Integrand size = 17, antiderivative size = 221 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=\frac {c^{2/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{\sqrt [6]{a} b^{5/6}}-\frac {c^{2/3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 \sqrt [6]{a} b^{5/6}}+\frac {c^{2/3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 \sqrt [6]{a} b^{5/6}}-\frac {\sqrt {3} c^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}\right )}{2 \sqrt [6]{a} b^{5/6}} \] Output:
c^(2/3)*arctan(b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(1/6)/b^(5/6)+1/2*c^ (2/3)*arctan(-3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(1/6)/b^(5/ 6)+1/2*c^(2/3)*arctan(3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^(1/ 6)/b^(5/6)-1/2*3^(1/2)*c^(2/3)*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^(1/6)/b^(5/6)
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.58 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=-\frac {(c x)^{2/3} \left (\arctan \left (\frac {\sqrt [6]{a}}{\sqrt [6]{b} \sqrt [3]{x}}-\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-2 \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}}\right )\right )}{2 \sqrt [6]{a} b^{5/6} x^{2/3}} \] Input:
Integrate[(c*x)^(2/3)/(a + b*x^2),x]
Output:
-1/2*((c*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))]))/(a^(1/6)*b^(5/6)*x^( 2/3))
Time = 0.52 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.42, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {266, 27, 824, 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.
| \(\displaystyle \int \frac {(c x)^{2/3}}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \int \frac {c^2 (c x)^{4/3}}{b x^2 c^2+a c^2}d\sqrt [3]{c x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 c \int \frac {(c x)^{4/3}}{b x^2 c^2+a c^2}d\sqrt [3]{c x}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle 3 c \left (\frac {\int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 c \left (\frac {\int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 3 c \left (-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 3 c \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 c \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 c \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )\) |
Input:
Int[(c*x)^(2/3)/(a + b*x^2),x]
Output:
3*c*(ArcTan[(b^(1/6)*(c*x)^(1/3))/(a^(1/6)*c^(1/3))]/(3*a^(1/6)*b^(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^(1/6)*b^(2/3) *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^(1/6)*b ^(2/3)*c^(1/3)))
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && 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 = 0.46 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(-\frac {c \left (\frac {\sqrt {3}\, \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}-\frac {\sqrt {3}\, \ln \left (\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}-\left (c x \right )^{\frac {2}{3}}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}-2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )-2 \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )-\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}+2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )\right )}{2 \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} b}\) | \(193\) |
| derivativedivides | \(3 c \left (\frac {\arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )\) | \(228\) |
| default | \(3 c \left (\frac {\arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )\) | \(228\) |
Input:
int((c*x)^(2/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2*c*(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)/b
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (145) = 290\).
Time = 0.07 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.50 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} c^{3} + \frac {1}{2} \, {\left (\sqrt {-3} a b^{4} + a b^{4}\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} c^{3} - \frac {1}{2} \, {\left (\sqrt {-3} a b^{4} + a b^{4}\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} c^{3} + \frac {1}{2} \, {\left (\sqrt {-3} a b^{4} - a b^{4}\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} c^{3} - \frac {1}{2} \, {\left (\sqrt {-3} a b^{4} - a b^{4}\right )} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}}\right ) + \frac {1}{2} \, \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (a b^{4} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} c^{3}\right ) - \frac {1}{2} \, \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-a b^{4} \left (-\frac {c^{4}}{a b^{5}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} c^{3}\right ) \] Input:
integrate((c*x)^(2/3)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/4*(sqrt(-3) - 1)*(-c^4/(a*b^5))^(1/6)*log((c*x)^(1/3)*c^3 + 1/2*(sqrt(- 3)*a*b^4 + a*b^4)*(-c^4/(a*b^5))^(5/6)) + 1/4*(sqrt(-3) - 1)*(-c^4/(a*b^5) )^(1/6)*log((c*x)^(1/3)*c^3 - 1/2*(sqrt(-3)*a*b^4 + a*b^4)*(-c^4/(a*b^5))^ (5/6)) - 1/4*(sqrt(-3) + 1)*(-c^4/(a*b^5))^(1/6)*log((c*x)^(1/3)*c^3 + 1/2 *(sqrt(-3)*a*b^4 - a*b^4)*(-c^4/(a*b^5))^(5/6)) + 1/4*(sqrt(-3) + 1)*(-c^4 /(a*b^5))^(1/6)*log((c*x)^(1/3)*c^3 - 1/2*(sqrt(-3)*a*b^4 - a*b^4)*(-c^4/( a*b^5))^(5/6)) + 1/2*(-c^4/(a*b^5))^(1/6)*log(a*b^4*(-c^4/(a*b^5))^(5/6) + (c*x)^(1/3)*c^3) - 1/2*(-c^4/(a*b^5))^(1/6)*log(-a*b^4*(-c^4/(a*b^5))^(5/ 6) + (c*x)^(1/3)*c^3)
Result contains complex when optimal does not.
