Integrand size = 17, antiderivative size = 305 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac {\left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \] Output:
1/3*d*arctan(c^(1/6)*x/a^(1/6))/a^(5/6)/c^(1/6)+1/6*(c^(1/2)*d+3^(1/2)*a^( 1/2)*e)*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(5/6)/c^(2/3)+1/6*(c^(1/2)* d-3^(1/2)*a^(1/2)*e)*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(5/6)/c^(2/3)-1 /6*e*ln(a^(1/3)+c^(1/3)*x^2)/a^(1/3)/c^(2/3)-1/12*(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^(5/6)/c^(2/3)+1/ 12*(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^(5/6)/c^(2/3)
Time = 0.11 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.10 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt [6]{a} \sqrt {c} d+\sqrt {3} a^{2/3} e\right ) \arctan \left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac {\left (\sqrt [6]{a} \sqrt {c} d-\sqrt {3} a^{2/3} e\right ) \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}-\frac {\left (-\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}} \] Input:
Integrate[(d + e*x^3)/(a + c*x^6),x]
Output:
(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) + ((a^(1/6)*Sqrt[c]*d + Sqrt[3]*a^(2/3)*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)])/( 6*a*c^(2/3)) + ((a^(1/6)*Sqrt[c]*d - Sqrt[3]*a^(2/3)*e)*ArcTan[(Sqrt[3]*a^ (1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(2/3)) - (e*Log[a^(1/3) + c^(1/3)*x^ 2])/(6*a^(1/3)*c^(2/3)) - ((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])/(12*a*c^(2/3)) - ((-(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])/(12*a*c^(2/3))
Time = 0.65 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {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.
| \(\displaystyle \int \frac {d+e x^3}{a+c x^6} \, dx\) |
| \(\Big \downarrow \) 1746 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {\sqrt [3]{c} d \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int -\frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}+\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\sqrt [6]{a} \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}+\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}+\frac {-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}\) |
Input:
Int[(d + e*x^3)/(a + c*x^6),x]
Output:
((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)*ArcTan[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) )
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.11
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c}\) | \(34\) |
| default | \(\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 a}\) | \(329\) |
Input:
int((e*x^3+d)/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
1/6/c*sum((_R^3*e+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. 1631 vs. \(2 (207) = 414\).
Time = 0.14 (sec) , antiderivative size = 1631, normalized size of antiderivative = 5.35 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\text {Too large to display} \] Input:
integrate((e*x^3+d)/(c*x^6+a),x, algorithm="fricas")
Output:
-1/12*(sqrt(-3) + 1)*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2* e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a* c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 - 3*a^2*c*d^2*e^2 + sqrt(-3)*( a*c^2*d^4 - 3*a^2*c*d^2*e^2) + (sqrt(-3)*a^4*c^2*e + a^4*c^2*e)*sqrt(-(c^2 *d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2*c^2 ))^(1/3)) + 1/12*(sqrt(-3) - 1)*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2 *d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 - 3*a^2*c*d^2*e^2 - sqrt(-3)*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) - (sqrt(-3)*a^4*c^2*e - a^4*c^2*e) *sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*((a^2*c^2*sqr t(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^ 3)/(a^2*c^2))^(1/3)) - 1/12*(sqrt(-3) + 1)*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a *c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/ 3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 - 3*a^2 *c*d^2*e^2 + sqrt(-3)*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) - (sqrt(-3)*a^4*c^2*e + a^4*c^2*e)*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*( -(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c *d^2*e + a*e^3)/(a^2*c^2))^(1/3)) + 1/12*(sqrt(-3) - 1)*(-(a^2*c^2*sqrt(-( c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e + a*e^3...
Time = 1.92 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{5} c^{4} + t^{3} \cdot \left (432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} c^{2} e - 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} - 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} + 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \] Input:
integrate((e*x**3+d)/(c*x**6+a),x)
Output:
RootSum(46656*_t**6*a**5*c**4 + _t**3*(432*a**4*c**2*e**3 - 1296*a**3*c**3 *d**2*e) + a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6 , Lambda(_t, _t*log(x + (-1296*_t**4*a**4*c**2*e - 6*_t*a**3*e**4 + 36*_t* a**2*c*d**2*e**2 - 6*_t*a*c**2*d**4)/(3*a**2*d*e**4 + 2*a*c*d**3*e**2 - c* *2*d**5))))
Time = 0.11 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=-\frac {e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{6 \, a^{\frac {1}{3}} c^{\frac {2}{3}}} + \frac {d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{3 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d + a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d - a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e - a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{6 \, a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e + a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{6 \, a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} \] Input:
integrate((e*x^3+d)/(c*x^6+a),x, algorithm="maxima")
Output:
-1/6*e*log(c^(1/3)*x^2 + a^(1/3))/(a^(1/3)*c^(2/3)) + 1/3*d*arctan(c^(1/3) *x/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*c^(1/3))) + 1/12*(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)) - 1/12*(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)) - 1/6*(sqr t(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))) + 1/6*(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)))
Time = 0.13 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=-\frac {e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (a c^{5}\right )^{\frac {1}{3}}} + \frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a c} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \] Input:
integrate((e*x^3+d)/(c*x^6+a),x, algorithm="giac")
Output:
-1/6*e*abs(c)*log(x^2 + (a/c)^(1/3))/(a*c^5)^(1/3) + 1/3*(a*c^5)^(1/6)*d*a rctan(x/(a/c)^(1/6))/(a*c) + 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*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*c^4) + 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*c^4) - 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*c^4)
Time = 0.95 (sec) , antiderivative size = 1331, normalized size of antiderivative = 4.36 \[ \int \frac {d+e x^3}{a+c x^6} \, dx =\text {Too large to display} \] Input:
int((d + e*x^3)/(a + c*x^6),x)
Output:
log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3* a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + e*x*(-a^5*c^5)^(1/2) + a^2*c^ 3*d*x)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e ^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c *d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5 *c^4))^(1/3) - e*x*(-a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 - c*d^3* (-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5* c^4))^(1/3) - log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3* c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - 2*e*x*(-a^5*c^5 )^(1/2) + 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3* c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d* x)*((3^(1/2)*1i)/2 + 1/2)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3* c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(e*x*(-a ^5*c^5)^(1/2) - (a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c ^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3))/2 + (3^(1/2)*a^3* c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2* (-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i)/2 + a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2 *(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d ^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^...
Time = 0.25 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a e +2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a e +4 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d -\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e +\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e +\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e}{12 c^{\frac {2}{3}} a^{\frac {4}{3}}} \] Input:
int((e*x^3+d)/(c*x^6+a),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)))*a*e + 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)))*a*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)*sq rt(a)*sqrt(3)*log(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)* d - 2*log(a**(1/3) + c**(1/3)*x**2)*a*e + log( - c**(1/6)*a**(1/6)*sqrt(3) *x + a**(1/3) + c**(1/3)*x**2)*a*e + log(c**(1/6)*a**(1/6)*sqrt(3)*x + a** (1/3) + c**(1/3)*x**2)*a*e)/(12*c**(2/3)*a**(1/3)*a)