Integrand size = 20, antiderivative size = 327 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=-\frac {d}{5 a x^5}-\frac {e}{2 a x^2}-\frac {c^{5/6} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{11/6}}+\frac {\sqrt [3]{c} \left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{11/6}}-\frac {\sqrt [3]{c} \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{11/6}}+\frac {\sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 a^{4/3}}+\frac {\sqrt [3]{c} \left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{11/6}}-\frac {\sqrt [3]{c} \left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{11/6}} \] Output:
-1/5*d/a/x^5-1/2*e/a/x^2-1/3*c^(5/6)*d*arctan(c^(1/6)*x/a^(1/6))/a^(11/6)- 1/6*c^(1/3)*(c^(1/2)*d+3^(1/2)*a^(1/2)*e)*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1 /6))/a^(11/6)-1/6*c^(1/3)*(c^(1/2)*d-3^(1/2)*a^(1/2)*e)*arctan(3^(1/2)+2*c ^(1/6)*x/a^(1/6))/a^(11/6)+1/6*c^(1/3)*e*ln(a^(1/3)+c^(1/3)*x^2)/a^(4/3)+1 /12*c^(1/3)*(3^(1/2)*c^(1/2)*d-a^(1/2)*e)*ln(a^(1/3)-3^(1/2)*a^(1/6)*c^(1/ 6)*x+c^(1/3)*x^2)/a^(11/6)-1/12*c^(1/3)*(3^(1/2)*c^(1/2)*d+a^(1/2)*e)*ln(a ^(1/3)+3^(1/2)*a^(1/6)*c^(1/6)*x+c^(1/3)*x^2)/a^(11/6)
Time = 0.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=\frac {-\frac {12 a d}{x^5}-\frac {30 a e}{x^2}-20 \sqrt [6]{a} c^{5/6} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )+10 \sqrt [6]{a} \sqrt [3]{c} \left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+10 \sqrt [3]{c} \left (-\sqrt [6]{a} \sqrt {c} d+\sqrt {3} a^{2/3} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+10 a^{2/3} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )+5 \sqrt [3]{c} \left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-5 \sqrt [3]{c} \left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d+a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{60 a^2} \] Input:
Integrate[(d + e*x^3)/(x^6*(a + c*x^6)),x]
Output:
((-12*a*d)/x^5 - (30*a*e)/x^2 - 20*a^(1/6)*c^(5/6)*d*ArcTan[(c^(1/6)*x)/a^ (1/6)] + 10*a^(1/6)*c^(1/3)*(Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)] + 10*c^(1/3)*(-(a^(1/6)*Sqrt[c]*d) + Sqrt[3]*a^( 2/3)*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)] + 10*a^(2/3)*c^(1/3)*e*Log [a^(1/3) + c^(1/3)*x^2] + 5*c^(1/3)*(Sqrt[3]*a^(1/6)*Sqrt[c]*d - a^(2/3)*e )*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] - 5*c^(1/3)*(Sqrt [3]*a^(1/6)*Sqrt[c]*d + a^(2/3)*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(60*a^2)
Time = 0.66 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {1829, 27, 1829, 27, 1746, 27, 452, 218, 240, 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.
