Integrand size = 11, antiderivative size = 152 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {x}{a^2}+\frac {2 b x}{3 a^2 \left (b+a x^{3/2}\right )}+\frac {10 b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {10 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )}{9 a^{8/3}}-\frac {5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{9 a^{8/3}} \] Output:
x/a^2+2/3*b*x/a^2/(b+a*x^(3/2))+10/9*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^(8/3)+10/9*b^(2/3)*ln(b^(1/3)+a^(1/3) *x^(1/2))/a^(8/3)-5/9*b^(2/3)*ln(b^(2/3)-a^(1/3)*b^(1/3)*x^(1/2)+a^(2/3)*x )/a^(8/3)
Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {\frac {3 a^{2/3} x \left (5 b+3 a x^{3/2}\right )}{b+a x^{3/2}}+10 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+10 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )-5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{9 a^{8/3}} \] Input:
Integrate[(a + b/x^(3/2))^(-2),x]
Output:
((3*a^(2/3)*x*(5*b + 3*a*x^(3/2)))/(b + a*x^(3/2)) + 10*Sqrt[3]*b^(2/3)*Ar cTan[(1 - (2*a^(1/3)*Sqrt[x])/b^(1/3))/Sqrt[3]] + 10*b^(2/3)*Log[b^(1/3) + a^(1/3)*Sqrt[x]] - 5*b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + a^(2 /3)*x])/(9*a^(8/3))
Time = 0.52 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {774, 795, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 2 \int \frac {\sqrt {x}}{\left (a+\frac {b}{x^{3/2}}\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle 2 \int \frac {x^{7/2}}{\left (a x^{3/2}+b\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 2 \left (\frac {5 \int \frac {x^2}{a x^{3/2}+b}d\sqrt {x}}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \int \frac {\sqrt {x}}{a x^{3/2}+b}d\sqrt {x}}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {5 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {\log \left (a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )}{3 a}-\frac {x^{5/2}}{3 a \left (a x^{3/2}+b\right )}\right )\) |
Input:
Int[(a + b/x^(3/2))^(-2),x]
Output:
2*(-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)*S qrt[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))
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 = 119, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {x}{a^{2}}-\frac {2 b \left (-\frac {x}{3 \left (b +a \,x^{\frac {3}{2}}\right )}-\frac {5 \ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{2}}\) | \(119\) |
| default | \(\frac {x}{a^{2}}-\frac {2 b \left (-\frac {x}{3 \left (b +a \,x^{\frac {3}{2}}\right )}-\frac {5 \ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{2}}\) | \(119\) |
Input:
int(1/(a+b/x^(3/2))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*x-2*b/a^2*(-1/3*x/(b+a*x^(3/2))-5/9/a/(b/a)^(1/3)*ln(x^(1/2)+(b/a)^( 1/3))+5/18/a/(b/a)^(1/3)*ln(x-(b/a)^(1/3)*x^(1/2)+(b/a)^(2/3))+5/9*3^(1/2) /a/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*x^(1/2)-1)))
Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {9 \, a^{2} x^{4} + 6 \, a b x^{\frac {5}{2}} - 15 \, b^{2} x - 10 \, \sqrt {3} {\left (a^{2} x^{3} - b^{2}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left (a^{2} x^{3} - b^{2}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (-a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b x + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, {\left (a^{2} x^{3} - b^{2}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \sqrt {x}\right )}{9 \, {\left (a^{4} x^{3} - a^{2} b^{2}\right )}} \] Input:
integrate(1/(a+b/x^(3/2))^2,x, algorithm="fricas")
Output:
1/9*(9*a^2*x^4 + 6*a*b*x^(5/2) - 15*b^2*x - 10*sqrt(3)*(a^2*x^3 - b^2)*(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) - 5*(a^2*x^3 - b^2)*(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)) + 10*(a^2*x^3 - b^2)*(b^2/a^2)^(1/3)*log(a*(b^2/a^2 )^(2/3) + b*sqrt(x)))/(a^4*x^3 - a^2*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (146) = 292\).
