 a^{11/3}} \] Output:
x/a^3-1/3*b^2*x/a^3/(b+a*x^(3/2))^2+14/9*b*x/a^3/(b+a*x^(3/2))+40/27*b^(2/ 3)*arctan(1/3*(b^(1/3)-2*a^(1/3)*x^(1/2))*3^(1/2)/b^(1/3))*3^(1/2)/a^(11/3 )+40/27*b^(2/3)*ln(b^(1/3)+a^(1/3)*x^(1/2))/a^(11/3)-20/27*b^(2/3)*ln(b^(2 /3)-a^(1/3)*b^(1/3)*x^(1/2)+a^(2/3)*x)/a^(11/3)
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {\frac {3 a^{2/3} \left (20 b^2 x+32 a b x^{5/2}+9 a^2 x^4\right )}{\left (b+a x^{3/2}\right )^2}+40 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+40 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )-20 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{27 a^{11/3}} \] Input:
Integrate[(a + b/x^(3/2))^(-3),x]
Output:
((3*a^(2/3)*(20*b^2*x + 32*a*b*x^(5/2) + 9*a^2*x^4))/(b + a*x^(3/2))^2 + 4 0*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*a^(1/3)*Sqrt[x])/b^(1/3))/Sqrt[3]] + 40*b ^(2/3)*Log[b^(1/3) + a^(1/3)*Sqrt[x]] - 20*b^(2/3)*Log[b^(2/3) - a^(1/3)*b ^(1/3)*Sqrt[x] + a^(2/3)*x])/(27*a^(11/3))
Time = 0.57 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {774, 795, 817, 817, 843, 821, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+\frac {b}{x^{3/2}}\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle 2 \int \frac {x^5}{\left (a x^{3/2}+b\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 2 \left (\frac {4 \int \frac {x^{7/2}}{\left (a x^{3/2}+b\right )^2}d\sqrt {x}}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \int \frac {x^2}{a x^{3/2}+b}d\sqrt {x}}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \int \frac {\sqrt {x}}{a x^{3/2}+b}d\sqrt {x}}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {4 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {\log \left (a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )}{3 a}-\frac {x^4}{6 a \left (a x^{3/2}+b\right )^2}\right )\) |
Input:
Int[(a + b/x^(3/2))^(-3),x]
Output:
2*(-1/6*x^4/(a*(b + a*x^(3/2))^2) + (4*(-1/3*x^(5/2)/(a*(b + a*x^(3/2))) + (5*(x/(2*a) - (b*(-1/3*Log[b^(1/3) + a^(1/3)*Sqrt[x]]/(a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*Sqrt[x])/b^(1/3))/Sqrt[3]])/a^(1/3)) + Log[b^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + a^(2/3)*x]/(2*a^(1/3)))/(3*a^(1/3 )*b^(1/3))))/a))/(3*a)))/(3*a))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {x}{a^{3}}-\frac {2 b \left (\frac {-\frac {7 a \,x^{\frac {5}{2}}}{9}-\frac {11 b x}{18}}{\left (b +a \,x^{\frac {3}{2}}\right )^{2}}-\frac {20 \ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {10 \ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{3}}\) | \(128\) |
| default | \(\frac {x}{a^{3}}-\frac {2 b \left (\frac {-\frac {7 a \,x^{\frac {5}{2}}}{9}-\frac {11 b x}{18}}{\left (b +a \,x^{\frac {3}{2}}\right )^{2}}-\frac {20 \ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {10 \ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{3}}\) | \(128\) |
Input:
int(1/(a+b/x^(3/2))^3,x,method=_RETURNVERBOSE)
Output:
x/a^3-2/a^3*b*((-7/9*a*x^(5/2)-11/18*b*x)/(b+a*x^(3/2))^2-20/27/a/(b/a)^(1 /3)*ln(x^(1/2)+(b/a)^(1/3))+10/27/a/(b/a)^(1/3)*ln(x-(b/a)^(1/3)*x^(1/2)+( b/a)^(2/3))+20/27*3^(1/2)/a/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)* x^(1/2)-1)))
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (123) = 246\).
