Integrand size = 11, antiderivative size = 130 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {x}{a}+\frac {2 b^{2/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {2 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{3 a^{5/3}} \] Output:
x/a+2/3*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^(5/3)+2/3*b^(2/3)*ln(b^(1/3)+a^(1/3)*x^(1/2))/a^(5/3)-1/3*b^(2/3)* ln(b^(2/3)-a^(1/3)*b^(1/3)*x^(1/2)+a^(2/3)*x)/a^(5/3)
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {3 a^{2/3} x+2 \sqrt {3} b^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} \sqrt {x}\right )-b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3} x\right )}{3 a^{5/3}} \] Input:
Integrate[(a + b/x^(3/2))^(-1),x]
Output:
(3*a^(2/3)*x + 2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*a^(1/3)*Sqrt[x])/b^(1/3))/ Sqrt[3]] + 2*b^(2/3)*Log[b^(1/3) + a^(1/3)*Sqrt[x]] - b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + a^(2/3)*x])/(3*a^(5/3))
Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {774, 795, 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.
| \(\displaystyle \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 2 \int \frac {\sqrt {x}}{a+\frac {b}{x^{3/2}}}d\sqrt {x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle 2 \int \frac {x^2}{a x^{3/2}+b}d\sqrt {x}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \int \frac {\sqrt {x}}{a x^{3/2}+b}d\sqrt {x}}{a}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\int \frac {\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}\right )}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{2 \sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{b} \int \frac {1}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}\right )}{\sqrt [3]{a}}-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b}-2 \sqrt [3]{a} \sqrt {x}}{a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {x}{2 a}-\frac {b \left (\frac {\frac {\log \left (a^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3}\right )}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} \sqrt {x}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a} \sqrt {x}+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}\right )\) |
Input:
Int[(a + b/x^(3/2))^(-1),x]
Output:
2*(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)) + L og[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)
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[(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 = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {x}{a}-\frac {2 \left (-\frac {\ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(107\) |
| default | \(\frac {x}{a}-\frac {2 \left (-\frac {\ln \left (\sqrt {x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(107\) |
Input:
int(1/(a+b/x^(3/2)),x,method=_RETURNVERBOSE)
Output:
x/a-2*(-1/3/a/(b/a)^(1/3)*ln(x^(1/2)+(b/a)^(1/3))+1/6/a/(b/a)^(1/3)*ln(x-( b/a)^(1/3)*x^(1/2)+(b/a)^(2/3))+1/3*3^(1/2)/a/(b/a)^(1/3)*arctan(1/3*3^(1/ 2)*(2/(b/a)^(1/3)*x^(1/2)-1)))*b/a
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) + \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (-a \sqrt {x} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b x + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \sqrt {x}\right ) - 3 \, x}{3 \, a} \] Input:
integrate(1/(a+b/x^(3/2)),x, algorithm="fricas")
Output:
-1/3*(2*sqrt(3)*(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) + (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)) - 2*(b^2/a^2)^(1/3)*log(a*(b^2/a^2)^(2/3) + b*sqrt (x)) - 3*x)/a
