Integrand size = 17, antiderivative size = 239 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {3}{5 a c (c x)^{5/3}}-\frac {b^{5/6} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{a^{11/6} c^{8/3}}+\frac {b^{5/6} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{11/6} c^{8/3}}-\frac {b^{5/6} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{11/6} c^{8/3}}-\frac {\sqrt {3} b^{5/6} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}\right )}{2 a^{11/6} c^{8/3}} \] Output:
-3/5/a/c/(c*x)^(5/3)-b^(5/6)*arctan(b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a ^(11/6)/c^(8/3)-1/2*b^(5/6)*arctan(-3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/ c^(1/3))/a^(11/6)/c^(8/3)-1/2*b^(5/6)*arctan(3^(1/2)+2*b^(1/6)*(c*x)^(1/3) /a^(1/6)/c^(1/3))/a^(11/6)/c^(8/3)-1/2*3^(1/2)*b^(5/6)*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^ (11/6)/c^(8/3)
Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {x \left (-6 a^{5/6}+5 b^{5/6} x^{5/3} \arctan \left (\frac {\sqrt [6]{a}}{\sqrt [6]{b} \sqrt [3]{x}}-\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-10 b^{5/6} x^{5/3} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-5 \sqrt {3} b^{5/6} x^{5/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}}\right )\right )}{10 a^{11/6} (c x)^{8/3}} \] Input:
Integrate[1/((c*x)^(8/3)*(a + b*x^2)),x]
Output:
(x*(-6*a^(5/6) + 5*b^(5/6)*x^(5/3)*ArcTan[a^(1/6)/(b^(1/6)*x^(1/3)) - (b^( 1/6)*x^(1/3))/a^(1/6)] - 10*b^(5/6)*x^(5/3)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/ 6)] - 5*Sqrt[3]*b^(5/6)*x^(5/3)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3))/ (a^(1/3) + b^(1/3)*x^(2/3))]))/(10*a^(11/6)*(c*x)^(8/3))
Time = 0.54 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.37, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {264, 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.
| \(\displaystyle \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {b \int \frac {1}{(c x)^{2/3} \left (b x^2+a\right )}dx}{a c^2}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {3 b \int \frac {1}{b x^2+a}d\sqrt [3]{c x}}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle -\frac {3 b \left (\frac {c^{2/3} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 a^{2/3}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 a^{5/6}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b \left (\frac {c^{2/3} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 a^{2/3}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \int \frac {2 \sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 b \left (\frac {\sqrt [3]{c} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{a c^3}-\frac {3}{5 a c (c x)^{5/3}}\) |
Input:
Int[1/((c*x)^(8/3)*(a + b*x^2)),x]
Output:
-3/(5*a*c*(c*x)^(5/3)) - (3*b*((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))))/(a*c^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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.64 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(3 c \left (-\frac {1}{5 a \,c^{2} \left (c x \right )^{\frac {5}{3}}}-\frac {\left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a \,c^{2}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a \,c^{2}}\right ) b}{a \,c^{2}}\right )\) | \(260\) |
| default | \(3 c \left (-\frac {1}{5 a \,c^{2} \left (c x \right )^{\frac {5}{3}}}-\frac {\left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 a \,c^{2}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a \,c^{2}}\right ) b}{a \,c^{2}}\right )\) | \(260\) |
| pseudoelliptic | \(\frac {-\frac {b \sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (c x \right )^{\frac {5}{3}}}{2}+\frac {b \sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}-\left (c x \right )^{\frac {2}{3}}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (c x \right )^{\frac {5}{3}}}{2}+b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}-2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {5}{3}}-2 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {5}{3}}-b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}+2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {5}{3}}-\frac {6 a \,c^{2}}{5}}{2 \left (c x \right )^{\frac {5}{3}} a^{2} c^{3}}\) | \(277\) |
Input:
int(1/(c*x)^(8/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
3*c*(-1/5/a/c^2/(c*x)^(5/3)-(1/12/a/c^2*3^(1/2)*(a*c^2/b)^(1/6)*ln((c*x)^( 2/3)+3^(1/2)*(a*c^2/b)^(1/6)*(c*x)^(1/3)+(a*c^2/b)^(1/3))+1/6/a/c^2*(a*c^2 /b)^(1/6)*arctan(2*(c*x)^(1/3)/(a*c^2/b)^(1/6)+3^(1/2))+1/3/a/c^2*(a*c^2/b )^(1/6)*arctan((c*x)^(1/3)/(a*c^2/b)^(1/6))-1/12/a/c^2*3^(1/2)*(a*c^2/b)^( 1/6)*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 /6/a/c^2*(a*c^2/b)^(1/6)*arctan(2*(c*x)^(1/3)/(a*c^2/b)^(1/6)-3^(1/2)))*b/ a/c^2)
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (159) = 318\).
