Integrand size = 17, antiderivative size = 221 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{a^{5/6} \sqrt [6]{b} c^{2/3}}-\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{5/6} \sqrt [6]{b} c^{2/3}}+\frac {\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{5/6} \sqrt [6]{b} c^{2/3}}+\frac {\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}\right )}{2 a^{5/6} \sqrt [6]{b} c^{2/3}} \] Output:
arctan(b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(5/6)/b^(1/6)/c^(2/3)+1/2*ar ctan(-3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(5/6)/b^(1/6)/c^(2/ 3)+1/2*arctan(3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(5/6)/b^(1/ 6)/c^(2/3)+1/2*3^(1/2)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*c^(1/3)*(c*x)^(1/3) /(a^(1/3)*c^(2/3)+b^(1/3)*(c*x)^(2/3)))/a^(5/6)/b^(1/6)/c^(2/3)
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.59 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {x^{2/3} \left (-\arctan \left (\frac {\sqrt [6]{a}}{\sqrt [6]{b} \sqrt [3]{x}}-\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}}\right )\right )}{2 a^{5/6} \sqrt [6]{b} (c x)^{2/3}} \] Input:
Integrate[1/((c*x)^(2/3)*(a + b*x^2)),x]
Output:
(x^(2/3)*(-ArcTan[a^(1/6)/(b^(1/6)*x^(1/3)) - (b^(1/6)*x^(1/3))/a^(1/6)] + 2*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] + Sqrt[3]*ArcTanh[(Sqrt[3]*a^(1/6)*b^ (1/6)*x^(1/3))/(a^(1/3) + b^(1/3)*x^(2/3))]))/(2*a^(5/6)*b^(1/6)*(c*x)^(2/ 3))
Time = 0.51 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.38, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {266, 753, 27, 218, 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.
| \(\displaystyle \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \int \frac {1}{b x^2+a}d\sqrt [3]{c x}}{c}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {3 \left (\frac {c^{2/3} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 a^{2/3}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 a^{5/6}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {c^{2/3} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 a^{2/3}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}\right )}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt [3]{c} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{c}\) |
Input:
Int[1/((c*x)^(2/3)*(a + b*x^2)),x]
Output:
(3*((c^(1/3)*ArcTan[(b^(1/6)*(c*x)^(1/3))/(a^(1/6)*c^(1/3))])/(3*a^(5/6)*b ^(1/6)) + (c^(1/3)*(-(ArcTan[Sqrt[3]*(1 - (2*b^(1/6)*(c*x)^(1/3))/(Sqrt[3] *a^(1/6)*c^(1/3)))]/b^(1/6)) - (Sqrt[3]*Log[a^(1/3)*c^(2/3) - Sqrt[3]*a^(1 /6)*b^(1/6)*c^(1/3)*(c*x)^(1/3) + b^(1/3)*(c*x)^(2/3)])/(2*b^(1/6))))/(6*a ^(5/6)) + (c^(1/3)*(ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*(c*x)^(1/3))/(Sqrt[3]*a ^(1/6)*c^(1/3)))]/b^(1/6) + (Sqrt[3]*Log[a^(1/3)*c^(2/3) + Sqrt[3]*a^(1/6) *b^(1/6)*c^(1/3)*(c*x)^(1/3) + b^(1/3)*(c*x)^(2/3)])/(2*b^(1/6))))/(6*a^(5 /6))))/c
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
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 = 0.51 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(-\frac {\left (-\frac {\sqrt {3}\, \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}+\frac {\sqrt {3}\, \ln \left (\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}-\left (c x \right )^{\frac {2}{3}}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}-2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )-2 \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )-\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}+2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )\right ) \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}{2 a c}\) | \(195\) |
| derivativedivides | \(3 c \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a \,c^{2}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a \,c^{2}}\right )\) | \(237\) |
| default | \(3 c \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a \,c^{2}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a \,c^{2}}\right )\) | \(237\) |
Input:
