Integrand size = 22, antiderivative size = 417 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {c \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{5/6} \sqrt [6]{b} e^{2/3}}-\frac {\left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{2/3} e^{2/3}}+\frac {\left (\sqrt {b} c-\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{2/3} e^{2/3}}-\frac {d \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [3]{a} b^{2/3} e^{2/3}}-\frac {\left (\sqrt {3} \sqrt {b} c-\sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} b^{2/3} e^{2/3}}+\frac {\left (\sqrt {3} \sqrt {b} c+\sqrt {a} d\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} b^{2/3} e^{2/3}} \] Output:
c*arctan(b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(1/6)/e^(2/3)+1/2* (b^(1/2)*c+3^(1/2)*a^(1/2)*d)*arctan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6 )/e^(1/3))/a^(5/6)/b^(2/3)/e^(2/3)+1/2*(b^(1/2)*c-3^(1/2)*a^(1/2)*d)*arcta n(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(2/3)/e^(2/3)-1 /2*d*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/a^(1/3)/b^(2/3)/e^(2/3)-1/4*( 3^(1/2)*b^(1/2)*c-a^(1/2)*d)*ln(a^(1/3)*e^(2/3)-3^(1/2)*a^(1/6)*b^(1/6)*e^ (1/3)*(e*x)^(1/3)+b^(1/3)*(e*x)^(2/3))/a^(5/6)/b^(2/3)/e^(2/3)+1/4*(3^(1/2 )*b^(1/2)*c+a^(1/2)*d)*ln(a^(1/3)*e^(2/3)+3^(1/2)*a^(1/6)*b^(1/6)*e^(1/3)* (e*x)^(1/3)+b^(1/3)*(e*x)^(2/3))/a^(5/6)/b^(2/3)/e^(2/3)
Time = 0.33 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {x^{2/3} \left (-2 \left (\sqrt {b} c+\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \left (\sqrt {b} c-\sqrt {3} \sqrt {a} d\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+4 \sqrt {b} c \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )-2 \sqrt {a} d \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )-\sqrt {3} \sqrt {b} c \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {a} d \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {3} \sqrt {b} c \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\sqrt {a} d \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{4 a^{5/6} b^{2/3} (e x)^{2/3}} \] Input:
Integrate[(c + d*x)/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(x^(2/3)*(-2*(Sqrt[b]*c + Sqrt[3]*Sqrt[a]*d)*ArcTan[Sqrt[3] - (2*b^(1/6)*x ^(1/3))/a^(1/6)] + 2*(Sqrt[b]*c - Sqrt[3]*Sqrt[a]*d)*ArcTan[Sqrt[3] + (2*b ^(1/6)*x^(1/3))/a^(1/6)] + 4*Sqrt[b]*c*ArcTan[(b^(1/6)*x^(1/3))/a^(1/6)] - 2*Sqrt[a]*d*Log[a^(1/3) + b^(1/3)*x^(2/3)] - Sqrt[3]*Sqrt[b]*c*Log[a^(1/3 ) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[a]*d*Log[a^( 1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqrt[3]*Sqrt[b ]*c*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)] + Sqr t[a]*d*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3)*x^(2/3)]))/ (4*a^(5/6)*b^(2/3)*(e*x)^(2/3))
Time = 1.38 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {557, 266, 27, 753, 27, 218, 807, 821, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle c \int \frac {1}{(e x)^{2/3} \left (b x^2+a\right )}dx+\frac {d \int \frac {\sqrt [3]{e x}}{b x^2+a}dx}{e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 c \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+\frac {3 d \int \frac {e^3 x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c \int \frac {1}{b x^2+a}d\sqrt [3]{e x}}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {3 c \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{2 \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}\right )}d\sqrt [3]{e x}}{3 a^{5/6}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c \left (\frac {e^{2/3} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{3 a^{2/3}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+3 d \int \frac {e x}{b x^2 e^2+a e^2}d\sqrt [3]{e x}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \int \frac {(e x)^{2/3}}{a e^2+b x e}d(e x)^{2/3}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}-\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \int \frac {2 \sqrt [6]{a} \sqrt [3]{e}+\sqrt {3} \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\int \frac {\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}\right )}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt [6]{a} \sqrt [3]{e} \int \frac {1}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3}{2} \sqrt [3]{a} e^{2/3} \int \frac {1}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\int \frac {1}{-(e x)^{2/3}-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )}{\sqrt {3} \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}+\frac {3}{2} d \left (\frac {\frac {3 \int \frac {1}{\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}-4}d\left (1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} d \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a} e^{2/3}-2 \sqrt [3]{b} (e x)^{2/3}}{a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}d(e x)^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 c \left (\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}-2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}+2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e x} \sqrt [3]{e}+\sqrt [3]{b} (e x)^{2/3}}d\sqrt [3]{e x}+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} d \left (\frac {\frac {\log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b} e^{2/3}}-\frac {\log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{3 \sqrt [3]{a} b^{2/3} e^{2/3}}\right )+\frac {3 c \left (\frac {\sqrt [3]{e} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{e}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [6]{b}}\right )}{6 a^{5/6}}+\frac {\sqrt [3]{e} \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{e}\) |
