Integrand size = 17, antiderivative size = 237 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=-\frac {3}{a c \sqrt [3]{c x}}-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{a^{7/6} c^{4/3}}+\frac {\sqrt [6]{b} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{7/6} c^{4/3}}-\frac {\sqrt [6]{b} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{2 a^{7/6} c^{4/3}}+\frac {\sqrt {3} \sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}\right )}{2 a^{7/6} c^{4/3}} \] Output:
-3/a/c/(c*x)^(1/3)-b^(1/6)*arctan(b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^(1/3))/a^( 7/6)/c^(4/3)-1/2*b^(1/6)*arctan(-3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^(1/6)/c^( 1/3))/a^(7/6)/c^(4/3)-1/2*b^(1/6)*arctan(3^(1/2)+2*b^(1/6)*(c*x)^(1/3)/a^( 1/6)/c^(1/3))/a^(7/6)/c^(4/3)+1/2*3^(1/2)*b^(1/6)*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^(7/6) /c^(4/3)
Time = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=\frac {x \left (-6 \sqrt [6]{a}+\sqrt [6]{b} \sqrt [3]{x} \arctan \left (\frac {\sqrt [6]{a}}{\sqrt [6]{b} \sqrt [3]{x}}-\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-2 \sqrt [6]{b} \sqrt [3]{x} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+\sqrt {3} \sqrt [6]{b} \sqrt [3]{x} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}}\right )\right )}{2 a^{7/6} (c x)^{4/3}} \] Input:
Integrate[1/((c*x)^(4/3)*(a + b*x^2)),x]
Output:
(x*(-6*a^(1/6) + b^(1/6)*x^(1/3)*ArcTan[a^(1/6)/(b^(1/6)*x^(1/3)) - (b^(1/ 6)*x^(1/3))/a^(1/6)] - 2*b^(1/6)*x^(1/3)*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] + Sqrt[3]*b^(1/6)*x^(1/3)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3))/(a^(1 /3) + b^(1/3)*x^(2/3))]))/(2*a^(7/6)*(c*x)^(4/3))
Time = 0.55 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.41, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {264, 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.
| \(\displaystyle \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {b \int \frac {(c x)^{2/3}}{b x^2+a}dx}{a c^2}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {3 b \int \frac {c^2 (c x)^{4/3}}{b x^2 c^2+a c^2}d\sqrt [3]{c x}}{a c^3}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b \int \frac {(c x)^{4/3}}{b x^2 c^2+a c^2}d\sqrt [3]{c x}}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle -\frac {3 b \left (\frac {\int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\int -\frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{2 \left (\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}\right )}d\sqrt [3]{c x}}{3 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b \left (\frac {\int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {3 b \left (-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\int \frac {\sqrt [6]{a} \sqrt [3]{c}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 b \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 b \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}\right )}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {1}{2} \sqrt [6]{a} \sqrt [3]{c} \int \frac {1}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {3 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\int \frac {1}{-(c x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}-2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}+2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [3]{a} c^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c x} \sqrt [3]{c}+\sqrt [3]{b} (c x)^{2/3}}d\sqrt [3]{c x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 b \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{c} \sqrt [3]{c x}+\sqrt [3]{a} c^{2/3}+\sqrt [3]{b} (c x)^{2/3}\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{c x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{c}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3} \sqrt [3]{c}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{c x}}{\sqrt [6]{a} \sqrt [3]{c}}\right )}{3 \sqrt [6]{a} b^{5/6} \sqrt [3]{c}}\right )}{a c}-\frac {3}{a c \sqrt [3]{c x}}\) |
