Integrand size = 20, antiderivative size = 322 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {d x}{c}+\frac {e x^4}{4 c}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\sqrt [6]{a} \left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 c^{5/3}}-\frac {\sqrt [6]{a} \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 c^{5/3}}+\frac {a^{2/3} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 c^{5/3}}+\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 c^{5/3}}-\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 c^{5/3}} \] Output:
d*x/c+1/4*e*x^4/c-1/3*a^(1/6)*d*arctan(c^(1/6)*x/a^(1/6))/c^(7/6)-1/6*a^(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))/c^( 5/3)-1/6*a^(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))/c^(5/3)+1/6*a^(2/3)*e*ln(a^(1/3)+c^(1/3)*x^2)/c^(5/3)+1/12*a^(1/6 )*(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)/c^(5/3)-1/12*a^(1/6)*(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)/c^(5/3)
Time = 0.14 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.92 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {12 c^{2/3} d x+3 c^{2/3} e x^4-4 \sqrt [6]{a} \sqrt {c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \left (-\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+2 a^{2/3} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )+\left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-\left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d+a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 c^{5/3}} \] Input:
Integrate[(x^6*(d + e*x^3))/(a + c*x^6),x]
Output:
(12*c^(2/3)*d*x + 3*c^(2/3)*e*x^4 - 4*a^(1/6)*Sqrt[c]*d*ArcTan[(c^(1/6)*x) /a^(1/6)] + 2*a^(1/6)*(Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] - (2* c^(1/6)*x)/a^(1/6)] + 2*a^(1/6)*(-(Sqrt[c]*d) + Sqrt[3]*Sqrt[a]*e)*ArcTan[ Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)] + 2*a^(2/3)*e*Log[a^(1/3) + c^(1/3)*x^2] + (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] - (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*c^(5/3))
Time = 0.75 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {1827, 27, 1827, 25, 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.
| \(\displaystyle \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {\int \frac {4 x^3 \left (a e-c d x^3\right )}{c x^6+a}dx}{4 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {\int \frac {x^3 \left (a e-c d x^3\right )}{c x^6+a}dx}{c}\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {-\frac {\int -\frac {a c \left (e x^3+d\right )}{c x^6+a}dx}{c}-d x}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {\frac {\int \frac {a c \left (e x^3+d\right )}{c x^6+a}dx}{c}-d x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \int \frac {e x^3+d}{c x^6+a}dx-d x}{c}\) |
| \(\Big \downarrow \) 1746 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{6 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\int \frac {\sqrt [3]{c} d-\sqrt [3]{a} e x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\sqrt [3]{c} d \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\sqrt [3]{a} e \int \frac {x}{\sqrt [3]{c} x^2+\sqrt [3]{a}}dx}{3 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{c} d-\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\int \frac {2 \sqrt [3]{c} d+\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int -\frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt [6]{c} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}+\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{a} \left (\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}dx}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\sqrt [6]{a} \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\sqrt [3]{a} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{c} x}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \int \frac {2 \sqrt [6]{c} x+\sqrt {3} \sqrt [6]{a}}{\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+1}dx}{2 \sqrt [6]{a} \sqrt [6]{c}}+\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}\right )-d x}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {e x^4}{4 c}-\frac {a \left (\frac {\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{a}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{3 a^{2/3} \sqrt [3]{c}}+\frac {-\frac {a^{2/3} \left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{\sqrt [3]{c}}}{6 a \sqrt [3]{c}}+\frac {\frac {\sqrt [6]{a} \arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{\sqrt [3]{c}}+\frac {\sqrt [6]{a} \left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{c}}}{6 a \sqrt [3]{c}}\right )-d x}{c}\) |
Input:
Int[(x^6*(d + e*x^3))/(a + c*x^6),x]
Output:
(e*x^4)/(4*c) - (-(d*x) + a*(((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)/Sq rt[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[Sqr t[3]*(1 + (2*c^(1/6)*x)/(Sqrt[3]*a^(1/6)))])/c^(1/3) + (a^(1/6)*(Sqrt[3]*S qrt[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))))/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[((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 = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.17
| method | result | size |
| risch | \(\frac {e \,x^{4}}{4 c}+\frac {d x}{c}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (-e \,\textit {\_R}^{3}-d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6 c^{2}}\) | \(54\) |
| default | \(\frac {\frac {1}{4} x^{4} e +d x}{c}-\frac {\left (\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 a}\right ) a}{c}\) | \(350\) |
Input:
int(x^6*(e*x^3+d)/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
1/4*e*x^4/c+d*x/c+1/6/c^2*a*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. 1582 vs. \(2 (222) = 444\).