Time = 19.94 (sec) , antiderivative size = 620, normalized size of antiderivative = 4.08 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\begin {cases} \tilde {\infty } x^{4} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x^{4}}{4 b^{2}} & \text {for}\: a = 0 \\\frac {x}{a^{2}} & \text {for}\: b = 0 \\\frac {9 a^{2} x^{\frac {5}{2}} \sqrt [3]{- \frac {b}{a}}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt [3]{- \frac {b}{a}} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} + \frac {5 a b x^{\frac {3}{2}} \log {\left (4 \sqrt {x} \sqrt [3]{- \frac {b}{a}} + 4 x + 4 \left (- \frac {b}{a}\right )^{\frac {2}{3}} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 \sqrt {3} a b x^{\frac {3}{2}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [3]{- \frac {b}{a}}} + \frac {\sqrt {3}}{3} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 a b x^{\frac {3}{2}} \log {\left (2 \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} + \frac {15 a b x \sqrt [3]{- \frac {b}{a}}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 b^{2} \log {\left (\sqrt {x} - \sqrt [3]{- \frac {b}{a}} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} + \frac {5 b^{2} \log {\left (4 \sqrt {x} \sqrt [3]{- \frac {b}{a}} + 4 x + 4 \left (- \frac {b}{a}\right )^{\frac {2}{3}} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 \sqrt {3} b^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [3]{- \frac {b}{a}}} + \frac {\sqrt {3}}{3} \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} - \frac {10 b^{2} \log {\left (2 \right )}}{9 a^{4} x^{\frac {3}{2}} \sqrt [3]{- \frac {b}{a}} + 9 a^{3} b \sqrt [3]{- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(3/2))**2,x)
Output:
Piecewise((zoo*x**4, Eq(a, 0) & Eq(b, 0)), (x**4/(4*b**2), Eq(a, 0)), (x/a **2, Eq(b, 0)), (9*a**2*x**(5/2)*(-b/a)**(1/3)/(9*a**4*x**(3/2)*(-b/a)**(1 /3) + 9*a**3*b*(-b/a)**(1/3)) - 10*a*b*x**(3/2)*log(sqrt(x) - (-b/a)**(1/3 ))/(9*a**4*x**(3/2)*(-b/a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) + 5*a*b*x**(3/ 2)*log(4*sqrt(x)*(-b/a)**(1/3) + 4*x + 4*(-b/a)**(2/3))/(9*a**4*x**(3/2)*( -b/a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) - 10*sqrt(3)*a*b*x**(3/2)*atan(2*sq rt(3)*sqrt(x)/(3*(-b/a)**(1/3)) + sqrt(3)/3)/(9*a**4*x**(3/2)*(-b/a)**(1/3 ) + 9*a**3*b*(-b/a)**(1/3)) - 10*a*b*x**(3/2)*log(2)/(9*a**4*x**(3/2)*(-b/ a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) + 15*a*b*x*(-b/a)**(1/3)/(9*a**4*x**(3 /2)*(-b/a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) - 10*b**2*log(sqrt(x) - (-b/a) **(1/3))/(9*a**4*x**(3/2)*(-b/a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) + 5*b**2 *log(4*sqrt(x)*(-b/a)**(1/3) + 4*x + 4*(-b/a)**(2/3))/(9*a**4*x**(3/2)*(-b /a)**(1/3) + 9*a**3*b*(-b/a)**(1/3)) - 10*sqrt(3)*b**2*atan(2*sqrt(3)*sqrt (x)/(3*(-b/a)**(1/3)) + sqrt(3)/3)/(9*a**4*x**(3/2)*(-b/a)**(1/3) + 9*a**3 *b*(-b/a)**(1/3)) - 10*b**2*log(2)/(9*a**4*x**(3/2)*(-b/a)**(1/3) + 9*a**3 *b*(-b/a)**(1/3)), True))
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {3 \, a + \frac {5 \, b}{x^{\frac {3}{2}}}}{3 \, {\left (\frac {a^{3}}{x} + \frac {a^{2} b}{x^{\frac {5}{2}}}\right )}} + \frac {10 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2}{\sqrt {x}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{3}}}{\sqrt {x}} + \frac {1}{x}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {10 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {1}{\sqrt {x}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(a+b/x^(3/2))^2,x, algorithm="maxima")