Input:
Int[(c + d*x)/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(3*c*((e^(1/3)*ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(3*a^(5/6) *b^(1/6)) + (e^(1/3)*(-(ArcTan[Sqrt[3]*(1 - (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[ 3]*a^(1/6)*e^(1/3)))]/b^(1/6)) - (Sqrt[3]*Log[a^(1/3)*e^(2/3) - Sqrt[3]*a^ (1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6))))/(6 *a^(5/6)) + (e^(1/3)*(ArcTan[Sqrt[3]*(1 + (2*b^(1/6)*(e*x)^(1/3))/(Sqrt[3] *a^(1/6)*e^(1/3)))]/b^(1/6) + (Sqrt[3]*Log[a^(1/3)*e^(2/3) + Sqrt[3]*a^(1/ 6)*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(2*b^(1/6))))/(6*a^ (5/6))))/e + (3*d*(-1/3*Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)]/(a^(1/3 )*b^(2/3)*e^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3))/(a^(1 /3)*e^(2/3)))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3)*e^(4/3) + b^(2/3)*(e*x)^(2/ 3) - a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3)* e^(2/3))))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((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 = 0.75 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) d \sqrt {\frac {a \,e^{2}}{b}}+2 \sqrt {\frac {a \,e^{2}}{b}}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) \sqrt {3}\, d +\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, c e -\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, c e +2 \sqrt {\frac {a \,e^{2}}{b}}\, d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )-\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {\frac {a \,e^{2}}{b}}\, d -\sqrt {\frac {a \,e^{2}}{b}}\, \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) d -4 c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) e +2 \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) c e -2 \arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right ) c e \right )}{4 a \,e^{2}}\) | \(438\) |
| derivativedivides | \(-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a e}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c}{4 a e}+\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) c}{2 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{2 a \,e^{2}}+\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {7}{6}} \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, c}{4 a^{2} e^{3}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) d}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{2 a e}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{2 a \,e^{2}}\) | \(482\) |
| default | \(-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d \ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a e}-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} c}{4 a e}+\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} d}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) c}{2 a e}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{2 a \,e^{2}}+\frac {b \left (\frac {a \,e^{2}}{b}\right )^{\frac {7}{6}} \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, c}{4 a^{2} e^{3}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right ) d}{4 a \,e^{2}}+\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) c}{2 a e}-\frac {\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \arctan \left (\frac {2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{2 a \,e^{2}}\) | \(482\) |
Input:
int((d*x+c)/(e*x)^(2/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/4*(a*e^2/b)^(1/6)*(2*3^(1/2)*arctan((3^(1/2)*(a*e^2/b)^(1/6)-2*(e*x)^(1 /3))/(a*e^2/b)^(1/6))*d*(a*e^2/b)^(1/2)+2*(a*e^2/b)^(1/2)*arctan((3^(1/2)* (a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6))*3^(1/2)*d+ln(3^(1/2)*(a*e^ 2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))*3^(1/2)*c*e-ln((e*x)^( 2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3))*3^(1/2)*c*e+2*(a *e^2/b)^(1/2)*d*ln((e*x)^(2/3)+(a*e^2/b)^(1/3))-ln(3^(1/2)*(a*e^2/b)^(1/6) *(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b)^(1/3))*(a*e^2/b)^(1/2)*d-(a*e^2/b)^(1/2 )*ln((e*x)^(2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^(1/3))*d-4* c*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))*e+2*arctan((3^(1/2)*(a*e^2/b)^(1/6)- 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))*c*e-2*arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e *x)^(1/3))/(a*e^2/b)^(1/6))*c*e)/a/e^2
Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (277) = 554\).