| \(\displaystyle \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx\) |
| \(\Big \downarrow \) 1829 |
| \(\displaystyle -\frac {\int -\frac {5 \left (a e-c d x^3\right )}{x^3 \left (c x^6+a\right )}dx}{5 a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a e-c d x^3}{x^3 \left (c x^6+a\right )}dx}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 1829 |
| \(\displaystyle \frac {-\frac {\int \frac {2 a c \left (e x^3+d\right )}{c x^6+a}dx}{2 a}-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \int \frac {e x^3+d}{c x^6+a}dx-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 1746 |
| \(\displaystyle \frac {-c \left (\frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \left (\frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {-c \left (\frac {\sqrt [3]{c} d \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {-c \left (\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int -\frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-c \left (\frac {\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}+\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\sqrt [6]{a} \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}+\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-c \left (\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}+\frac {-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}\right )-\frac {e}{2 x^2}}{a}-\frac {d}{5 a x^5}\) |
Input:
Int[(d + e*x^3)/(x^6*(a + c*x^6)),x]
Output:
-1/5*d/(a*x^5) + (-1/2*e/x^2 - c*(((c^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)]) /a^(1/6) - (a^(1/3)*e*Log[a^(1/3) + c^(1/3)*x^2])/(2*c^(1/3)))/(3*a^(2/3)* c^(1/3)) + (-((a^(1/6)*(Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3]*(1 - (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))])/c^(1/3)) - (a^(2/3)*((Sqrt[3]*Sqrt[c]* d)/Sqrt[a] - e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(2 *c^(1/3)))/(6*a*c^(1/3)) + ((a^(1/6)*(Sqrt[c]*d - Sqrt[3]*Sqrt[a]*e)*ArcTa n[Sqrt[3]*(1 + (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))])/c^(1/3) + (a^(1/6)*(Sqrt [3]*Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/ 3)*x^2])/(2*c^(1/3)))/(6*a*c^(1/3))))/a
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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
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]
Int[((d_) + (e_.)*(x_)^3)/((a_) + (c_.)*(x_)^6), x_Symbol] :> With[{q = Rt[ c/a, 6]}, Simp[1/(3*a*q^2) Int[(q^2*d - e*x)/(1 + q^2*x^2), x], x] + (Sim p[1/(6*a*q^2) Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x + q^ 2*x^2), x], x] + Simp[1/(6*a*q^2) Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/( 1 + Sqrt[3]*q*x + q^2*x^2), x], x])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^( p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + c*x^(2*n))^(p + 1)/(a*f*(m + 1 ))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + c*x^(2*n))^p*(a*e *(m + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {-\frac {e \,x^{3}}{2 a}-\frac {d}{5 a}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{6}+\left (-2 a^{7} c \,e^{3}+6 a^{6} c^{2} d^{2} e \right ) \textit {\_Z}^{3}+a^{3} c^{2} e^{6}+3 a^{2} c^{3} d^{2} e^{4}+3 a \,c^{4} d^{4} e^{2}+c^{5} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (7 \textit {\_R}^{6} a^{11}+\left (-13 a^{7} c \,e^{3}+39 a^{6} c^{2} d^{2} e \right ) \textit {\_R}^{3}+6 a^{3} c^{2} e^{6}+18 a^{2} c^{3} d^{2} e^{4}+18 a \,c^{4} d^{4} e^{2}+6 c^{5} d^{6}\right ) x +2 a^{8} c d e \,\textit {\_R}^{4}+\left (a^{4} c^{2} d \,e^{4}+2 a^{3} c^{3} d^{3} e^{2}+a^{2} c^{4} d^{5}\right ) \textit {\_R} \right )\right )}{6}\) | \(240\) |
| default | \(-\frac {d}{5 a \,x^{5}}-\frac {e}{2 a \,x^{2}}-\frac {\left (\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 a}\right ) c}{a}\) | \(354\) |
Input:
int((e*x^3+d)/x^6/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
(-1/2*e/a*x^3-1/5*d/a)/x^5+1/6*sum(_R*ln((7*_R^6*a^11+(-13*a^7*c*e^3+39*a^ 6*c^2*d^2*e)*_R^3+6*a^3*c^2*e^6+18*a^2*c^3*d^2*e^4+18*a*c^4*d^4*e^2+6*c^5* d^6)*x+2*a^8*c*d*e*_R^4+(a^4*c^2*d*e^4+2*a^3*c^3*d^3*e^2+a^2*c^4*d^5)*_R), _R=RootOf(a^11*_Z^6+(-2*a^7*c*e^3+6*a^6*c^2*d^2*e)*_Z^3+a^3*c^2*e^6+3*a^2* c^3*d^2*e^4+3*a*c^4*d^4*e^2+c^5*d^6))
Leaf count of result is larger than twice the leaf count of optimal. 1734 vs. \(2 (225) = 450\).
