Integrand size = 20, antiderivative size = 301 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {d x^2}{2 c}+\frac {e x^5}{5 c}-\frac {a^{5/6} e \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{11/6}}+\frac {a^{5/6} e \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 c^{11/6}}-\frac {a^{5/6} e \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 c^{11/6}}+\frac {\sqrt [3]{a} d \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} c^{4/3}}+\frac {a^{5/6} e \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{2 \sqrt {3} c^{11/6}}-\frac {\sqrt [3]{a} d \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 c^{4/3}}+\frac {\sqrt [3]{a} d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{12 c^{4/3}} \] Output:
1/2*d*x^2/c+1/5*e*x^5/c-1/3*a^(5/6)*e*arctan(c^(1/6)*x/a^(1/6))/c^(11/6)-1 /6*a^(5/6)*e*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/c^(11/6)-1/6*a^(5/6)*e*a rctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/c^(11/6)+1/6*a^(1/3)*d*arctan(1/3*(a^(1 /3)-2*c^(1/3)*x^2)*3^(1/2)/a^(1/3))*3^(1/2)/c^(4/3)+1/6*a^(5/6)*e*arctanh( 3^(1/2)*a^(1/6)*c^(1/6)*x/(a^(1/3)+c^(1/3)*x^2))*3^(1/2)/c^(11/6)-1/6*a^(1 /3)*d*ln(a^(1/3)+c^(1/3)*x^2)/c^(4/3)+1/12*a^(1/3)*d*ln(a^(2/3)-a^(1/3)*c^ (1/3)*x^2+c^(2/3)*x^4)/c^(4/3)
Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.99 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {30 c^{5/6} d x^2+12 c^{5/6} e x^5-20 a^{5/6} e \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )+10 \sqrt [3]{a} \left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+10 \sqrt [3]{a} \left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )-10 \sqrt [3]{a} \sqrt {c} d \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )-5 \left (-\sqrt [3]{a} \sqrt {c} d+\sqrt {3} a^{5/6} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+5 \left (\sqrt [3]{a} \sqrt {c} d+\sqrt {3} a^{5/6} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{60 c^{11/6}} \] Input:
Integrate[(x^7*(d + e*x^3))/(a + c*x^6),x]
Output:
(30*c^(5/6)*d*x^2 + 12*c^(5/6)*e*x^5 - 20*a^(5/6)*e*ArcTan[(c^(1/6)*x)/a^( 1/6)] + 10*a^(1/3)*(Sqrt[3]*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[Sqrt[3] - (2*c^( 1/6)*x)/a^(1/6)] + 10*a^(1/3)*(Sqrt[3]*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[Sqrt[ 3] + (2*c^(1/6)*x)/a^(1/6)] - 10*a^(1/3)*Sqrt[c]*d*Log[a^(1/3) + c^(1/3)*x ^2] - 5*(-(a^(1/3)*Sqrt[c]*d) + Sqrt[3]*a^(5/6)*e)*Log[a^(1/3) - Sqrt[3]*a ^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + 5*(a^(1/3)*Sqrt[c]*d + Sqrt[3]*a^(5/6)*e )*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(60*c^(11/6))
Time = 0.62 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {1827, 27, 1827, 27, 1835, 27, 821, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {\int \frac {5 x^4 \left (a e-c d x^3\right )}{c x^6+a}dx}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {\int \frac {x^4 \left (a e-c d x^3\right )}{c x^6+a}dx}{c}\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {-\frac {\int -\frac {2 a c x \left (e x^3+d\right )}{c x^6+a}dx}{2 c}-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \int \frac {x \left (e x^3+d\right )}{c x^6+a}dx-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 1835 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {1}{2} \left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^3\right )}dx+\frac {1}{2} \left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {c} x^3+\sqrt {-a}\right )}dx\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \int \frac {x}{\sqrt {-a}-\sqrt {c} x^3}dx}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \int \frac {x}{\sqrt {c} x^3+\sqrt {-a}}dx}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {1}{\sqrt [6]{c} x+\sqrt [6]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (\frac {\int \frac {1}{\sqrt [6]{-a}-\sqrt [6]{c} x}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx+\frac {\int -\frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {e x^5}{5 c}-\frac {a \left (\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}+e\right ) \left (\frac {\frac {\log \left (-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {a \sqrt {c} d}{(-a)^{3/2}}-e\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{2 \sqrt {c}}\right )-\frac {d x^2}{2}}{c}\) |
Input:
Int[(x^7*(d + e*x^3))/(a + c*x^6),x]
Output:
(e*x^5)/(5*c) - (-1/2*(d*x^2) + a*((((a*Sqrt[c]*d)/(-a)^(3/2) + e)*(-1/3*L og[(-a)^(1/6) + c^(1/6)*x]/((-a)^(1/6)*c^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*c^(1/6)*x)/(-a)^(1/6))/Sqrt[3]])/c^(1/6)) + Log[(-a)^(1/3) - (-a)^(1/6) *c^(1/6)*x + c^(1/3)*x^2]/(2*c^(1/6)))/(3*(-a)^(1/6)*c^(1/6))))/(2*Sqrt[c] ) + (((a*Sqrt[c]*d)/(-a)^(3/2) - e)*(-1/3*Log[(-a)^(1/6) - c^(1/6)*x]/((-a )^(1/6)*c^(1/3)) - ((Sqrt[3]*ArcTan[(1 + (2*c^(1/6)*x)/(-a)^(1/6))/Sqrt[3] ])/c^(1/6) - Log[(-a)^(1/3) + (-a)^(1/6)*c^(1/6)*x + c^(1/3)*x^2]/(2*c^(1/ 6)))/(3*(-a)^(1/6)*c^(1/6))))/(2*Sqrt[c])))/c
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (c_.)*(x_)^(n2_)) , x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[-(e/2 + c*(d/(2*q))) Int[(f *x)^m/(q - c*x^n), x], x] + Simp[(e/2 - c*(d/(2*q))) Int[(f*x)^m/(q + c*x ^n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.19
| method | result | size |
| risch | \(\frac {e \,x^{5}}{5 c}+\frac {d \,x^{2}}{2 c}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (-\textit {\_R}^{4} e -d \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6 c^{2}}\) | \(58\) |
| default | \(\frac {\frac {1}{5} e \,x^{5}+\frac {1}{2} d \,x^{2}}{c}-\frac {\left (\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} e}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} d}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d}{6 a}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} e}{12 a}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} d}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} d \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {e \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 c \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right ) a}{c}\) | \(352\) |
Input:
int(x^7*(e*x^3+d)/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
1/5*e*x^5/c+1/2*d/c*x^2+1/6/c^2*a*sum((-_R^4*e-_R*d)/_R^5*ln(x-_R),_R=Root Of(_Z^6*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 1931 vs. \(2 (210) = 420\).
Time = 0.14 (sec) , antiderivative size = 1931, normalized size of antiderivative = 6.42 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^7*(e*x^3+d)/(c*x^6+a),x, algorithm="fricas")
Output:
