Integrand size = 17, antiderivative size = 311 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \] Output:
d*x/c-1/3*a^(1/6)*d*arctan(c^(1/6)*x/a^(1/6))/c^(7/6)-1/6*(a^(1/2)*d-3^(1/ 2)*c^(1/2)*e)*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(1/3)/c^(7/6)-1/6*(a^ (1/2)*d+3^(1/2)*c^(1/2)*e)*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(1/3)/c^( 7/6)-1/6*e*ln(a^(1/3)+c^(1/3)*x^2)/a^(1/3)/c^(2/3)+1/12*(3^(1/2)*a^(1/2)*d +c^(1/2)*e)*ln(a^(1/3)-3^(1/2)*a^(1/6)*c^(1/6)*x+c^(1/3)*x^2)/a^(1/3)/c^(7 /6)-1/12*(3^(1/2)*a^(1/2)*d-c^(1/2)*e)*ln(a^(1/3)+3^(1/2)*a^(1/6)*c^(1/6)* x+c^(1/3)*x^2)/a^(1/3)/c^(7/6)
Time = 0.11 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.11 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (-a^{7/6} \sqrt {c} d+\sqrt {3} a^{2/3} c e\right ) \arctan \left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}+\frac {\left (-a^{7/6} \sqrt {c} d-\sqrt {3} a^{2/3} c e\right ) \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (-\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}}-\frac {\left (\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}} \] Input:
Integrate[(d + e/x^3)/(c + a/x^6),x]
Output:
(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((-(a^(7/6 )*Sqrt[c]*d) + Sqrt[3]*a^(2/3)*c*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6) *x)/a^(1/6)])/(6*a*c^(5/3)) + ((-(a^(7/6)*Sqrt[c]*d) - Sqrt[3]*a^(2/3)*c*e )*ArcTan[(Sqrt[3]*a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) - (e*Log[ a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((-(Sqrt[3]*a^(7/6)*Sqrt[c]* d) - a^(2/3)*c*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/ (12*a*c^(5/3)) - ((Sqrt[3]*a^(7/6)*Sqrt[c]*d - a^(2/3)*c*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(5/3))
Time = 0.68 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {1728, 1827, 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.
\(\displaystyle \int \frac {d+\frac {e}{x^3}}{\frac {a}{x^6}+c} \, dx\) |
\(\Big \downarrow \) 1728 |
\(\displaystyle \int \frac {x^3 \left (d x^3+e\right )}{a+c x^6}dx\) |
\(\Big \downarrow \) 1827 |
\(\displaystyle \frac {d x}{c}-\frac {\int \frac {a d-c e x^3}{c x^6+a}dx}{c}\) |
\(\Big \downarrow \) 1746 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {\sqrt [3]{a} \sqrt [3]{c} \left (a^{2/3} d+c^{2/3} e x\right )}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {\sqrt [3]{c} \left (2 a^{2/3} d-\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) x\right )}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {\sqrt [3]{c} \left (2 a^{2/3} d+\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) x\right )}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3} \sqrt [3]{c}}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {a^{2/3} d+c^{2/3} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 \sqrt [3]{a}}+\frac {\int \frac {2 a^{2/3} d-\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\int \frac {2 a^{2/3} d+\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}}{c}\) |
\(\Big \downarrow \) 452 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {a^{2/3} d \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx+c^{2/3} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 \sqrt [3]{a}}+\frac {\int \frac {2 a^{2/3} d-\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\int \frac {2 a^{2/3} d+\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}}{c}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {2 a^{2/3} d-\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\int \frac {2 a^{2/3} d+\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {c^{2/3} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx+\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {2 a^{2/3} d-\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\int \frac {2 a^{2/3} d+\sqrt [6]{c} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx-\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int -\frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int \frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {1}{2} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\frac {1}{2} \sqrt [6]{a} \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {1}{2} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {1}{2} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {\sqrt [3]{a} \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {1}{2} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx-\frac {\sqrt [3]{a} \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {1}{2} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx-\frac {\sqrt [3]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right )}{\sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {1}{2} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx+\frac {\sqrt [3]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right )}{\sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {d x}{c}-\frac {\frac {-\frac {\sqrt [3]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right )}{\sqrt [6]{c}}-\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt [3]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right )}{\sqrt [6]{c}}+\frac {\sqrt [3]{a} \left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{6 a^{2/3}}+\frac {\frac {\sqrt {a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}+\frac {1}{2} \sqrt [3]{c} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 \sqrt [3]{a}}}{c}\) |