Time = 0.27 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.40 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(e*x)^(2/3)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/4*(sqrt(-3) + 1)*((a^2*b^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c ^2*d^4)/(a^5*b^3*e^4)) + 3*b*c^2*d - a*d^3)/(a^2*b^2*e^2))^(1/3)*log(-(b^2 *c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4)*(e*x)^(1/3) + 1/2*(sqrt(-3)*(a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e + (a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e + (sqrt(-3)*a^4*b^ 2*d*e^3 + a^4*b^2*d*e^3)*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/( a^5*b^3*e^4)))*((a^2*b^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^ 4)/(a^5*b^3*e^4)) + 3*b*c^2*d - a*d^3)/(a^2*b^2*e^2))^(1/3)) + 1/4*(sqrt(- 3) - 1)*((a^2*b^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5 *b^3*e^4)) + 3*b*c^2*d - a*d^3)/(a^2*b^2*e^2))^(1/3)*log(-(b^2*c^5 - 2*a*b *c^3*d^2 - 3*a^2*c*d^4)*(e*x)^(1/3) - 1/2*(sqrt(-3)*(a*b^2*c^4 - 3*a^2*b*c ^2*d^2)*e - (a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e + (sqrt(-3)*a^4*b^2*d*e^3 - a^ 4*b^2*d*e^3)*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b^3*e^4) ))*((a^2*b^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b^3* e^4)) + 3*b*c^2*d - a*d^3)/(a^2*b^2*e^2))^(1/3)) - 1/4*(sqrt(-3) + 1)*(-(a ^2*b^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b^3*e^4)) - 3*b*c^2*d + a*d^3)/(a^2*b^2*e^2))^(1/3)*log(-(b^2*c^5 - 2*a*b*c^3*d^2 - 3*a^2*c*d^4)*(e*x)^(1/3) + 1/2*(sqrt(-3)*(a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e + (a*b^2*c^4 - 3*a^2*b*c^2*d^2)*e - (sqrt(-3)*a^4*b^2*d*e^3 + a^4*b^2*d*e^3 )*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b^3*e^4)))*(-(a^2*b ^2*e^2*sqrt(-(b^2*c^6 - 6*a*b*c^4*d^2 + 9*a^2*c^2*d^4)/(a^5*b^3*e^4)) -...
\[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {2}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)/(e*x)**(2/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)/((e*x)**(2/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)/(e*x)^(2/3)/(b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.12 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=-\frac {\left (a b^{5} e^{2}\right )^{\frac {1}{6}} d {\left | b \right |} {\left | e \right |} \log \left (\left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2 \, \sqrt {a b} b^{2} e^{2}} + \frac {\left (a b^{5} e^{2}\right )^{\frac {1}{6}} c \arctan \left (\frac {\left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{a b e} + \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e - \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} + 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a b^{4} e^{2}} + \frac {{\left (\left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e + \sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} - 2 \, \left (e x\right )^{\frac {1}{3}}}{\left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}}}\right )}{2 \, a b^{4} e^{2}} + \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e + \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \log \left (\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a b^{4} e^{2}} - \frac {{\left (\sqrt {3} \left (a b^{5} e^{2}\right )^{\frac {1}{6}} b^{3} c e - \left (a b^{5} e^{2}\right )^{\frac {2}{3}} d\right )} \log \left (-\sqrt {3} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{6}} \left (e x\right )^{\frac {1}{3}} + \left (e x\right )^{\frac {2}{3}} + \left (\frac {a e^{2}}{b}\right )^{\frac {1}{3}}\right )}{4 \, a b^{4} e^{2}} \] Input:
integrate((d*x+c)/(e*x)^(2/3)/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*(a*b^5*e^2)^(1/6)*d*abs(b)*abs(e)*log((e*x)^(2/3) + (a*e^2/b)^(1/3))/ (sqrt(a*b)*b^2*e^2) + (a*b^5*e^2)^(1/6)*c*arctan((e*x)^(1/3)/(a*e^2/b)^(1/ 6))/(a*b*e) + 1/2*((a*b^5*e^2)^(1/6)*b^3*c*e - sqrt(3)*(a*b^5*e^2)^(2/3)*d )*arctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a*b^4 *e^2) + 1/2*((a*b^5*e^2)^(1/6)*b^3*c*e + sqrt(3)*(a*b^5*e^2)^(2/3)*d)*arct an(-(sqrt(3)*(a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a*b^4*e^2) + 1/4*(sqrt(3)*(a*b^5*e^2)^(1/6)*b^3*c*e + (a*b^5*e^2)^(2/3)*d)*log(sqrt( 3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a*b^4*e^2 ) - 1/4*(sqrt(3)*(a*b^5*e^2)^(1/6)*b^3*c*e - (a*b^5*e^2)^(2/3)*d)*log(-sqr t(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/(a*b^4*e ^2)