Time = 0.08 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {27 \, a^{4} x^{7} - 105 \, a^{2} b^{2} x^{4} + 60 \, b^{4} x - 40 \, \sqrt {3} {\left (a^{4} x^{6} - 2 \, a^{2} b^{2} x^{3} + b^{4}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) - 20 \, {\left (a^{4} x^{6} - 2 \, a^{2} b^{2} x^{3} + b^{4}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (-a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b x + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 40 \, {\left (a^{4} x^{6} - 2 \, a^{2} b^{2} x^{3} + b^{4}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \sqrt {x}\right ) + 6 \, {\left (7 \, a^{3} b x^{5} - 4 \, a b^{3} x^{2}\right )} \sqrt {x}}{27 \, {\left (a^{7} x^{6} - 2 \, a^{5} b^{2} x^{3} + a^{3} b^{4}\right )}} \] Input:
integrate(1/(a+b/x^(3/2))^3,x, algorithm="fricas")
Output:
1/27*(27*a^4*x^7 - 105*a^2*b^2*x^4 + 60*b^4*x - 40*sqrt(3)*(a^4*x^6 - 2*a^ 2*b^2*x^3 + b^4)*(b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*sqrt(x)*(b^2/a^2) ^(1/3) - sqrt(3)*b)/b) - 20*(a^4*x^6 - 2*a^2*b^2*x^3 + b^4)*(b^2/a^2)^(1/3 )*log(-a*sqrt(x)*(b^2/a^2)^(2/3) + b*x + b*(b^2/a^2)^(1/3)) + 40*(a^4*x^6 - 2*a^2*b^2*x^3 + b^4)*(b^2/a^2)^(1/3)*log(a*(b^2/a^2)^(2/3) + b*sqrt(x)) + 6*(7*a^3*b*x^5 - 4*a*b^3*x^2)*sqrt(x))/(a^7*x^6 - 2*a^5*b^2*x^3 + a^3*b^ 4)
Leaf count of result is larger than twice the leaf count of optimal. 1255 vs. \(2 (167) = 334\).
Time = 111.68 (sec) , antiderivative size = 1255, normalized size of antiderivative = 7.21 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x**(3/2))**3,x)
Output:
Piecewise((zoo*x**(11/2), Eq(a, 0) & Eq(b, 0)), (2*x**(11/2)/(11*b**3), Eq (a, 0)), (x/a**3, Eq(b, 0)), (27*a**3*x**4*(-b/a)**(1/3)/(27*a**6*x**3*(-b /a)**(1/3) + 54*a**5*b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3) ) + 96*a**2*b*x**(5/2)*(-b/a)**(1/3)/(27*a**6*x**3*(-b/a)**(1/3) + 54*a**5 *b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) - 40*a**2*b*x**3*l og(sqrt(x) - (-b/a)**(1/3))/(27*a**6*x**3*(-b/a)**(1/3) + 54*a**5*b*x**(3/ 2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) + 20*a**2*b*x**3*log(4*sqrt (x)*(-b/a)**(1/3) + 4*x + 4*(-b/a)**(2/3))/(27*a**6*x**3*(-b/a)**(1/3) + 5 4*a**5*b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) - 40*sqrt(3) *a**2*b*x**3*atan(2*sqrt(3)*sqrt(x)/(3*(-b/a)**(1/3)) + sqrt(3)/3)/(27*a** 6*x**3*(-b/a)**(1/3) + 54*a**5*b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b /a)**(1/3)) - 40*a**2*b*x**3*log(2)/(27*a**6*x**3*(-b/a)**(1/3) + 54*a**5* b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) - 80*a*b**2*x**(3/2 )*log(sqrt(x) - (-b/a)**(1/3))/(27*a**6*x**3*(-b/a)**(1/3) + 54*a**5*b*x** (3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) + 40*a*b**2*x**(3/2)*log (4*sqrt(x)*(-b/a)**(1/3) + 4*x + 4*(-b/a)**(2/3))/(27*a**6*x**3*(-b/a)**(1 /3) + 54*a**5*b*x**(3/2)*(-b/a)**(1/3) + 27*a**4*b**2*(-b/a)**(1/3)) - 80* sqrt(3)*a*b**2*x**(3/2)*atan(2*sqrt(3)*sqrt(x)/(3*(-b/a)**(1/3)) + sqrt(3) /3)/(27*a**6*x**3*(-b/a)**(1/3) + 54*a**5*b*x**(3/2)*(-b/a)**(1/3) + 27*a* *4*b**2*(-b/a)**(1/3)) - 80*a*b**2*x**(3/2)*log(2)/(27*a**6*x**3*(-b/a)...