Time = 0.20 (sec) , antiderivative size = 1734, normalized size of antiderivative = 5.30 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x^3+d)/x^6/(c*x^6+a),x, algorithm="fricas")
Output:
1/60*(10*a*x^5*(-(a^5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4 )/a^11) + 3*c^2*d^2*e - a*c*e^3)/a^5)^(1/3)*log(-(c^4*d^5 - 2*a*c^3*d^3*e^ 2 - 3*a^2*c^2*d*e^4)*x + (a^8*e*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c ^3*d^2*e^4)/a^11) + a^2*c^3*d^4 - 3*a^3*c^2*d^2*e^2)*(-(a^5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4)/a^11) + 3*c^2*d^2*e - a*c*e^3)/a^5 )^(1/3)) + 10*a*x^5*((a^5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2 *e^4)/a^11) - 3*c^2*d^2*e + a*c*e^3)/a^5)^(1/3)*log(-(c^4*d^5 - 2*a*c^3*d^ 3*e^2 - 3*a^2*c^2*d*e^4)*x - (a^8*e*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a ^2*c^3*d^2*e^4)/a^11) - a^2*c^3*d^4 + 3*a^3*c^2*d^2*e^2)*((a^5*sqrt(-(c^5* d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4)/a^11) - 3*c^2*d^2*e + a*c*e^3)/ a^5)^(1/3)) - 30*e*x^3 - 5*(sqrt(-3)*a*x^5 + a*x^5)*(-(a^5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4)/a^11) + 3*c^2*d^2*e - a*c*e^3)/a^5) ^(1/3)*log(-(c^4*d^5 - 2*a*c^3*d^3*e^2 - 3*a^2*c^2*d*e^4)*x - 1/2*(a^2*c^3 *d^4 - 3*a^3*c^2*d^2*e^2 + sqrt(-3)*(a^2*c^3*d^4 - 3*a^3*c^2*d^2*e^2) + (s qrt(-3)*a^8*e + a^8*e)*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^ 4)/a^11))*(-(a^5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4)/a^1 1) + 3*c^2*d^2*e - a*c*e^3)/a^5)^(1/3)) + 5*(sqrt(-3)*a*x^5 - a*x^5)*(-(a^ 5*sqrt(-(c^5*d^6 - 6*a*c^4*d^4*e^2 + 9*a^2*c^3*d^2*e^4)/a^11) + 3*c^2*d^2* e - a*c*e^3)/a^5)^(1/3)*log(-(c^4*d^5 - 2*a*c^3*d^3*e^2 - 3*a^2*c^2*d*e^4) *x - 1/2*(a^2*c^3*d^4 - 3*a^3*c^2*d^2*e^2 - sqrt(-3)*(a^2*c^3*d^4 - 3*a...