Time = 8.95 (sec) , antiderivative size = 1830, normalized size of antiderivative = 4.39 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)/((e*x)^(2/3)*(a + b*x^2)),x)
Output:
log((2916*a^2*b^4*c*d^2*e^5 - 972*a*b^5*c^3*e^5 + 3888*a^3*b^5*d*e^6*(e*x) ^(1/3)*(-(a^4*b^2*d^3 + b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d ^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2))^(2/3))*(-(a^4*b^2*d^3 + b*c^3*(-a^5* b^5)^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2) )^(1/3) - (e*x)^(1/3)*(486*b^5*c^4*e^4 - 486*a^2*b^3*d^4*e^4))*(-(a^4*b^2* d^3 + b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d^2*(-a^5*b^5)^(1/2 ))/(8*a^5*b^4*e^2))^(1/3) + log((2916*a^2*b^4*c*d^2*e^5 - 972*a*b^5*c^3*e^ 5 + 3888*a^3*b^5*d*e^6*(e*x)^(1/3)*(-(a^4*b^2*d^3 - b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d + 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2))^(2/3))*( -(a^4*b^2*d^3 - b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d + 3*a*c*d^2*(-a^5 *b^5)^(1/2))/(8*a^5*b^4*e^2))^(1/3) - (e*x)^(1/3)*(486*b^5*c^4*e^4 - 486*a ^2*b^3*d^4*e^4))*(-(a^4*b^2*d^3 - b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d + 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2))^(1/3) - log((e*x)^(1/3)*(4 86*b^5*c^4*e^4 - 486*a^2*b^3*d^4*e^4) + ((3^(1/2)*1i)/2 + 1/2)*(2916*a^2*b ^4*c*d^2*e^5 - 972*a*b^5*c^3*e^5 + 3888*a^3*b^5*d*e^6*((3^(1/2)*1i)/2 + 1/ 2)^2*(e*x)^(1/3)*(-(a^4*b^2*d^3 + b*c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2))^(2/3))*(-(a^4*b^2*d^3 + b* c^3*(-a^5*b^5)^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^ 5*b^4*e^2))^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(-(a^4*b^2*d^3 + b*c^3*(-a^5*b^5 )^(1/2) - 3*a^3*b^3*c^2*d - 3*a*c*d^2*(-a^5*b^5)^(1/2))/(8*a^5*b^4*e^2)...
Time = 0.21 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.71 \[ \int \frac {c+d x}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c -2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a d +2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c -2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 x^{\frac {1}{3}} b^{\frac {1}{3}}}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a d +4 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b^{\frac {1}{6}}}{a^{\frac {1}{6}}}\right ) c -\sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c +\sqrt {b}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) c -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a d +\mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a d +\mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+a^{\frac {1}{3}}+x^{\frac {2}{3}} b^{\frac {1}{3}}\right ) a d}{4 e^{\frac {2}{3}} b^{\frac {2}{3}} a^{\frac {4}{3}}} \] Input:
int((d*x+c)/(e*x)^(2/3)/(b*x^2+a),x)
Output:
( - 2*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3 ))/(b**(1/6)*a**(1/6)))*c - 2*sqrt(3)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2* x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*d + 2*sqrt(b)*sqrt(a)*atan((b**( 1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*c - 2*sq rt(3)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a** (1/6)))*a*d + 4*sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6 )))*c - sqrt(b)*sqrt(a)*sqrt(3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*c + sqrt(b)*sqrt(a)*sqrt(3)*log(x**(1/3)*b **(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*c - 2*log(a**(1/3 ) + x**(2/3)*b**(1/3))*a*d + log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a **(1/3) + x**(2/3)*b**(1/3))*a*d + log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*a*d)/(4*e**(2/3)*b**(2/3)*a**(1/3)*a)