Integrand size = 11, antiderivative size = 146 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\frac {2 x}{3 a \left (a+b x^{3/2}\right )}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{9 a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{9 a^{4/3} b^{2/3}} \] Output:
2/3*x/a/(a+b*x^(3/2))-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/2))*3^(1/2)/a ^(1/3))*3^(1/2)/a^(4/3)/b^(2/3)-2/9*ln(a^(1/3)+b^(1/3)*x^(1/2))/a^(4/3)/b^ (2/3)+1/9*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/2)+b^(2/3)*x)/a^(4/3)/b^(2/3)
Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\frac {\frac {6 \sqrt [3]{a} x}{a+b x^{3/2}}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{b^{2/3}}}{9 a^{4/3}} \] Input:
Integrate[(a + b*x^(3/2))^(-2),x]
Output:
((6*a^(1/3)*x)/(a + b*x^(3/2)) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Sqrt[x] )/a^(1/3))/Sqrt[3]])/b^(2/3) - (2*Log[a^(1/3) + b^(1/3)*Sqrt[x]])/b^(2/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + b^(2/3)*x]/b^(2/3))/(9*a^(4/3))
Time = 0.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {774, 819, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 2 \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 2 \left (\frac {\int \frac {\sqrt {x}}{b x^{3/2}+a}d\sqrt {x}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )\) |
Input:
Int[(a + b*x^(3/2))^(-2),x]
Output:
2*(x/(3*a*(a + b*x^(3/2))) + (-1/3*Log[a^(1/3) + b^(1/3)*Sqrt[x]]/(a^(1/3) *b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Sqrt[x])/a^(1/3))/Sqrt[3]]) /b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + b^(2/3)*x]/(2*b^(1/3)) )/(3*a^(1/3)*b^(1/3)))/(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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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[((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 = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {2 x}{3 a \left (a +b \,x^{\frac {3}{2}}\right )}+\frac {-\frac {2 \ln \left (\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{a}\) | \(116\) |
| default | \(-\frac {2 x}{3 \left (b^{2} x^{3}-a^{2}\right )}-\frac {\ln \left (x -\left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right )}{9 b^{2} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} x +\left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{18 b^{2} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{9 b^{2} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}}+\frac {x}{3 a \left (b \,x^{\frac {3}{2}}-a \right )}+\frac {\ln \left (\sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {x}{3 a \left (a +b \,x^{\frac {3}{2}}\right )}-\frac {\ln \left (\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(361\) |
Input:
int(1/(a+b*x^(3/2))^2,x,method=_RETURNVERBOSE)
Output:
2/3*x/a/(a+b*x^(3/2))+2/3/a*(-1/3/b/(a/b)^(1/3)*ln(x^(1/2)+(a/b)^(1/3))+1/ 6/b/(a/b)^(1/3)*ln(x-(a/b)^(1/3)*x^(1/2)+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^ (1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/2)-1)))
Time = 0.08 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.49 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\left [\frac {6 \, a b^{3} x^{\frac {5}{2}} - 6 \, a^{2} b^{2} x + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} - a^{3} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{3} x^{3} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} b x^{2} + a^{2} b + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} a x + \left (a b^{2}\right )^{\frac {1}{3}} a^{2} + {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} b x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a b x - a^{2} b\right )} \sqrt {x}\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (a b^{2} x - \left (a b^{2}\right )^{\frac {2}{3}} a\right )} \sqrt {x}}{b^{2} x^{3} - a^{2}}\right ) + {\left (b^{2} x^{3} - a^{2}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x - \left (a b^{2}\right )^{\frac {1}{3}} b \sqrt {x} + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{3} - a^{2}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b \sqrt {x} + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a^{2} b^{4} x^{3} - a^{4} b^{2}\right )}}, \frac {6 \, a b^{3} x^{\frac {5}{2}} - 6 \, a^{2} b^{2} x + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} - a^{3} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b \sqrt {x} - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b^{2} x^{3} - a^{2}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x - \left (a b^{2}\right )^{\frac {1}{3}} b \sqrt {x} + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b^{2} x^{3} - a^{2}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b \sqrt {x} + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a^{2} b^{4} x^{3} - a^{4} b^{2}\right )}}\right ] \] Input:
integrate(1/(a+b*x^(3/2))^2,x, algorithm="fricas")
Output:
[1/9*(6*a*b^3*x^(5/2) - 6*a^2*b^2*x + 3*sqrt(1/3)*(a*b^3*x^3 - a^3*b)*sqrt (-(a*b^2)^(1/3)/a)*log((2*b^3*x^3 - 3*(a*b^2)^(2/3)*b*x^2 + a^2*b + 3*sqrt (1/3)*(a*b^2*x^2 - 2*(a*b^2)^(2/3)*a*x + (a*b^2)^(1/3)*a^2 + (2*(a*b^2)^(2 /3)*b*x^2 - (a*b^2)^(1/3)*a*b*x - a^2*b)*sqrt(x))*sqrt(-(a*b^2)^(1/3)/a) - 3*(a*b^2*x - (a*b^2)^(2/3)*a)*sqrt(x))/(b^2*x^3 - a^2)) + (b^2*x^3 - a^2) *(a*b^2)^(2/3)*log(b^2*x - (a*b^2)^(1/3)*b*sqrt(x) + (a*b^2)^(2/3)) - 2*(b ^2*x^3 - a^2)*(a*b^2)^(2/3)*log(b*sqrt(x) + (a*b^2)^(1/3)))/(a^2*b^4*x^3 - a^4*b^2), 1/9*(6*a*b^3*x^(5/2) - 6*a^2*b^2*x + 6*sqrt(1/3)*(a*b^3*x^3 - a ^3*b)*sqrt((a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*sqrt(x) - (a*b^2)^(1/3)) *sqrt((a*b^2)^(1/3)/a)/b) + (b^2*x^3 - a^2)*(a*b^2)^(2/3)*log(b^2*x - (a*b ^2)^(1/3)*b*sqrt(x) + (a*b^2)^(2/3)) - 2*(b^2*x^3 - a^2)*(a*b^2)^(2/3)*log (b*sqrt(x) + (a*b^2)^(1/3)))/(a^2*b^4*x^3 - a^4*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (136) = 272\).
Time = 18.99 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.14 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{2 b^{2} x^{2}} & \text {for}\: a = 0 \\\frac {6 a x}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 a \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (\sqrt {x} - \sqrt [3]{- \frac {a}{b}} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} + \frac {a \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (4 \sqrt {x} \sqrt [3]{- \frac {a}{b}} + 4 x + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 \sqrt {3} a \left (- \frac {a}{b}\right )^{\frac {2}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 a \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (2 \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 b x^{\frac {3}{2}} \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (\sqrt {x} - \sqrt [3]{- \frac {a}{b}} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} + \frac {b x^{\frac {3}{2}} \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (4 \sqrt {x} \sqrt [3]{- \frac {a}{b}} + 4 x + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 \sqrt {3} b x^{\frac {3}{2}} \left (- \frac {a}{b}\right )^{\frac {2}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} - \frac {2 b x^{\frac {3}{2}} \left (- \frac {a}{b}\right )^{\frac {2}{3}} \log {\left (2 \right )}}{9 a^{3} + 9 a^{2} b x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(3/2))**2,x)
Output:
Piecewise((zoo/x**2, Eq(a, 0) & Eq(b, 0)), (x/a**2, Eq(b, 0)), (-1/(2*b**2 *x**2), Eq(a, 0)), (6*a*x/(9*a**3 + 9*a**2*b*x**(3/2)) - 2*a*(-a/b)**(2/3) *log(sqrt(x) - (-a/b)**(1/3))/(9*a**3 + 9*a**2*b*x**(3/2)) + a*(-a/b)**(2/ 3)*log(4*sqrt(x)*(-a/b)**(1/3) + 4*x + 4*(-a/b)**(2/3))/(9*a**3 + 9*a**2*b *x**(3/2)) - 2*sqrt(3)*a*(-a/b)**(2/3)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**( 1/3)) + sqrt(3)/3)/(9*a**3 + 9*a**2*b*x**(3/2)) - 2*a*(-a/b)**(2/3)*log(2) /(9*a**3 + 9*a**2*b*x**(3/2)) - 2*b*x**(3/2)*(-a/b)**(2/3)*log(sqrt(x) - ( -a/b)**(1/3))/(9*a**3 + 9*a**2*b*x**(3/2)) + b*x**(3/2)*(-a/b)**(2/3)*log( 4*sqrt(x)*(-a/b)**(1/3) + 4*x + 4*(-a/b)**(2/3))/(9*a**3 + 9*a**2*b*x**(3/ 2)) - 2*sqrt(3)*b*x**(3/2)*(-a/b)**(2/3)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)* *(1/3)) + sqrt(3)/3)/(9*a**3 + 9*a**2*b*x**(3/2)) - 2*b*x**(3/2)*(-a/b)**( 2/3)*log(2)/(9*a**3 + 9*a**2*b*x**(3/2)), True))
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\frac {2 \, x}{3 \, {\left (a b x^{\frac {3}{2}} + a^{2}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x - \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (\sqrt {x} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(1/(a+b*x^(3/2))^2,x, algorithm="maxima")
Output:
2/3*x/(a*b*x^(3/2) + a^2) + 2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*sqrt(x) - (a /b)^(1/3))/(a/b)^(1/3))/(a*b*(a/b)^(1/3)) + 1/9*log(x - sqrt(x)*(a/b)^(1/3 ) + (a/b)^(2/3))/(a*b*(a/b)^(1/3)) - 2/9*log(sqrt(x) + (a/b)^(1/3))/(a*b*( a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {2 \, x}{3 \, {\left (b x^{\frac {3}{2}} + a\right )} a} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a^{2} b^{2}} \] Input:
integrate(1/(a+b*x^(3/2))^2,x, algorithm="giac")
Output:
-2/9*(-a/b)^(2/3)*log(abs(sqrt(x) - (-a/b)^(1/3)))/a^2 + 2/3*x/((b*x^(3/2) + a)*a) - 2/9*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-a/ b)^(1/3))/(-a/b)^(1/3))/(a^2*b^2) + 1/9*(-a*b^2)^(2/3)*log(x + sqrt(x)*(-a /b)^(1/3) + (-a/b)^(2/3))/(a^2*b^2)
Time = 0.35 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\frac {2\,x}{3\,a\,\left (a+b\,x^{3/2}\right )}-\frac {\ln \left (\frac {b^{2/3}\,{\left ({\left (-1\right )}^{1/3}+{\left (-1\right )}^{1/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}^2}{9\,a^{5/3}}+\frac {4\,b\,\sqrt {x}}{9\,a^2}\right )\,\left ({\left (-1\right )}^{1/3}+{\left (-1\right )}^{1/3}\,\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{4/3}\,b^{2/3}}+\frac {2\,{\left (-1\right )}^{1/3}\,\ln \left (\frac {4\,{\left (-1\right )}^{2/3}\,b^{2/3}}{9\,a^{5/3}}+\frac {4\,b\,\sqrt {x}}{9\,a^2}\right )}{9\,a^{4/3}\,b^{2/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {4\,b\,\sqrt {x}}{9\,a^2}+\frac {9\,{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2}{a^{5/3}}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{4/3}\,b^{2/3}} \] Input:
int(1/(a + b*x^(3/2))^2,x)
Output:
(2*x)/(3*a*(a + b*x^(3/2))) - (log((b^(2/3)*((-1)^(1/3)*3^(1/2)*1i + (-1)^ (1/3))^2)/(9*a^(5/3)) + (4*b*x^(1/2))/(9*a^2))*((-1)^(1/3)*3^(1/2)*1i + (- 1)^(1/3)))/(9*a^(4/3)*b^(2/3)) + (2*(-1)^(1/3)*log((4*(-1)^(2/3)*b^(2/3))/ (9*a^(5/3)) + (4*b*x^(1/2))/(9*a^2)))/(9*a^(4/3)*b^(2/3)) + ((-1)^(1/3)*lo g((4*b*x^(1/2))/(9*a^2) + (9*(-1)^(2/3)*b^(2/3)*((3^(1/2)*1i)/9 - 1/9)^2)/ a^(5/3))*((3^(1/2)*1i)/9 - 1/9))/(a^(4/3)*b^(2/3))
Time = 0.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^2} \, dx=\frac {-2 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b x -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a +6 b^{\frac {2}{3}} a^{\frac {1}{3}} x +\sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) b x -2 \sqrt {x}\, \mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) b x +\mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a}{9 b^{\frac {2}{3}} a^{\frac {4}{3}} \left (\sqrt {x}\, b x +a \right )} \] Input:
int(1/(a+b*x^(3/2))^2,x)
Output:
( - 2*sqrt(x)*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt( 3)))*b*x - 2*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt(3 )))*a + 6*b**(2/3)*a**(1/3)*x + sqrt(x)*log(a**(2/3) - sqrt(x)*b**(1/3)*a* *(1/3) + b**(2/3)*x)*b*x - 2*sqrt(x)*log(a**(1/3) + sqrt(x)*b**(1/3))*b*x + log(a**(2/3) - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3)*x)*a - 2*log(a**(1/3 ) + sqrt(x)*b**(1/3))*a)/(9*b**(2/3)*a**(1/3)*a*(sqrt(x)*b*x + a))