Time = 0.15 (sec) , antiderivative size = 1582, normalized size of antiderivative = 4.91 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^6*(e*x^3+d)/(c*x^6+a),x, algorithm="fricas")
Output:
1/12*(3*e*x^4 + 2*c*(-(c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2* e^4)/c^9) + 3*a*c*d^2*e - a^2*e^3)/c^5)^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^ 2 - 3*a^2*d*e^4)*x + (c^6*e*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2 *e^4)/c^9) + c^3*d^4 - 3*a*c^2*d^2*e^2)*(-(c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c* d^4*e^2 + 9*a^3*d^2*e^4)/c^9) + 3*a*c*d^2*e - a^2*e^3)/c^5)^(1/3)) - (sqrt (-3)*c + c)*(-(c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9 ) + 3*a*c*d^2*e - a^2*e^3)/c^5)^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^ 2*d*e^4)*x - 1/2*(c^3*d^4 - 3*a*c^2*d^2*e^2 + sqrt(-3)*(c^3*d^4 - 3*a*c^2* d^2*e^2) + (sqrt(-3)*c^6*e + c^6*e)*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9 *a^3*d^2*e^4)/c^9))*(-(c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2* e^4)/c^9) + 3*a*c*d^2*e - a^2*e^3)/c^5)^(1/3)) + (sqrt(-3)*c - c)*(-(c^5*s qrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9) + 3*a*c*d^2*e - a^ 2*e^3)/c^5)^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x - 1/2*(c^ 3*d^4 - 3*a*c^2*d^2*e^2 - sqrt(-3)*(c^3*d^4 - 3*a*c^2*d^2*e^2) - (sqrt(-3) *c^6*e - c^6*e)*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9))* (-(c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9) + 3*a*c*d^ 2*e - a^2*e^3)/c^5)^(1/3)) + 2*c*((c^5*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9) - 3*a*c*d^2*e + a^2*e^3)/c^5)^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x - (c^6*e*sqrt(-(a*c^2*d^6 - 6*a^2*c*d^4*e^2 + 9*a^3*d^2*e^4)/c^9) - c^3*d^4 + 3*a*c^2*d^2*e^2)*((c^5*sqrt(-(a*c^2*...
Time = 1.72 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.54 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} c^{10} + t^{3} \left (- 432 a^{2} c^{5} e^{3} + 1296 a c^{6} d^{2} e\right ) + a^{4} e^{6} + 3 a^{3} c d^{2} e^{4} + 3 a^{2} c^{2} d^{4} e^{2} + a c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} c^{6} e + 6 t a^{2} c e^{4} - 36 t a c^{2} d^{2} e^{2} + 6 t c^{3} d^{4}}{3 a^{2} d e^{4} + 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} + \frac {d x}{c} + \frac {e x^{4}}{4 c} \] Input:
integrate(x**6*(e*x**3+d)/(c*x**6+a),x)
Output:
RootSum(46656*_t**6*c**10 + _t**3*(-432*a**2*c**5*e**3 + 1296*a*c**6*d**2* e) + a**4*e**6 + 3*a**3*c*d**2*e**4 + 3*a**2*c**2*d**4*e**2 + a*c**3*d**6, Lambda(_t, _t*log(x + (-1296*_t**4*c**6*e + 6*_t*a**2*c*e**4 - 36*_t*a*c* *2*d**2*e**2 + 6*_t*c**3*d**4)/(3*a**2*d*e**4 + 2*a*c*d**3*e**2 - c**2*d** 5)))) + d*x/c + e*x**4/(4*c)
Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {a {\left (\frac {2 \, e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}} c^{\frac {2}{3}}} - \frac {4 \, d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d + a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d - a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e - a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e + a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}}{12 \, c} + \frac {e x^{4} + 4 \, d x}{4 \, c} \] Input:
integrate(x^6*(e*x^3+d)/(c*x^6+a),x, algorithm="maxima")
Output:
1/12*a*(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))))/ c + 1/4*(e*x^4 + 4*d*x)/c