int(1/(c*x)^(2/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2*(-1/2*3^(1/2)*ln((c*x)^(2/3)+3^(1/2)*(a*c^2/b)^(1/6)*(c*x)^(1/3)+(a*c ^2/b)^(1/3))+1/2*3^(1/2)*ln(3^(1/2)*(a*c^2/b)^(1/6)*(c*x)^(1/3)-(c*x)^(2/3 )-(a*c^2/b)^(1/3))+arctan((3^(1/2)*(a*c^2/b)^(1/6)-2*(c*x)^(1/3))/(a*c^2/b )^(1/6))-2*arctan((c*x)^(1/3)/(a*c^2/b)^(1/6))-arctan((3^(1/2)*(a*c^2/b)^( 1/6)+2*(c*x)^(1/3))/(a*c^2/b)^(1/6)))*(a*c^2/b)^(1/6)/a/c
Time = 0.07 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a c + a c\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a c + a c\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a c - a c\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a c - a c\right )} \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) + \frac {1}{2} \, \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (a c \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) - \frac {1}{2} \, \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} \log \left (-a c \left (-\frac {1}{a^{5} b c^{4}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}}\right ) \] Input:
integrate(1/(c*x)^(2/3)/(b*x^2+a),x, algorithm="fricas")
Output:
1/4*(sqrt(-3) + 1)*(-1/(a^5*b*c^4))^(1/6)*log(1/2*(sqrt(-3)*a*c + a*c)*(-1 /(a^5*b*c^4))^(1/6) + (c*x)^(1/3)) - 1/4*(sqrt(-3) + 1)*(-1/(a^5*b*c^4))^( 1/6)*log(-1/2*(sqrt(-3)*a*c + a*c)*(-1/(a^5*b*c^4))^(1/6) + (c*x)^(1/3)) + 1/4*(sqrt(-3) - 1)*(-1/(a^5*b*c^4))^(1/6)*log(1/2*(sqrt(-3)*a*c - a*c)*(- 1/(a^5*b*c^4))^(1/6) + (c*x)^(1/3)) - 1/4*(sqrt(-3) - 1)*(-1/(a^5*b*c^4))^ (1/6)*log(-1/2*(sqrt(-3)*a*c - a*c)*(-1/(a^5*b*c^4))^(1/6) + (c*x)^(1/3)) + 1/2*(-1/(a^5*b*c^4))^(1/6)*log(a*c*(-1/(a^5*b*c^4))^(1/6) + (c*x)^(1/3)) - 1/2*(-1/(a^5*b*c^4))^(1/6)*log(-a*c*(-1/(a^5*b*c^4))^(1/6) + (c*x)^(1/3 ))
Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} + \frac {i \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} + \frac {e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {5 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} - \frac {e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {7 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} - \frac {i \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {3 i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} - \frac {e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {11 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {1}{6}\right )}{12 a^{\frac {5}{6}} \sqrt [6]{b} c^{\frac {2}{3}} \Gamma \left (\frac {7}{6}\right )} \] Input:
integrate(1/(c*x)**(2/3)/(b*x**2+a),x)
Output:
exp(5*I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(I*pi/6)/a**(1/6))*gamma( 1/6)/(12*a**(5/6)*b**(1/6)*c**(2/3)*gamma(7/6)) + I*log(1 - b**(1/6)*x**(1 /3)*exp_polar(I*pi/2)/a**(1/6))*gamma(1/6)/(12*a**(5/6)*b**(1/6)*c**(2/3)* gamma(7/6)) + exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(5*I*pi/6)/a* *(1/6))*gamma(1/6)/(12*a**(5/6)*b**(1/6)*c**(2/3)*gamma(7/6)) - exp(5*I*pi /6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(7*I*pi/6)/a**(1/6))*gamma(1/6)/(12 *a**(5/6)*b**(1/6)*c**(2/3)*gamma(7/6)) - I*log(1 - b**(1/6)*x**(1/3)*exp_ polar(3*I*pi/2)/a**(1/6))*gamma(1/6)/(12*a**(5/6)*b**(1/6)*c**(2/3)*gamma( 7/6)) - exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(11*I*pi/6)/a**(1/6 ))*gamma(1/6)/(12*a**(5/6)*b**(1/6)*c**(2/3)*gamma(7/6))
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (145) = 290\).
Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {\frac {\sqrt {3} c^{2} \log \left (\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} c^{2} \log \left (-\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, c^{2} \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\left (a c^{2}\right )^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \left (a c^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \left (a c^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}}{4 \, c} \] Input:
integrate(1/(c*x)^(2/3)/(b*x^2+a),x, algorithm="maxima")
Output:
1/4*(sqrt(3)*c^2*log(sqrt(3)*(a*c^2)^(1/6)*(c*x)^(1/3)*b^(1/6) + (c*x)^(2/ 3)*b^(1/3) + (a*c^2)^(1/3))/((a*c^2)^(5/6)*b^(1/6)) - sqrt(3)*c^2*log(-sqr t(3)*(a*c^2)^(1/6)*(c*x)^(1/3)*b^(1/6) + (c*x)^(2/3)*b^(1/3) + (a*c^2)^(1/ 3))/((a*c^2)^(5/6)*b^(1/6)) + 4*c^2*arctan((c*x)^(1/3)*b^(1/3)/sqrt((a*c^2 )^(1/3)*b^(1/3)))/((a*c^2)^(2/3)*sqrt((a*c^2)^(1/3)*b^(1/3))) + 2*(a*c^2)^ (1/3)*arctan((sqrt(3)*(a*c^2)^(1/6)*b^(1/6) + 2*(c*x)^(1/3)*b^(1/3))/sqrt( (a*c^2)^(1/3)*b^(1/3)))/(a*sqrt((a*c^2)^(1/3)*b^(1/3))) + 2*(a*c^2)^(1/3)* arctan(-(sqrt(3)*(a*c^2)^(1/6)*b^(1/6) - 2*(c*x)^(1/3)*b^(1/3))/sqrt((a*c^ 2)^(1/3)*b^(1/3)))/(a*sqrt((a*c^2)^(1/3)*b^(1/3))))/c