Time = 1.91 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.58 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{11} + t^{3} \left (- 432 a^{7} c e^{3} + 1296 a^{6} c^{2} d^{2} e\right ) + a^{3} c^{2} e^{6} + 3 a^{2} c^{3} d^{2} e^{4} + 3 a c^{4} d^{4} e^{2} + c^{5} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{8} e + 6 t a^{4} c e^{4} - 36 t a^{3} c^{2} d^{2} e^{2} + 6 t a^{2} c^{3} d^{4}}{3 a^{2} c^{2} d e^{4} + 2 a c^{3} d^{3} e^{2} - c^{4} d^{5}} \right )} \right )\right )} + \frac {- 2 d - 5 e x^{3}}{10 a x^{5}} \] Input:
integrate((e*x**3+d)/x**6/(c*x**6+a),x)
Output:
RootSum(46656*_t**6*a**11 + _t**3*(-432*a**7*c*e**3 + 1296*a**6*c**2*d**2* e) + a**3*c**2*e**6 + 3*a**2*c**3*d**2*e**4 + 3*a*c**4*d**4*e**2 + c**5*d* *6, Lambda(_t, _t*log(x + (-1296*_t**4*a**8*e + 6*_t*a**4*c*e**4 - 36*_t*a **3*c**2*d**2*e**2 + 6*_t*a**2*c**3*d**4)/(3*a**2*c**2*d*e**4 + 2*a*c**3*d **3*e**2 - c**4*d**5)))) + (-2*d - 5*e*x**3)/(10*a*x**5)
Time = 0.12 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=\frac {c {\left (\frac {2 \, e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}} c^{\frac {2}{3}}} - \frac {4 \, d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d + a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d - a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e - a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e + a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}}{12 \, a} - \frac {5 \, e x^{3} + 2 \, d}{10 \, a x^{5}} \] Input:
integrate((e*x^3+d)/x^6/(c*x^6+a),x, algorithm="maxima")
Output:
1/12*c*(2*e*log(c^(1/3)*x^2 + a^(1/3))/(a^(1/3)*c^(2/3)) - 4*d*arctan(c^(1 /3)*x/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*c^(1/3))) - (sqrt(3)*a^ (1/6)*sqrt(c)*d + a^(2/3)*e)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a*c^(2/3)) + (sqrt(3)*a^(1/6)*sqrt(c)*d - a^(2/3)*e)*log(c^(1/3 )*x^2 - sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a*c^(2/3)) + 2*(sqrt(3)*a^(5 /6)*c^(1/6)*e - a^(1/3)*c^(2/3)*d)*arctan((2*c^(1/3)*x + sqrt(3)*a^(1/6)*c ^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(2/3)*sqrt(a^(1/3)*c^(1/3))) - 2*(sqrt (3)*a^(5/6)*c^(1/6)*e + a^(1/3)*c^(2/3)*d)*arctan((2*c^(1/3)*x - sqrt(3)*a ^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(2/3)*sqrt(a^(1/3)*c^(1/3))))/ a - 1/10*(5*e*x^3 + 2*d)/(a*x^5)
Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=-\frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} + \frac {\left (a c^{5}\right )^{\frac {2}{3}} e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, a^{2} c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} c^{3}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} c^{3}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} c^{3}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} c^{3}} - \frac {5 \, e x^{3} + 2 \, d}{10 \, a x^{5}} \] Input:
integrate((e*x^3+d)/x^6/(c*x^6+a),x, algorithm="giac")
Output:
-1/3*(a*c^5)^(1/6)*d*arctan(x/(a/c)^(1/6))/a^2 + 1/6*(a*c^5)^(2/3)*e*abs(c )*log(x^2 + (a/c)^(1/3))/(a^2*c^4) - 1/6*((a*c^5)^(1/6)*c^3*d - sqrt(3)*(a *c^5)^(2/3)*e)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^3) - 1/6*((a*c^5)^(1/6)*c^3*d + sqrt(3)*(a*c^5)^(2/3)*e)*arctan((2*x - sqrt(3) *(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^3) - 1/12*(sqrt(3)*(a*c^5)^(1/6)*c^3*d + (a*c^5)^(2/3)*e)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^3) + 1/12*(sqrt(3)*(a*c^5)^(1/6)*c^3*d - (a*c^5)^(2/3)*e)*log(x^2 - sqrt(3)* x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^3) - 1/10*(5*e*x^3 + 2*d)/(a*x^5)