Time = 0.08 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {10 \, a c^{3} x^{2} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (a^{2} c^{3} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) - 10 \, a c^{3} x^{2} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (-a^{2} c^{3} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) + 5 \, {\left (\sqrt {-3} a c^{3} x^{2} + a c^{3} x^{2}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} b + \frac {1}{2} \, {\left (\sqrt {-3} a^{2} c^{3} + a^{2} c^{3}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}}\right ) - 5 \, {\left (\sqrt {-3} a c^{3} x^{2} + a c^{3} x^{2}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} b - \frac {1}{2} \, {\left (\sqrt {-3} a^{2} c^{3} + a^{2} c^{3}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}}\right ) + 5 \, {\left (\sqrt {-3} a c^{3} x^{2} - a c^{3} x^{2}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} b + \frac {1}{2} \, {\left (\sqrt {-3} a^{2} c^{3} - a^{2} c^{3}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}}\right ) - 5 \, {\left (\sqrt {-3} a c^{3} x^{2} - a c^{3} x^{2}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}} \log \left (\left (c x\right )^{\frac {1}{3}} b - \frac {1}{2} \, {\left (\sqrt {-3} a^{2} c^{3} - a^{2} c^{3}\right )} \left (-\frac {b^{5}}{a^{11} c^{16}}\right )^{\frac {1}{6}}\right ) + 12 \, \left (c x\right )^{\frac {1}{3}}}{20 \, a c^{3} x^{2}} \] Input:
integrate(1/(c*x)^(8/3)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/20*(10*a*c^3*x^2*(-b^5/(a^11*c^16))^(1/6)*log(a^2*c^3*(-b^5/(a^11*c^16) )^(1/6) + (c*x)^(1/3)*b) - 10*a*c^3*x^2*(-b^5/(a^11*c^16))^(1/6)*log(-a^2* c^3*(-b^5/(a^11*c^16))^(1/6) + (c*x)^(1/3)*b) + 5*(sqrt(-3)*a*c^3*x^2 + a* c^3*x^2)*(-b^5/(a^11*c^16))^(1/6)*log((c*x)^(1/3)*b + 1/2*(sqrt(-3)*a^2*c^ 3 + a^2*c^3)*(-b^5/(a^11*c^16))^(1/6)) - 5*(sqrt(-3)*a*c^3*x^2 + a*c^3*x^2 )*(-b^5/(a^11*c^16))^(1/6)*log((c*x)^(1/3)*b - 1/2*(sqrt(-3)*a^2*c^3 + a^2 *c^3)*(-b^5/(a^11*c^16))^(1/6)) + 5*(sqrt(-3)*a*c^3*x^2 - a*c^3*x^2)*(-b^5 /(a^11*c^16))^(1/6)*log((c*x)^(1/3)*b + 1/2*(sqrt(-3)*a^2*c^3 - a^2*c^3)*( -b^5/(a^11*c^16))^(1/6)) - 5*(sqrt(-3)*a*c^3*x^2 - a*c^3*x^2)*(-b^5/(a^11* c^16))^(1/6)*log((c*x)^(1/3)*b - 1/2*(sqrt(-3)*a^2*c^3 - a^2*c^3)*(-b^5/(a ^11*c^16))^(1/6)) + 12*(c*x)^(1/3))/(a*c^3*x^2)
Result contains complex when optimal does not.