Time = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {1}{6}} \log \left (\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a b c} - \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a b c} + \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a b c} + \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a b c} + \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a b c} \] Input:
integrate(1/(c*x)^(2/3)/(b*x^2+a),x, algorithm="giac")
Output:
1/4*sqrt(3)*(a*b^5*c^2)^(1/6)*log(sqrt(3)*(a*c^2/b)^(1/6)*(c*x)^(1/3) + (c *x)^(2/3) + (a*c^2/b)^(1/3))/(a*b*c) - 1/4*sqrt(3)*(a*b^5*c^2)^(1/6)*log(- sqrt(3)*(a*c^2/b)^(1/6)*(c*x)^(1/3) + (c*x)^(2/3) + (a*c^2/b)^(1/3))/(a*b* c) + 1/2*(a*b^5*c^2)^(1/6)*arctan((sqrt(3)*(a*c^2/b)^(1/6) + 2*(c*x)^(1/3) )/(a*c^2/b)^(1/6))/(a*b*c) + 1/2*(a*b^5*c^2)^(1/6)*arctan(-(sqrt(3)*(a*c^2 /b)^(1/6) - 2*(c*x)^(1/3))/(a*c^2/b)^(1/6))/(a*b*c) + (a*b^5*c^2)^(1/6)*ar ctan((c*x)^(1/3)/(a*c^2/b)^(1/6))/(a*b*c)
Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=-\frac {\mathrm {atanh}\left (\frac {b^{1/6}\,{\left (c\,x\right )}^{1/3}}{{\left (-a\right )}^{1/6}\,c^{1/3}}\right )}{{\left (-a\right )}^{5/6}\,b^{1/6}\,c^{2/3}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,c^{10/3}\,{\left (c\,x\right )}^{1/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {243\,b^{14/3}\,c^{11/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,c^{11/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}-\frac {243\,\sqrt {3}\,b^{29/6}\,c^{10/3}\,{\left (c\,x\right )}^{1/3}}{{\left (-a\right )}^{5/6}\,\left (\frac {243\,b^{14/3}\,c^{11/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,c^{11/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{5/6}\,b^{1/6}\,c^{2/3}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,c^{10/3}\,{\left (c\,x\right )}^{1/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {243\,b^{14/3}\,c^{11/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,c^{11/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}+\frac {243\,\sqrt {3}\,b^{29/6}\,c^{10/3}\,{\left (c\,x\right )}^{1/3}}{{\left (-a\right )}^{5/6}\,\left (\frac {243\,b^{14/3}\,c^{11/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,c^{11/3}\,243{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{5/6}\,b^{1/6}\,c^{2/3}} \] Input:
int(1/((c*x)^(2/3)*(a + b*x^2)),x)
Output:
(atan((b^(29/6)*c^(10/3)*(c*x)^(1/3)*243i)/((-a)^(5/6)*((243*b^(14/3)*c^(1 1/3))/(-a)^(2/3) + (3^(1/2)*b^(14/3)*c^(11/3)*243i)/(-a)^(2/3))) + (243*3^ (1/2)*b^(29/6)*c^(10/3)*(c*x)^(1/3))/((-a)^(5/6)*((243*b^(14/3)*c^(11/3))/ (-a)^(2/3) + (3^(1/2)*b^(14/3)*c^(11/3)*243i)/(-a)^(2/3))))*(3^(1/2)*1i - 1)*1i)/(2*(-a)^(5/6)*b^(1/6)*c^(2/3)) - (atan((b^(29/6)*c^(10/3)*(c*x)^(1/ 3)*243i)/((-a)^(5/6)*((243*b^(14/3)*c^(11/3))/(-a)^(2/3) - (3^(1/2)*b^(14/ 3)*c^(11/3)*243i)/(-a)^(2/3))) - (243*3^(1/2)*b^(29/6)*c^(10/3)*(c*x)^(1/3 ))/((-a)^(5/6)*((243*b^(14/3)*c^(11/3))/(-a)^(2/3) - (3^(1/2)*b^(14/3)*c^( 11/3)*243i)/(-a)^(2/3))))*(3^(1/2)*1i + 1)*1i)/(2*(-a)^(5/6)*b^(1/6)*c^(2/ 3)) - atanh((b^(1/6)*(c*x)^(1/3))/((-a)^(1/6)*c^(1/3)))/((-a)^(5/6)*b^(1/6 )*c^(2/3))
Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {-2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+4 \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )-\sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )+\sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )}{4 b^{\frac {1}{6}} a^{\frac {5}{6}} c^{\frac {2}{3}}} \] Input:
int(1/(c*x)^(2/3)/(b*x^2+a),x)
Output:
(b**(1/6)*a**(1/6)*( - 2*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**( 1/3))/(b**(1/6)*a**(1/6))) + 2*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3 )*b**(1/3))/(b**(1/6)*a**(1/6))) + 4*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a* *(1/6))) - sqrt(3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3)) + sqrt(3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**( 1/3) + x**(2/3)*b**(1/3))))/(4*c**(2/3)*b**(1/3)*a)