Input:
Int[(d + e/x^3)/(c + a/x^6),x]
Output:
(d*x)/c - (((Sqrt[a]*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/c^(1/6) + (c^(1/3)*e*L og[a^(1/3) + c^(1/3)*x^2])/2)/(3*a^(1/3)) + (-((a^(1/3)*(Sqrt[a]*d - Sqrt[ 3]*Sqrt[c]*e)*ArcTan[Sqrt[3]*(1 - (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))])/c^(1/ 6)) - (a^(1/3)*(Sqrt[3]*Sqrt[a]*d + Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/ 6)*c^(1/6)*x + c^(1/3)*x^2])/(2*c^(1/6)))/(6*a^(2/3)) + ((a^(1/3)*(Sqrt[a] *d + Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3]*(1 + (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6) ))])/c^(1/6) + (a^(1/3)*(Sqrt[3]*Sqrt[a]*d - Sqrt[c]*e)*Log[a^(1/3) + Sqrt [3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(2*c^(1/6)))/(6*a^(2/3)))/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[(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[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> Int[x^(n*(2*p + q))*(e + d/x^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
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[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*(m + n*(2*p + 1) + 1)) In t[(f*x)^(m - n)*(a + c*x^(2*n))^p*(a*e*(m - n + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.14
method | result | size |
risch | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\) | \(45\) |
default | \(\frac {d x}{c}+\frac {\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3}}{c}\) | \(333\) |
Input:
int((d+e/x^3)/(c+a/x^6),x,method=_RETURNVERBOSE)
Output:
d*x/c+1/6/c^2*sum((_R^3*c*e-a*d)/_R^5*ln(x-_R),_R=RootOf(_Z^6*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (213) = 426\).
Time = 0.16 (sec) , antiderivative size = 1608, normalized size of antiderivative = 5.17 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\text {Too large to display} \] Input:
integrate((d+e/x^3)/(c+a/x^6),x, algorithm="fricas")
Output:
1/12*(2*c*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2 *d*e^4)*x + (a*c^5*e*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^ 7)) + a^2*c*d^4 - 3*a*c^2*d^2*e^2)*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)) - (sqrt(-3) *c + c)*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d *e^4)*x - 1/2*(a^2*c*d^4 - 3*a*c^2*d^2*e^2 + sqrt(-3)*(a^2*c*d^4 - 3*a*c^2 *d^2*e^2) + (sqrt(-3)*a*c^5*e + a*c^5*e)*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)))*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^ 2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)) + (sqrt(-3)*c - c)*(( a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - 1/2*(a^2*c*d^4 - 3*a*c^2*d^2*e^2 - sqrt(-3)*(a^2*c*d^4 - 3*a*c^2*d^2*e^2) - (sqrt(-3)*a*c^5*e - a*c^5*e)*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2* e^4)/(a*c^7)))*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a* c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)) + 2*c*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) - 3*a*d^2*e + c*e^3)/(a*c^3))^(1/ 3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - (a*c^5*e*sqrt(-(a^2*d^ 6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) - a^2*c*d^4 + 3*a*c^2*d^2*e...
Time = 1.67 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.54 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{2} c^{7} + t^{3} \left (- 1296 a^{2} c^{4} d^{2} e + 432 a c^{5} e^{3}\right ) + a^{3} d^{6} + 3 a^{2} c d^{4} e^{2} + 3 a c^{2} d^{2} e^{4} + c^{3} e^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a c^{5} e - 6 t a^{2} c d^{4} + 36 t a c^{2} d^{2} e^{2} - 6 t c^{3} e^{4}}{a^{2} d^{5} - 2 a c d^{3} e^{2} - 3 c^{2} d e^{4}} \right )} \right )\right )} + \frac {d x}{c} \] Input:
integrate((d+e/x**3)/(c+a/x**6),x)
Output:
RootSum(46656*_t**6*a**2*c**7 + _t**3*(-1296*a**2*c**4*d**2*e + 432*a*c**5 *e**3) + a**3*d**6 + 3*a**2*c*d**4*e**2 + 3*a*c**2*d**2*e**4 + c**3*e**6, Lambda(_t, _t*log(x + (-1296*_t**4*a*c**5*e - 6*_t*a**2*c*d**4 + 36*_t*a*c **2*d**2*e**2 - 6*_t*c**3*e**4)/(a**2*d**5 - 2*a*c*d**3*e**2 - 3*c**2*d*e* *4)))) + d*x/c
Time = 0.11 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.95 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c} - \frac {\frac {2 \, c^{\frac {1}{3}} e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} + \frac {4 \, a^{\frac {1}{3}} d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d - a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d + a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e + a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e - a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{12 \, c} \] Input:
integrate((d+e/x^3)/(c+a/x^6),x, algorithm="maxima")
Output:
d*x/c - 1/12*(2*c^(1/3)*e*log(c^(1/3)*x^2 + a^(1/3))/a^(1/3) + 4*a^(1/3)*d *arctan(c^(1/3)*x/sqrt(a^(1/3)*c^(1/3)))/sqrt(a^(1/3)*c^(1/3)) + (sqrt(3)* a^(7/6)*sqrt(c)*d - a^(2/3)*c*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^(7/6)*sqrt(c)*d + a^(2/3)*c*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^(7/6)*e + a^(4/3)*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^(7/6)*e - a^(4/3)*c^(2/3)*d)*arctan((2*c^(1/3)*x - sqr t(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))))/c
Time = 0.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.93 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=-\frac {e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (a c^{5}\right )^{\frac {1}{3}}} + \frac {d x}{c} - \frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c^{2}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \] Input:
integrate((d+e/x^3)/(c+a/x^6),x, algorithm="giac")
Output:
-1/6*e*abs(c)*log(x^2 + (a/c)^(1/3))/(a*c^5)^(1/3) + d*x/c - 1/3*(a*c^5)^( 1/6)*d*arctan(x/(a/c)^(1/6))/c^2 - 1/6*((a*c^5)^(1/6)*a*c^2*d + sqrt(3)*(a *c^5)^(2/3)*e)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a*c^4) - 1 /6*((a*c^5)^(1/6)*a*c^2*d - sqrt(3)*(a*c^5)^(2/3)*e)*arctan((2*x - sqrt(3) *(a/c)^(1/6))/(a/c)^(1/6))/(a*c^4) - 1/12*(sqrt(3)*(a*c^5)^(1/6)*a*c^2*d - (a*c^5)^(2/3)*e)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a*c^4) + 1/12*(sqrt(3)*(a*c^5)^(1/6)*a*c^2*d + (a*c^5)^(2/3)*e)*log(x^2 - sqrt(3)* x*(a/c)^(1/6) + (a/c)^(1/3))/(a*c^4)
Time = 21.28 (sec) , antiderivative size = 1308, normalized size of antiderivative = 4.21 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx =\text {Too large to display} \] Input:
int((d + e/x^3)/(c + a/x^6),x)
Output:
log(e*x*(-a^3*c^7)^(1/2) - a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) + a^2*c^3* d*x)*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*( -a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(e*x*(-a^3*c^7)^(1/2) + a^2*c^4 *(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3 *c^7)^(1/2))/(a^2*c^7))^(1/3) - a^2*c^3*d*x)*(-(a*c^5*e^3 - a*d^3*(-a^3*c^ 7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1 /3) + log(2*e*x*(-a^3*c^7)^(1/2) + a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7) ^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - 3^(1/2)*a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1i + 2*a^2*c^3*d*x)*((3^(1/2) *1i)/2 - 1/2)*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3* c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) - log(2*e*x*(-a^3*c^7)^(1/2 ) + a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c* d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) + 3^(1/2)*a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^ 2*c^7))^(1/3)*1i + 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 + 1/2)*(-(a*c^5*e^3 + a* d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216* a^2*c^7))^(1/3) - log(a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^ 2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - 2*e*x*(-a^...
Time = 0.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.92 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c e -2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) c e -4 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d +\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +12 c^{\frac {2}{3}} a^{\frac {1}{3}} d x -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c e +\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c e +\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c e}{12 c^{\frac {5}{3}} a^{\frac {1}{3}}} \] Input:
int((d+e/x^3)/(c+a/x^6),x)
Output:
(2*sqrt(c)*sqrt(a)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/ 6)*a**(1/6)))*d - 2*sqrt(3)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x )/(c**(1/6)*a**(1/6)))*c*e - 2*sqrt(c)*sqrt(a)*atan((c**(1/6)*a**(1/6)*sqr t(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*d - 2*sqrt(3)*atan((c**(1/6)*a** (1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*e - 4*sqrt(c)*sqrt(a) *atan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*d + sqrt(c)*sqrt(a)*sqrt(3)*log( - c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*d - sqrt(c)*sqrt( a)*sqrt(3)*log(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*d + 12*c**(2/3)*a**(1/3)*d*x - 2*log(a**(1/3) + c**(1/3)*x**2)*c*e + log( - c **(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*e + log(c**(1/6)* a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*e)/(12*c**(2/3)*a**(1/3)* c)