Time = 22.33 (sec) , antiderivative size = 1261, normalized size of antiderivative = 3.86 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx =\text {Too large to display} \] Input:
int((d + e*x^3)/(x^6*(a + c*x^6)),x)
Output:
log(e*x*(-a^11*c^3)^(1/2) - a^7*c*((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/a^11)^(1/3) + a^5*c^2*d*x)* ((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11 *c^3)^(1/2))/(216*a^11))^(1/3) + log(e*x*(-a^11*c^3)^(1/2) + a^7*c*((a^7*c *e^3 - c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e + 3*a*d*e^2*(-a^11*c^3)^( 1/2))/a^11)^(1/3) - a^5*c^2*d*x)*((a^7*c*e^3 - c*d^3*(-a^11*c^3)^(1/2) - 3 *a^6*c^2*d^2*e + 3*a*d*e^2*(-a^11*c^3)^(1/2))/(216*a^11))^(1/3) - (d/(5*a) + (e*x^3)/(2*a))/x^5 + log(2*e*x*(-a^11*c^3)^(1/2) + a^7*c*((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/a ^11)^(1/3) - 3^(1/2)*a^7*c*((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c ^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/a^11)^(1/3)*1i + 2*a^5*c^2*d*x)*(( 3^(1/2)*1i)/2 - 1/2)*((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2 *e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/(216*a^11))^(1/3) - log(2*e*x*(-a^11*c^3 )^(1/2) + a^7*c*((a^7*c*e^3 + c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/a^11)^(1/3) + 3^(1/2)*a^7*c*((a^7*c*e^3 + c*d ^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/a^11 )^(1/3)*1i + 2*a^5*c^2*d*x)*((3^(1/2)*1i)/2 + 1/2)*((a^7*c*e^3 + c*d^3*(-a ^11*c^3)^(1/2) - 3*a^6*c^2*d^2*e - 3*a*d*e^2*(-a^11*c^3)^(1/2))/(216*a^11) )^(1/3) - log(a^7*c*((a^7*c*e^3 - c*d^3*(-a^11*c^3)^(1/2) - 3*a^6*c^2*d^2* e + 3*a*d*e^2*(-a^11*c^3)^(1/2))/a^11)^(1/3) - 2*e*x*(-a^11*c^3)^(1/2) ...
Time = 0.19 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.05 \[ \int \frac {d+e x^3}{x^6 \left (a+c x^6\right )} \, dx=\frac {10 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c d \,x^{5}+10 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a c e \,x^{5}-10 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c d \,x^{5}+10 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a c e \,x^{5}-20 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) c d \,x^{5}+5 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c d \,x^{5}-5 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c d \,x^{5}-12 c^{\frac {2}{3}} a^{\frac {4}{3}} d -30 c^{\frac {2}{3}} a^{\frac {4}{3}} e \,x^{3}+10 \,\mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a c e \,x^{5}-5 \,\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a c e \,x^{5}-5 \,\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a c e \,x^{5}}{60 c^{\frac {2}{3}} a^{\frac {7}{3}} x^{5}} \] Input:
int((e*x^3+d)/x^6/(c*x^6+a),x)
Output:
(10*sqrt(c)*sqrt(a)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1 /6)*a**(1/6)))*c*d*x**5 + 10*sqrt(3)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c **(1/3)*x)/(c**(1/6)*a**(1/6)))*a*c*e*x**5 - 10*sqrt(c)*sqrt(a)*atan((c**( 1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d*x**5 + 10*s qrt(3)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)) )*a*c*e*x**5 - 20*sqrt(c)*sqrt(a)*atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c *d*x**5 + 5*sqrt(c)*sqrt(a)*sqrt(3)*log( - c**(1/6)*a**(1/6)*sqrt(3)*x + a **(1/3) + c**(1/3)*x**2)*c*d*x**5 - 5*sqrt(c)*sqrt(a)*sqrt(3)*log(c**(1/6) *a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*d*x**5 - 12*c**(2/3)*a** (1/3)*a*d - 30*c**(2/3)*a**(1/3)*a*e*x**3 + 10*log(a**(1/3) + c**(1/3)*x** 2)*a*c*e*x**5 - 5*log( - c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3) *x**2)*a*c*e*x**5 - 5*log(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3 )*x**2)*a*c*e*x**5)/(60*c**(2/3)*a**(1/3)*a**2*x**5)