Time = 1.22 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.66 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=\frac {5 c^{\frac {2}{3}} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} + \frac {5 i c^{\frac {2}{3}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} + \frac {5 c^{\frac {2}{3}} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {5 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} - \frac {5 c^{\frac {2}{3}} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {7 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} - \frac {5 i c^{\frac {2}{3}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {3 i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} - \frac {5 c^{\frac {2}{3}} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {11 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (\frac {5}{6}\right )}{12 \sqrt [6]{a} b^{\frac {5}{6}} \Gamma \left (\frac {11}{6}\right )} \] Input:
integrate((c*x)**(2/3)/(b*x**2+a),x)
Output:
5*c**(2/3)*exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(I*pi/6)/a**(1/6 ))*gamma(5/6)/(12*a**(1/6)*b**(5/6)*gamma(11/6)) + 5*I*c**(2/3)*log(1 - b* *(1/6)*x**(1/3)*exp_polar(I*pi/2)/a**(1/6))*gamma(5/6)/(12*a**(1/6)*b**(5/ 6)*gamma(11/6)) + 5*c**(2/3)*exp(5*I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_p olar(5*I*pi/6)/a**(1/6))*gamma(5/6)/(12*a**(1/6)*b**(5/6)*gamma(11/6)) - 5 *c**(2/3)*exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(7*I*pi/6)/a**(1/ 6))*gamma(5/6)/(12*a**(1/6)*b**(5/6)*gamma(11/6)) - 5*I*c**(2/3)*log(1 - b **(1/6)*x**(1/3)*exp_polar(3*I*pi/2)/a**(1/6))*gamma(5/6)/(12*a**(1/6)*b** (5/6)*gamma(11/6)) - 5*c**(2/3)*exp(5*I*pi/6)*log(1 - b**(1/6)*x**(1/3)*ex p_polar(11*I*pi/6)/a**(1/6))*gamma(5/6)/(12*a**(1/6)*b**(5/6)*gamma(11/6))
Time = 0.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.20 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=-\frac {1}{4} \, c {\left (\frac {\sqrt {3} \log \left (\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )} \] Input:
integrate((c*x)^(2/3)/(b*x^2+a),x, algorithm="maxima")
Output:
-1/4*c*(sqrt(3)*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)^(1/6)*b^(5/6)) - sqrt(3)*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)^(1/6)*b^(5/6)) - 2*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)))/(b^(2/3)*sqrt((a*c^2)^(1/3)*b ^(1/3))) - 2*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)))/(b^(2/3)*sqrt((a*c^2)^(1/3)*b^(1/3))) - 4* arctan((c*x)^(1/3)*b^(1/3)/sqrt((a*c^2)^(1/3)*b^(1/3)))/(b^(2/3)*sqrt((a*c ^2)^(1/3)*b^(1/3))))
Time = 0.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.16 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=\frac {\frac {4 \, \left (\frac {a c^{2}}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a} - \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \log \left (\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{a b^{5}} + \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \log \left (-\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{a b^{5}} + \frac {2 \, \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a b^{5}} + \frac {2 \, \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \arctan \left (-\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a b^{5}}}{4 \, c} \] Input:
integrate((c*x)^(2/3)/(b*x^2+a),x, algorithm="giac")
Output:
1/4*(4*(a*c^2/b)^(5/6)*arctan((c*x)^(1/3)/(a*c^2/b)^(1/6))/a - sqrt(3)*(a* b^5*c^2)^(5/6)*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^5) + sqrt(3)*(a*b^5*c^2)^(5/6)*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^5) + 2*(a*b^5*c^2)^ (5/6)*arctan((sqrt(3)*(a*c^2/b)^(1/6) + 2*(c*x)^(1/3))/(a*c^2/b)^(1/6))/(a *b^5) + 2*(a*b^5*c^2)^(5/6)*arctan(-(sqrt(3)*(a*c^2/b)^(1/6) - 2*(c*x)^(1/ 3))/(a*c^2/b)^(1/6))/(a*b^5))/c
Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.24 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=\frac {c^{2/3}\,\mathrm {atan}\left (\frac {b^{1/6}\,{\left (c\,x\right )}^{1/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}\,c^{1/3}}\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}\,b^{5/6}}-\frac {c^{2/3}\,\ln \left (972\,a^3\,b^3\,c^9-972\,{\left (-a\right )}^{17/6}\,b^{19/6}\,c^{26/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,{\left (-a\right )}^{1/6}\,b^{5/6}}-\frac {c^{2/3}\,\ln \left (972\,a^3\,b^3\,c^9-972\,{\left (-a\right )}^{17/6}\,b^{19/6}\,c^{26/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,{\left (-a\right )}^{1/6}\,b^{5/6}}+\frac {c^{2/3}\,\ln \left (972\,a^3\,b^3\,c^9+1944\,{\left (-a\right )}^{17/6}\,b^{19/6}\,c^{26/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{{\left (-a\right )}^{1/6}\,b^{5/6}}+\frac {c^{2/3}\,\ln \left (972\,a^3\,b^3\,c^9+1944\,{\left (-a\right )}^{17/6}\,b^{19/6}\,c^{26/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{{\left (-a\right )}^{1/6}\,b^{5/6}} \] Input:
int((c*x)^(2/3)/(a + b*x^2),x)
Output:
(c^(2/3)*atan((b^(1/6)*(c*x)^(1/3)*1i)/((-a)^(1/6)*c^(1/3)))*1i)/((-a)^(1/ 6)*b^(5/6)) - (c^(2/3)*log(972*a^3*b^3*c^9 - 972*(-a)^(17/6)*b^(19/6)*c^(2 6/3)*((3^(1/2)*1i)/2 - 1/2)*(c*x)^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(2*(-a)^( 1/6)*b^(5/6)) - (c^(2/3)*log(972*a^3*b^3*c^9 - 972*(-a)^(17/6)*b^(19/6)*c^ (26/3)*((3^(1/2)*1i)/2 + 1/2)*(c*x)^(1/3))*((3^(1/2)*1i)/2 + 1/2))/(2*(-a) ^(1/6)*b^(5/6)) + (c^(2/3)*log(972*a^3*b^3*c^9 + 1944*(-a)^(17/6)*b^(19/6) *c^(26/3)*((3^(1/2)*1i)/4 - 1/4)*(c*x)^(1/3))*((3^(1/2)*1i)/4 - 1/4))/((-a )^(1/6)*b^(5/6)) + (c^(2/3)*log(972*a^3*b^3*c^9 + 1944*(-a)^(17/6)*b^(19/6 )*c^(26/3)*((3^(1/2)*1i)/4 + 1/4)*(c*x)^(1/3))*((3^(1/2)*1i)/4 + 1/4))/((- a)^(1/6)*b^(5/6))
Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.62 \[ \int \frac {(c x)^{2/3}}{a+b x^2} \, dx=\frac {c^{\frac {2}{3}} \left (-2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+4 \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )+\sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )-\sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )\right )}{4 b^{\frac {5}{6}} a^{\frac {1}{6}}} \] Input:
int((c*x)^(2/3)/(b*x^2+a),x)
Output:
(c**(2/3)*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))) + 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*a**(1/3)*b)