Input:
Int[1/((c*x)^(4/3)*(a + b*x^2)),x]
Output:
-3/(a*c*(c*x)^(1/3)) - (3*b*(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) - S qrt[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))))/(a*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] :> 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[(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 = 0.56 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (c x \right )^{\frac {1}{3}}}{2}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}-\left (c x \right )^{\frac {2}{3}}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (c x \right )^{\frac {1}{3}}}{2}+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}-2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {1}{3}}-2 \arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {1}{3}}-\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}+2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) \left (c x \right )^{\frac {1}{3}}-6 \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}{2 \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}} a c}\) | \(238\) |
| risch | \(-\frac {3}{a c \left (c x \right )^{\frac {1}{3}}}+\frac {-\frac {\arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\frac {b \sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 a^{2} c^{2}}-\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 a \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\frac {b \sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}-\left (c x \right )^{\frac {2}{3}}-\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 a^{2} c^{2}}-\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 a \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}}{c}\) | \(248\) |
| derivativedivides | \(3 c \left (-\frac {1}{a \,c^{2} \left (c x \right )^{\frac {1}{3}}}-\frac {\left (\frac {\arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) b}{a \,c^{2}}\right )\) | \(251\) |
| default | \(3 c \left (-\frac {1}{a \,c^{2} \left (c x \right )^{\frac {1}{3}}}-\frac {\left (\frac {\arctan \left (\frac {\left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}-\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {5}{6}} \ln \left (\left (c x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}} \left (c x \right )^{\frac {1}{3}}+\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{3}}\right )}{12 a \,c^{2}}+\frac {\arctan \left (\frac {2 \left (c x \right )^{\frac {1}{3}}}{\left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a \,c^{2}}{b}\right )^{\frac {1}{6}}}\right ) b}{a \,c^{2}}\right )\) | \(251\) |
Input:
int(1/(c*x)^(4/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2/(a*c^2/b)^(1/6)*(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))*3^(1/2)*(c*x)^(1/3)-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))*3^(1/2)*(c*x)^(1/3)+arctan((3^(1/2)*( a*c^2/b)^(1/6)-2*(c*x)^(1/3))/(a*c^2/b)^(1/6))*(c*x)^(1/3)-2*arctan((c*x)^ (1/3)/(a*c^2/b)^(1/6))*(c*x)^(1/3)-arctan((3^(1/2)*(a*c^2/b)^(1/6)+2*(c*x) ^(1/3))/(a*c^2/b)^(1/6))*(c*x)^(1/3)-6*(a*c^2/b)^(1/6))/(c*x)^(1/3)/a/c
Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (159) = 318\).
Time = 0.08 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=-\frac {2 \, a c^{2} x \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (a^{6} c^{7} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) - 2 \, a c^{2} x \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (-a^{6} c^{7} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) - {\left (\sqrt {-3} a c^{2} x - a c^{2} x\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{6} c^{7} + a^{6} c^{7}\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) + {\left (\sqrt {-3} a c^{2} x - a c^{2} x\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{6} c^{7} + a^{6} c^{7}\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) - {\left (\sqrt {-3} a c^{2} x + a c^{2} x\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{6} c^{7} - a^{6} c^{7}\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) + {\left (\sqrt {-3} a c^{2} x + a c^{2} x\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{6} c^{7} - a^{6} c^{7}\right )} \left (-\frac {b}{a^{7} c^{8}}\right )^{\frac {5}{6}} + \left (c x\right )^{\frac {1}{3}} b\right ) + 12 \, \left (c x\right )^{\frac {2}{3}}}{4 \, a c^{2} x} \] Input:
integrate(1/(c*x)^(4/3)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/4*(2*a*c^2*x*(-b/(a^7*c^8))^(1/6)*log(a^6*c^7*(-b/(a^7*c^8))^(5/6) + (c *x)^(1/3)*b) - 2*a*c^2*x*(-b/(a^7*c^8))^(1/6)*log(-a^6*c^7*(-b/(a^7*c^8))^ (5/6) + (c*x)^(1/3)*b) - (sqrt(-3)*a*c^2*x - a*c^2*x)*(-b/(a^7*c^8))^(1/6) *log(1/2*(sqrt(-3)*a^6*c^7 + a^6*c^7)*(-b/(a^7*c^8))^(5/6) + (c*x)^(1/3)*b ) + (sqrt(-3)*a*c^2*x - a*c^2*x)*(-b/(a^7*c^8))^(1/6)*log(-1/2*(sqrt(-3)*a ^6*c^7 + a^6*c^7)*(-b/(a^7*c^8))^(5/6) + (c*x)^(1/3)*b) - (sqrt(-3)*a*c^2* x + a*c^2*x)*(-b/(a^7*c^8))^(1/6)*log(1/2*(sqrt(-3)*a^6*c^7 - a^6*c^7)*(-b /(a^7*c^8))^(5/6) + (c*x)^(1/3)*b) + (sqrt(-3)*a*c^2*x + a*c^2*x)*(-b/(a^7 *c^8))^(1/6)*log(-1/2*(sqrt(-3)*a^6*c^7 - a^6*c^7)*(-b/(a^7*c^8))^(5/6) + (c*x)^(1/3)*b) + 12*(c*x)^(2/3))/(a*c^2*x)
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=\frac {\Gamma \left (- \frac {1}{6}\right )}{2 a c^{\frac {4}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} + \frac {\sqrt [6]{b} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} + \frac {i \sqrt [6]{b} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} + \frac {\sqrt [6]{b} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {5 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} - \frac {\sqrt [6]{b} e^{\frac {i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {7 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} - \frac {i \sqrt [6]{b} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {3 i \pi }{2}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} - \frac {\sqrt [6]{b} e^{\frac {5 i \pi }{6}} \log {\left (1 - \frac {\sqrt [6]{b} \sqrt [3]{x} e^{\frac {11 i \pi }{6}}}{\sqrt [6]{a}} \right )} \Gamma \left (- \frac {1}{6}\right )}{12 a^{\frac {7}{6}} c^{\frac {4}{3}} \Gamma \left (\frac {5}{6}\right )} \] Input:
integrate(1/(c*x)**(4/3)/(b*x**2+a),x)
Output:
gamma(-1/6)/(2*a*c**(4/3)*x**(1/3)*gamma(5/6)) + b**(1/6)*exp(I*pi/6)*log( 1 - b**(1/6)*x**(1/3)*exp_polar(I*pi/6)/a**(1/6))*gamma(-1/6)/(12*a**(7/6) *c**(4/3)*gamma(5/6)) + I*b**(1/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(I*p i/2)/a**(1/6))*gamma(-1/6)/(12*a**(7/6)*c**(4/3)*gamma(5/6)) + b**(1/6)*ex p(5*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**(7/6)*c**(4/3)*gamma(5/6)) - b**(1/6)*exp(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**(7/6)*c**( 4/3)*gamma(5/6)) - I*b**(1/6)*log(1 - b**(1/6)*x**(1/3)*exp_polar(3*I*pi/2 )/a**(1/6))*gamma(-1/6)/(12*a**(7/6)*c**(4/3)*gamma(5/6)) - b**(1/6)*exp(5 *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**(7/6)*c**(4/3)*gamma(5/6))
Time = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=\frac {\frac {b {\left (\frac {\sqrt {3} \log \left (\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{6}} + \left (c x\right )^{\frac {2}{3}} b^{\frac {1}{3}} + \left (a c^{2}\right )^{\frac {1}{3}}\right )}{\left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} \left (a c^{2}\right )^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {\left (a c^{2}\right )^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{a} - \frac {12}{\left (c x\right )^{\frac {1}{3}} a}}{4 \, c} \] Input:
integrate(1/(c*x)^(4/3)/(b*x^2+a),x, algorithm="maxima")
Output:
1/4*(b*(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))))/a - 12/((c*x)^(1/3)*a))/c
Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=-\frac {\frac {12}{\left (c x\right )^{\frac {1}{3}} a} - \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \log \left (\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{a^{2} b^{4} c^{2}} + \frac {\sqrt {3} \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \log \left (-\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} \left (c x\right )^{\frac {1}{3}} + \left (c x\right )^{\frac {2}{3}} + \left (\frac {a c^{2}}{b}\right )^{\frac {1}{3}}\right )}{a^{2} b^{4} c^{2}} + \frac {2 \, \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} b^{4} c^{2}} + \frac {2 \, \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \arctan \left (-\frac {\sqrt {3} \left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} b^{4} c^{2}} + \frac {4 \, \left (a b^{5} c^{2}\right )^{\frac {5}{6}} \arctan \left (\frac {\left (c x\right )^{\frac {1}{3}}}{\left (\frac {a c^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a^{2} b^{4} c^{2}}}{4 \, c} \] Input:
integrate(1/(c*x)^(4/3)/(b*x^2+a),x, algorithm="giac")
Output:
-1/4*(12/((c*x)^(1/3)*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^2*b^4*c^2) + 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^2*b^4*c^2) + 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^2*b^4*c^2) + 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^2*b^4*c^2) + 4*(a*b^5*c^2)^(5/6)*arctan((c*x)^(1/3)/(a*c^2/b)^(1/6)) /(a^2*b^4*c^2))/c
Time = 0.31 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=-\frac {3}{a\,c\,{\left (c\,x\right )}^{1/3}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,{\left (c\,x\right )}^{1/3}\,1{}\mathrm {i}}{a^{1/6}\,c^{1/3}}\right )\,1{}\mathrm {i}}{a^{7/6}\,c^{4/3}}-\frac {{\left (-b\right )}^{1/6}\,\ln \left (972\,a^9\,b^6\,c^{12}-972\,a^{53/6}\,{\left (-b\right )}^{37/6}\,c^{35/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,a^{7/6}\,c^{4/3}}-\frac {{\left (-b\right )}^{1/6}\,\ln \left (972\,a^9\,b^6\,c^{12}-972\,a^{53/6}\,{\left (-b\right )}^{37/6}\,c^{35/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,a^{7/6}\,c^{4/3}}+\frac {{\left (-b\right )}^{1/6}\,\ln \left (972\,a^9\,b^6\,c^{12}+1944\,a^{53/6}\,{\left (-b\right )}^{37/6}\,c^{35/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{a^{7/6}\,c^{4/3}}+\frac {{\left (-b\right )}^{1/6}\,\ln \left (972\,a^9\,b^6\,c^{12}+1944\,a^{53/6}\,{\left (-b\right )}^{37/6}\,c^{35/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,x\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{a^{7/6}\,c^{4/3}} \] Input:
int(1/((c*x)^(4/3)*(a + b*x^2)),x)
Output:
((-b)^(1/6)*log(972*a^9*b^6*c^12 + 1944*a^(53/6)*(-b)^(37/6)*c^(35/3)*((3^ (1/2)*1i)/4 - 1/4)*(c*x)^(1/3))*((3^(1/2)*1i)/4 - 1/4))/(a^(7/6)*c^(4/3)) - ((-b)^(1/6)*atan(((-b)^(1/6)*(c*x)^(1/3)*1i)/(a^(1/6)*c^(1/3)))*1i)/(a^( 7/6)*c^(4/3)) - ((-b)^(1/6)*log(972*a^9*b^6*c^12 - 972*a^(53/6)*(-b)^(37/6 )*c^(35/3)*((3^(1/2)*1i)/2 - 1/2)*(c*x)^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(2* a^(7/6)*c^(4/3)) - ((-b)^(1/6)*log(972*a^9*b^6*c^12 - 972*a^(53/6)*(-b)^(3 7/6)*c^(35/3)*((3^(1/2)*1i)/2 + 1/2)*(c*x)^(1/3))*((3^(1/2)*1i)/2 + 1/2))/ (2*a^(7/6)*c^(4/3)) - 3/(a*c*(c*x)^(1/3)) + ((-b)^(1/6)*log(972*a^9*b^6*c^ 12 + 1944*a^(53/6)*(-b)^(37/6)*c^(35/3)*((3^(1/2)*1i)/4 + 1/4)*(c*x)^(1/3) )*((3^(1/2)*1i)/4 + 1/4))/(a^(7/6)*c^(4/3))
Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(c x)^{4/3} \left (a+b x^2\right )} \, dx=\frac {2 x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-2 x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-4 x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right )-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )+x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right )-12 a^{\frac {1}{3}}}{4 x^{\frac {1}{3}} c^{\frac {4}{3}} a^{\frac {4}{3}}} \] Input:
int(1/(c*x)^(4/3)/(b*x^2+a),x)
Output:
(2*x**(1/3)*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))) - 2*x**(1/3)*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))) - 4*x**(1 /3)*b**(1/6)*a**(1/6)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6))) - x**( 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)) + x**(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)) - 12*a**( 1/3))/(4*x**(1/3)*c**(1/3)*a**(1/3)*a*c)