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {9 \, a^{2} + \frac {32 \, a b}{x^{\frac {3}{2}}} + \frac {20 \, b^{2}}{x^{3}}}{9 \, {\left (\frac {a^{5}}{x} + \frac {2 \, a^{4} b}{x^{\frac {5}{2}}} + \frac {a^{3} b^{2}}{x^{4}}\right )}} + \frac {40 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2}{\sqrt {x}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {20 \, \log \left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{3}}}{\sqrt {x}} + \frac {1}{x}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {40 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {1}{\sqrt {x}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(a+b/x^(3/2))^3,x, algorithm="maxima")
Output:
1/9*(9*a^2 + 32*a*b/x^(3/2) + 20*b^2/x^3)/(a^5/x + 2*a^4*b/x^(5/2) + a^3*b ^2/x^4) + 40/27*sqrt(3)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2/sqrt(x))/(a/b )^(1/3))/(a^3*(a/b)^(2/3)) - 20/27*log((a/b)^(2/3) - (a/b)^(1/3)/sqrt(x) + 1/x)/(a^3*(a/b)^(2/3)) + 40/27*log((a/b)^(1/3) + 1/sqrt(x))/(a^3*(a/b)^(2 /3))
Time = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {40 \, \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3}} + \frac {x}{a^{3}} + \frac {40 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5}} - \frac {20 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{27 \, a^{5}} + \frac {14 \, a b x^{\frac {5}{2}} + 11 \, b^{2} x}{9 \, {\left (a x^{\frac {3}{2}} + b\right )}^{2} a^{3}} \] Input:
integrate(1/(a+b/x^(3/2))^3,x, algorithm="giac")
Output:
40/27*(-b/a)^(2/3)*log(abs(sqrt(x) - (-b/a)^(1/3)))/a^3 + x/a^3 + 40/27*sq rt(3)*(-a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-b/a)^(1/3))/(-b/a)^ (1/3))/a^5 - 20/27*(-a^2*b)^(2/3)*log(x + sqrt(x)*(-b/a)^(1/3) + (-b/a)^(2 /3))/a^5 + 1/9*(14*a*b*x^(5/2) + 11*b^2*x)/((a*x^(3/2) + b)^2*a^3)
Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {\frac {11\,b^2\,x}{9}+\frac {14\,a\,b\,x^{5/2}}{9}}{a^3\,b^2+a^5\,x^3+2\,a^4\,b\,x^{3/2}}+\frac {x}{a^3}+\frac {40\,b^{2/3}\,\ln \left (\frac {1600\,b^{7/3}}{81\,a^{16/3}}+\frac {1600\,b^2\,\sqrt {x}}{81\,a^5}\right )}{27\,a^{11/3}}+\frac {40\,b^{2/3}\,\ln \left (\frac {1600\,b^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,a^{16/3}}+\frac {1600\,b^2\,\sqrt {x}}{81\,a^5}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{11/3}}-\frac {40\,b^{2/3}\,\ln \left (\frac {1600\,b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,a^{16/3}}+\frac {1600\,b^2\,\sqrt {x}}{81\,a^5}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{11/3}} \] Input:
int(1/(a + b/x^(3/2))^3,x)
Output:
((11*b^2*x)/9 + (14*a*b*x^(5/2))/9)/(a^3*b^2 + a^5*x^3 + 2*a^4*b*x^(3/2)) + x/a^3 + (40*b^(2/3)*log((1600*b^(7/3))/(81*a^(16/3)) + (1600*b^2*x^(1/2) )/(81*a^5)))/(27*a^(11/3)) + (40*b^(2/3)*log((1600*b^(7/3)*((3^(1/2)*1i)/2 - 1/2)^2)/(81*a^(16/3)) + (1600*b^2*x^(1/2))/(81*a^5))*((3^(1/2)*1i)/2 - 1/2))/(27*a^(11/3)) - (40*b^(2/3)*log((1600*b^(7/3)*((3^(1/2)*1i)/2 + 1/2) ^2)/(81*a^(16/3)) + (1600*b^2*x^(1/2))/(81*a^5))*((3^(1/2)*1i)/2 + 1/2))/( 27*a^(11/3))
Time = 0.18 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^3} \, dx=\frac {-80 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{2} x -40 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b \,x^{3}-40 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) b^{3}+96 \sqrt {x}\, b^{\frac {4}{3}} a^{\frac {5}{3}} x^{2}+27 b^{\frac {1}{3}} a^{\frac {8}{3}} x^{4}+60 b^{\frac {7}{3}} a^{\frac {2}{3}} x -40 \sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) a \,b^{2} x +80 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) a \,b^{2} x -20 \,\mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) a^{2} b \,x^{3}-20 \,\mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) b^{3}+40 \,\mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) a^{2} b \,x^{3}+40 \,\mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) b^{3}}{27 b^{\frac {1}{3}} a^{\frac {11}{3}} \left (2 \sqrt {x}\, a b x +a^{2} x^{3}+b^{2}\right )} \] Input:
int(1/(a+b/x^(3/2))^3,x)
Output:
( - 80*sqrt(x)*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/(b**(1/3)*sqrt (3)))*a*b**2*x - 40*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/(b**(1/3) *sqrt(3)))*a**2*b*x**3 - 40*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/( b**(1/3)*sqrt(3)))*b**3 + 96*sqrt(x)*b**(1/3)*a**(2/3)*a*b*x**2 + 27*b**(1 /3)*a**(2/3)*a**2*x**4 + 60*b**(1/3)*a**(2/3)*b**2*x - 40*sqrt(x)*log(a**( 2/3)*x - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3))*a*b**2*x + 80*sqrt(x)*log(s qrt(x)*a**(1/3) + b**(1/3))*a*b**2*x - 20*log(a**(2/3)*x - sqrt(x)*b**(1/3 )*a**(1/3) + b**(2/3))*a**2*b*x**3 - 20*log(a**(2/3)*x - sqrt(x)*b**(1/3)* a**(1/3) + b**(2/3))*b**3 + 40*log(sqrt(x)*a**(1/3) + b**(1/3))*a**2*b*x** 3 + 40*log(sqrt(x)*a**(1/3) + b**(1/3))*b**3)/(27*b**(1/3)*a**(2/3)*a**3*( 2*sqrt(x)*a*b*x + a**2*x**3 + b**2))