Time = 1.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b} & \text {for}\: a = 0 \\\frac {x}{a} & \text {for}\: b = 0 \\\frac {x}{a} - \frac {2 b \log {\left (\sqrt {x} - \sqrt [3]{- \frac {b}{a}} \right )}}{3 a^{2} \sqrt [3]{- \frac {b}{a}}} + \frac {b \log {\left (4 \sqrt {x} \sqrt [3]{- \frac {b}{a}} + 4 x + 4 \left (- \frac {b}{a}\right )^{\frac {2}{3}} \right )}}{3 a^{2} \sqrt [3]{- \frac {b}{a}}} - \frac {2 \sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [3]{- \frac {b}{a}}} + \frac {\sqrt {3}}{3} \right )}}{3 a^{2} \sqrt [3]{- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(3/2)),x)
Output:
Piecewise((zoo*x**(5/2), Eq(a, 0) & Eq(b, 0)), (2*x**(5/2)/(5*b), Eq(a, 0) ), (x/a, Eq(b, 0)), (x/a - 2*b*log(sqrt(x) - (-b/a)**(1/3))/(3*a**2*(-b/a) **(1/3)) + b*log(4*sqrt(x)*(-b/a)**(1/3) + 4*x + 4*(-b/a)**(2/3))/(3*a**2* (-b/a)**(1/3)) - 2*sqrt(3)*b*atan(2*sqrt(3)*sqrt(x)/(3*(-b/a)**(1/3)) + sq rt(3)/3)/(3*a**2*(-b/a)**(1/3)), True))
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {x}{a} + \frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2}{\sqrt {x}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{3}}}{\sqrt {x}} + \frac {1}{x}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {1}{\sqrt {x}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(a+b/x^(3/2)),x, algorithm="maxima")
Output:
x/a + 2/3*sqrt(3)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2/sqrt(x))/(a/b)^(1/3 ))/(a*(a/b)^(2/3)) - 1/3*log((a/b)^(2/3) - (a/b)^(1/3)/sqrt(x) + 1/x)/(a*( a/b)^(2/3)) + 2/3*log((a/b)^(1/3) + 1/sqrt(x))/(a*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {2 \, \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {x}{a} + \frac {2 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} - \frac {\left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{3 \, a^{3}} \] Input:
integrate(1/(a+b/x^(3/2)),x, algorithm="giac")
Output:
2/3*(-b/a)^(2/3)*log(abs(sqrt(x) - (-b/a)^(1/3)))/a + x/a + 2/3*sqrt(3)*(- a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-b/a)^(1/3))/(-b/a)^(1/3))/a ^3 - 1/3*(-a^2*b)^(2/3)*log(x + sqrt(x)*(-b/a)^(1/3) + (-b/a)^(2/3))/a^3
Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {x}{a}+\frac {2\,b^{2/3}\,\ln \left (\frac {4\,b^{7/3}}{a^{4/3}}+\frac {4\,b^2\,\sqrt {x}}{a}\right )}{3\,a^{5/3}}+\frac {b^{2/3}\,\ln \left (\frac {9\,b^{7/3}\,{\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2}{a^{4/3}}+\frac {4\,b^2\,\sqrt {x}}{a}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}{a^{5/3}}-\frac {b^{2/3}\,\ln \left (\frac {9\,b^{7/3}\,{\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}^2}{a^{4/3}}+\frac {4\,b^2\,\sqrt {x}}{a}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )}{a^{5/3}} \] Input:
int(1/(a + b/x^(3/2)),x)
Output:
x/a + (2*b^(2/3)*log((4*b^(7/3))/a^(4/3) + (4*b^2*x^(1/2))/a))/(3*a^(5/3)) + (b^(2/3)*log((9*b^(7/3)*((3^(1/2)*1i)/3 - 1/3)^2)/a^(4/3) + (4*b^2*x^(1 /2))/a)*((3^(1/2)*1i)/3 - 1/3))/a^(5/3) - (b^(2/3)*log((9*b^(7/3)*((3^(1/2 )*1i)/3 + 1/3)^2)/a^(4/3) + (4*b^2*x^(1/2))/a)*((3^(1/2)*1i)/3 + 1/3))/a^( 5/3)
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int \frac {1}{a+\frac {b}{x^{3/2}}} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, a^{\frac {1}{3}}-b^{\frac {1}{3}}}{b^{\frac {1}{3}} \sqrt {3}}\right ) b +3 b^{\frac {1}{3}} a^{\frac {2}{3}} x -\mathrm {log}\left (a^{\frac {2}{3}} x -\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}}\right ) b +2 \,\mathrm {log}\left (\sqrt {x}\, a^{\frac {1}{3}}+b^{\frac {1}{3}}\right ) b}{3 b^{\frac {1}{3}} a^{\frac {5}{3}}} \] Input:
int(1/(a+b/x^(3/2)),x)
Output:
( - 2*sqrt(3)*atan((2*sqrt(x)*a**(1/3) - b**(1/3))/(b**(1/3)*sqrt(3)))*b + 3*b**(1/3)*a**(2/3)*x - log(a**(2/3)*x - sqrt(x)*b**(1/3)*a**(1/3) + b**( 2/3))*b + 2*log(sqrt(x)*a**(1/3) + b**(1/3))*b)/(3*b**(1/3)*a**(2/3)*a)