Time = 0.12 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=-\frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c^{2}} + \frac {\left (a c^{5}\right )^{\frac {2}{3}} e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, c^{6}} + \frac {c^{3} e x^{4} + 4 \, c^{3} d x}{4 \, c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, c^{5}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, c^{5}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, c^{5}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, c^{5}} \] Input:
integrate(x^6*(e*x^3+d)/(c*x^6+a),x, algorithm="giac")
Output:
-1/3*(a*c^5)^(1/6)*d*arctan(x/(a/c)^(1/6))/c^2 + 1/6*(a*c^5)^(2/3)*e*abs(c )*log(x^2 + (a/c)^(1/3))/c^6 + 1/4*(c^3*e*x^4 + 4*c^3*d*x)/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))/c^5 - 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))/c^5 - 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))/c^5 + 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))/c^5
Time = 22.54 (sec) , antiderivative size = 1147, normalized size of antiderivative = 3.56 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx =\text {Too large to display} \] Input:
int((x^6*(d + e*x^3))/(a + c*x^6),x)
Output:
log(e*x*(-a*c^11)^(1/2) - c^7*((c*d^3*(-a*c^11)^(1/2) + a^2*c^5*e^3 - 3*a* c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3) + c^6*d*x)*((c*d^3*(-a* c^11)^(1/2) + a^2*c^5*e^3 - 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/(21 6*c^10))^(1/3) + log(c^7*(-(c*d^3*(-a*c^11)^(1/2) - a^2*c^5*e^3 + 3*a*c^6* d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3) + e*x*(-a*c^11)^(1/2) - c^6 *d*x)*(-(c*d^3*(-a*c^11)^(1/2) - a^2*c^5*e^3 + 3*a*c^6*d^2*e - 3*a*d*e^2*( -a*c^11)^(1/2))/(216*c^10))^(1/3) + log(c^7*((c*d^3*(-a*c^11)^(1/2) + a^2* c^5*e^3 - 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3) + 2*e*x*( -a*c^11)^(1/2) - 3^(1/2)*c^7*((c*d^3*(-a*c^11)^(1/2) + a^2*c^5*e^3 - 3*a*c ^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3)*1i + 2*c^6*d*x)*((3^(1/2 )*1i)/2 - 1/2)*((c*d^3*(-a*c^11)^(1/2) + a^2*c^5*e^3 - 3*a*c^6*d^2*e - 3*a *d*e^2*(-a*c^11)^(1/2))/(216*c^10))^(1/3) - log(c^7*((c*d^3*(-a*c^11)^(1/2 ) + a^2*c^5*e^3 - 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3) + 2*e*x*(-a*c^11)^(1/2) + 3^(1/2)*c^7*((c*d^3*(-a*c^11)^(1/2) + a^2*c^5*e^3 - 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3)*1i + 2*c^6*d*x)* ((3^(1/2)*1i)/2 + 1/2)*((c*d^3*(-a*c^11)^(1/2) + a^2*c^5*e^3 - 3*a*c^6*d^2 *e - 3*a*d*e^2*(-a*c^11)^(1/2))/(216*c^10))^(1/3) + log(c^7*(-(c*d^3*(-a*c ^11)^(1/2) - a^2*c^5*e^3 + 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10 )^(1/3) - 2*e*x*(-a*c^11)^(1/2) - 3^(1/2)*c^7*(-(c*d^3*(-a*c^11)^(1/2) - a ^2*c^5*e^3 + 3*a*c^6*d^2*e - 3*a*d*e^2*(-a*c^11)^(1/2))/c^10)^(1/3)*1i ...
Time = 0.18 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.93 \[ \int \frac {x^6 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a e -2 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) a e -4 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d +\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -\sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +12 c^{\frac {2}{3}} a^{\frac {1}{3}} d x +3 c^{\frac {2}{3}} a^{\frac {1}{3}} e \,x^{4}+2 \,\mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e -\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e -\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e}{12 c^{\frac {5}{3}} a^{\frac {1}{3}}} \] Input:
int(x^6*(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)*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)))*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)*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 + 3*c**(2/3)*a**(1/3)*e*x**4 + 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)*c)