Time = 1.77 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {\Gamma \left (- \frac {5}{6}\right )}{2 a c^{\frac {8}{3}} x^{\frac {5}{3}} \Gamma \left (\frac {1}{6}\right )} + \frac {5 b^{\frac {5}{6}} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} + \frac {5 i b^{\frac {5}{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} + \frac {5 b^{\frac {5}{6}} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {5 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} - \frac {5 b^{\frac {5}{6}} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {7 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} - \frac {5 i b^{\frac {5}{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {3 i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} - \frac {5 b^{\frac {5}{6}} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {11 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {5}{6}\right )}{12 a^{\frac {11}{6}} c^{\frac {8}{3}} \Gamma \left (\frac {1}{6}\right )} \] Input:
integrate(1/(c*x)**(8/3)/(b*x**2+a),x)
Output:
gamma(-5/6)/(2*a*c**(8/3)*x**(5/3)*gamma(1/6)) + 5*b**(5/6)*exp(5*I*pi/6)* log(1 - b**(1/6)*x**(1/3)*exp_polar(I*pi/6)/a**(1/6))*gamma(-5/6)/(12*a**( 11/6)*c**(8/3)*gamma(1/6)) + 5*I*b**(5/6)*log(1 - b**(1/6)*x**(1/3)*exp_po lar(I*pi/2)/a**(1/6))*gamma(-5/6)/(12*a**(11/6)*c**(8/3)*gamma(1/6)) + 5*b **(5/6)*exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(5*I*pi/6)/a**(1/6) )*gamma(-5/6)/(12*a**(11/6)*c**(8/3)*gamma(1/6)) - 5*b**(5/6)*exp(5*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**(11/6)*c**(8/3)*gamma(1/6)) - 5*I*b**(5/6)*log(1 - b**(1/6)*x**(1/3)*ex p_polar(3*I*pi/2)/a**(1/6))*gamma(-5/6)/(12*a**(11/6)*c**(8/3)*gamma(1/6)) - 5*b**(5/6)*exp(I*pi/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(11*I*pi/6)/a **(1/6))*gamma(-5/6)/(12*a**(11/6)*c**(8/3)*gamma(1/6))
Time = 0.12 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {\frac {5 \, {\left (\frac {\sqrt {3} b^{\frac {5}{6}} \log \left (\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {5}{6}}} - \frac {\sqrt {3} b^{\frac {5}{6}} \log \left (-\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {5}{6}}} + \frac {4 \, b \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\left (a c^{2}\right )^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \left (a c^{2}\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a c^{2} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \left (a c^{2}\right )^{\frac {1}{3}} b \arctan \left (-\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a c^{2} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{a} + \frac {12}{\left (c x\right )^{\frac {5}{3}} a}}{20 \, c} \] Input:
integrate(1/(c*x)^(8/3)/(b*x^2+a),x, algorithm="maxima")
Output:
-1/20*(5*(sqrt(3)*b^(5/6)*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) - sqrt(3)*b^(5/6)*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) + 4*b*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)*b*a rctan((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*c^2*sqrt((a*c^2)^(1/3)*b^(1/3))) + 2*(a*c^2)^(1/3)*b*a rctan(-(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*c^2*sqrt((a*c^2)^(1/3)*b^(1/3))))/a + 12/((c*x)^(5/3) *a))/c
Time = 0.13 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {1}{6}} \log \left (\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} c^{3}} + \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a^{2} c^{3}} - \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} c^{3}} - \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a^{2} c^{3}} - \frac {\left (a b^{5} c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} c^{3}} - \frac {3}{5 \, \left (c x\right )^{\frac {2}{3}} a c^{2} x} \] Input:
integrate(1/(c*x)^(8/3)/(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^2*c^3) + 1/4*sqrt(3)*(a*b^5*c^2)^(1/6)*lo g(-sqrt(3)*(a*c^2/b)^(1/6)*(c*x)^(1/3) + (c*x)^(2/3) + (a*c^2/b)^(1/3))/(a ^2*c^3) - 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^2*c^3) - 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^2*c^3) - (a*b^5*c^2) ^(1/6)*arctan((c*x)^(1/3)/(a*c^2/b)^(1/6))/(a^2*c^3) - 3/5/((c*x)^(2/3)*a* c^2*x)
Time = 0.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=-\frac {3}{5\,a\,c\,{\left (c\,x\right )}^{5/3}}-\frac {{\left (-b\right )}^{5/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,{\left (c\,x\right )}^{1/3}\,1{}\mathrm {i}}{a^{1/6}\,c^{1/3}}\right )\,1{}\mathrm {i}}{a^{11/6}\,c^{8/3}}-\frac {{\left (-b\right )}^{5/6}\,\ln \left (486\,a^{31/6}\,{\left (-b\right )}^{53/6}\,c^{16/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+486\,a^5\,b^9\,c^5\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,a^{11/6}\,c^{8/3}}-\frac {{\left (-b\right )}^{5/6}\,\ln \left (486\,a^{31/6}\,{\left (-b\right )}^{53/6}\,c^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+486\,a^5\,b^9\,c^5\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,a^{11/6}\,c^{8/3}}+\frac {{\left (-b\right )}^{5/6}\,\ln \left (972\,a^{31/6}\,{\left (-b\right )}^{53/6}\,c^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-486\,a^5\,b^9\,c^5\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{a^{11/6}\,c^{8/3}}+\frac {{\left (-b\right )}^{5/6}\,\ln \left (972\,a^{31/6}\,{\left (-b\right )}^{53/6}\,c^{16/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-486\,a^5\,b^9\,c^5\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{a^{11/6}\,c^{8/3}} \] Input:
int(1/((c*x)^(8/3)*(a + b*x^2)),x)
Output:
((-b)^(5/6)*log(972*a^(31/6)*(-b)^(53/6)*c^(16/3)*((3^(1/2)*1i)/4 - 1/4) - 486*a^5*b^9*c^5*(c*x)^(1/3))*((3^(1/2)*1i)/4 - 1/4))/(a^(11/6)*c^(8/3)) - ((-b)^(5/6)*atan(((-b)^(1/6)*(c*x)^(1/3)*1i)/(a^(1/6)*c^(1/3)))*1i)/(a^(1 1/6)*c^(8/3)) - ((-b)^(5/6)*log(486*a^(31/6)*(-b)^(53/6)*c^(16/3)*((3^(1/2 )*1i)/2 - 1/2) + 486*a^5*b^9*c^5*(c*x)^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(2*a ^(11/6)*c^(8/3)) - ((-b)^(5/6)*log(486*a^(31/6)*(-b)^(53/6)*c^(16/3)*((3^( 1/2)*1i)/2 + 1/2) + 486*a^5*b^9*c^5*(c*x)^(1/3))*((3^(1/2)*1i)/2 + 1/2))/( 2*a^(11/6)*c^(8/3)) - 3/(5*a*c*(c*x)^(5/3)) + ((-b)^(5/6)*log(972*a^(31/6) *(-b)^(53/6)*c^(16/3)*((3^(1/2)*1i)/4 + 1/4) - 486*a^5*b^9*c^5*(c*x)^(1/3) )*((3^(1/2)*1i)/4 + 1/4))/(a^(11/6)*c^(8/3))
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )} \, dx=\frac {10 x^{\frac {5}{3}} b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-10 x^{\frac {5}{3}} b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-20 x^{\frac {5}{3}} b^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )+5 x^{\frac {5}{3}} b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )-5 x^{\frac {5}{3}} b^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )-12 b^{\frac {1}{3}} a}{20 c^{\frac {8}{3}} x^{\frac {5}{3}} b^{\frac {1}{3}} a^{2}} \] Input:
int(1/(c*x)^(8/3)/(b*x^2+a),x)
Output:
(c**(1/3)*(10*x**(2/3)*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*x - 10*x**(2/3)*b**(1/6)*a**( 1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**( 1/6)))*b*x - 20*x**(2/3)*b**(1/6)*a**(1/6)*atan((x**(1/3)*b**(1/3))/(b**(1 /6)*a**(1/6)))*b*x + 5*x**(2/3)*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))*b*x - 5*x**(2/3) *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))*b*x - 12*b**(1/3)*a))/(20*x**(2/3)*b**(1/3)*a**2*c **3*x)