Output:
1/3*(3*a + 5*b/x^(3/2))/(a^3/x + a^2*b/x^(5/2)) + 10/9*sqrt(3)*arctan(-1/3 *sqrt(3)*((a/b)^(1/3) - 2/sqrt(x))/(a/b)^(1/3))/(a^2*(a/b)^(2/3)) - 5/9*lo g((a/b)^(2/3) - (a/b)^(1/3)/sqrt(x) + 1/x)/(a^2*(a/b)^(2/3)) + 10/9*log((a /b)^(1/3) + 1/sqrt(x))/(a^2*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {10 \, \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {x}{a^{2}} + \frac {2 \, b x}{3 \, {\left (a x^{\frac {3}{2}} + b\right )} a^{2}} + \frac {10 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} - \frac {5 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{9 \, a^{4}} \] Input:
integrate(1/(a+b/x^(3/2))^2,x, algorithm="giac")
Output:
10/9*(-b/a)^(2/3)*log(abs(sqrt(x) - (-b/a)^(1/3)))/a^2 + x/a^2 + 2/3*b*x/( (a*x^(3/2) + b)*a^2) + 10/9*sqrt(3)*(-a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*s qrt(x) + (-b/a)^(1/3))/(-b/a)^(1/3))/a^4 - 5/9*(-a^2*b)^(2/3)*log(x + sqrt (x)*(-b/a)^(1/3) + (-b/a)^(2/3))/a^4
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {x}{a^2}+\frac {10\,b^{2/3}\,\ln \left (\frac {100\,b^{7/3}}{9\,a^{10/3}}+\frac {100\,b^2\,\sqrt {x}}{9\,a^3}\right )}{9\,a^{8/3}}+\frac {2\,b\,x}{3\,\left (a^2\,b+a^3\,x^{3/2}\right )}+\frac {10\,b^{2/3}\,\ln \left (\frac {100\,b^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{10/3}}+\frac {100\,b^2\,\sqrt {x}}{9\,a^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}}-\frac {10\,b^{2/3}\,\ln \left (\frac {100\,b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{10/3}}+\frac {100\,b^2\,\sqrt {x}}{9\,a^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}} \] Input:
int(1/(a + b/x^(3/2))^2,x)
Output:
x/a^2 + (10*b^(2/3)*log((100*b^(7/3))/(9*a^(10/3)) + (100*b^2*x^(1/2))/(9* a^3)))/(9*a^(8/3)) + (2*b*x)/(3*(a^2*b + a^3*x^(3/2))) + (10*b^(2/3)*log(( 100*b^(7/3)*((3^(1/2)*1i)/2 - 1/2)^2)/(9*a^(10/3)) + (100*b^2*x^(1/2))/(9* a^3))*((3^(1/2)*1i)/2 - 1/2))/(9*a^(8/3)) - (10*b^(2/3)*log((100*b^(7/3)*( (3^(1/2)*1i)/2 + 1/2)^2)/(9*a^(10/3)) + (100*b^2*x^(1/2))/(9*a^3))*((3^(1/ 2)*1i)/2 + 1/2))/(9*a^(8/3))
Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+\frac {b}{x^{3/2}}\right )^2} \, dx=\frac {-10 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) a b x -10 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) b^{2}+9 \sqrt {x}\, b^{\frac {1}{3}} a^{\frac {5}{3}} x^{2}+15 b^{\frac {4}{3}} a^{\frac {2}{3}} x -5 \sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) a b x +10 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) a b x -5 \,\mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) b^{2}+10 \,\mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) b^{2}}{9 b^{\frac {1}{3}} a^{\frac {8}{3}} \left (\sqrt {x}\, a x +b \right )} \] Input:
int(1/(a+b/x^(3/2))^2,x)
Output:
( - 10*sqrt(x)*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/(b**(1/3)*sqrt (3)))*a*b*x - 10*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/(b**(1/3)*sq rt(3)))*b**2 + 9*sqrt(x)*b**(1/3)*a**(2/3)*a*x**2 + 15*b**(1/3)*a**(2/3)*b *x - 5*sqrt(x)*log(a**(2/3)*x - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3))*a*b* x + 10*sqrt(x)*log(sqrt(x)*a**(1/3) + b**(1/3))*a*b*x - 5*log(a**(2/3)*x - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3))*b**2 + 10*log(sqrt(x)*a**(1/3) + b* *(1/3))*b**2)/(9*b**(1/3)*a**(2/3)*a**2*(sqrt(x)*a*x + b))