1/60*(12*e*x^5 + 30*d*x^2 + 5*(sqrt(-3)*c - c)*(-(c^5*sqrt(-(9*a^3*c^2*d^4 *e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11) + a*c*d^3 - 3*a^2*d*e^2)/c^5)^(1/3 )*log(-(3*a^2*c^3*d^6*e + 5*a^3*c^2*d^4*e^3 + a^4*c*d^2*e^5 - a^5*e^7)*x + 1/2*(6*a^2*c^5*d^3*e^2 - 2*a^3*c^4*d*e^4 + 2*sqrt(-3)*(3*a^2*c^5*d^3*e^2 - a^3*c^4*d*e^4) + (c^10*d^2 - a*c^9*e^2 + sqrt(-3)*(c^10*d^2 - a*c^9*e^2) )*sqrt(-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11))*(-(c^5*sqrt (-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11) + a*c*d^3 - 3*a^2* d*e^2)/c^5)^(2/3)) - 5*(sqrt(-3)*c + c)*(-(c^5*sqrt(-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11) + a*c*d^3 - 3*a^2*d*e^2)/c^5)^(1/3)*log(- (3*a^2*c^3*d^6*e + 5*a^3*c^2*d^4*e^3 + a^4*c*d^2*e^5 - a^5*e^7)*x + 1/2*(6 *a^2*c^5*d^3*e^2 - 2*a^3*c^4*d*e^4 - 2*sqrt(-3)*(3*a^2*c^5*d^3*e^2 - a^3*c ^4*d*e^4) + (c^10*d^2 - a*c^9*e^2 - sqrt(-3)*(c^10*d^2 - a*c^9*e^2))*sqrt( -(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11))*(-(c^5*sqrt(-(9*a^ 3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11) + a*c*d^3 - 3*a^2*d*e^2)/ c^5)^(2/3)) + 10*c*(-(c^5*sqrt(-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5 *e^6)/c^11) + a*c*d^3 - 3*a^2*d*e^2)/c^5)^(1/3)*log(-(3*a^2*c^3*d^6*e + 5* a^3*c^2*d^4*e^3 + a^4*c*d^2*e^5 - a^5*e^7)*x - (6*a^2*c^5*d^3*e^2 - 2*a^3* c^4*d*e^4 + (c^10*d^2 - a*c^9*e^2)*sqrt(-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2* e^4 + a^5*e^6)/c^11))*(-(c^5*sqrt(-(9*a^3*c^2*d^4*e^2 - 6*a^4*c*d^2*e^4 + a^5*e^6)/c^11) + a*c*d^3 - 3*a^2*d*e^2)/c^5)^(2/3)) + 5*(sqrt(-3)*c - c...
Time = 1.61 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.74 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} c^{11} + t^{3} \left (- 1296 a^{2} c^{6} d e^{2} + 432 a c^{7} d^{3}\right ) + a^{5} e^{6} + 3 a^{4} c d^{2} e^{4} + 3 a^{3} c^{2} d^{4} e^{2} + a^{2} c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 7776 t^{5} a c^{9} e^{2} + 7776 t^{5} c^{10} d^{2} + 180 t^{2} a^{3} c^{4} d e^{4} - 360 t^{2} a^{2} c^{5} d^{3} e^{2} + 36 t^{2} a c^{6} d^{5}}{a^{5} e^{7} - a^{4} c d^{2} e^{5} - 5 a^{3} c^{2} d^{4} e^{3} - 3 a^{2} c^{3} d^{6} e} \right )} \right )\right )} + \frac {d x^{2}}{2 c} + \frac {e x^{5}}{5 c} \] Input:
integrate(x**7*(e*x**3+d)/(c*x**6+a),x)
Output:
RootSum(46656*_t**6*c**11 + _t**3*(-1296*a**2*c**6*d*e**2 + 432*a*c**7*d** 3) + a**5*e**6 + 3*a**4*c*d**2*e**4 + 3*a**3*c**2*d**4*e**2 + a**2*c**3*d* *6, Lambda(_t, _t*log(x + (-7776*_t**5*a*c**9*e**2 + 7776*_t**5*c**10*d**2 + 180*_t**2*a**3*c**4*d*e**4 - 360*_t**2*a**2*c**5*d**3*e**2 + 36*_t**2*a *c**6*d**5)/(a**5*e**7 - a**4*c*d**2*e**5 - 5*a**3*c**2*d**4*e**3 - 3*a**2 *c**3*d**6*e)))) + d*x**2/(2*c) + e*x**5/(5*c)
Time = 0.12 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.02 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=-\frac {a {\left (\frac {2 \, d \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c^{\frac {1}{3}}} + \frac {4 \, e \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {1}{6}} e + c^{\frac {2}{3}} d\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} + \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {1}{6}} e - c^{\frac {2}{3}} d\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} - \frac {2 \, {\left (\sqrt {3} a^{\frac {1}{6}} c^{\frac {5}{6}} d - a^{\frac {2}{3}} c^{\frac {1}{3}} e\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {1}{6}} c^{\frac {5}{6}} d + a^{\frac {2}{3}} c^{\frac {1}{3}} e\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}}{12 \, c} + \frac {2 \, e x^{5} + 5 \, d x^{2}}{10 \, c} \] Input:
integrate(x^7*(e*x^3+d)/(c*x^6+a),x, algorithm="maxima")
Output:
-1/12*a*(2*d*log(c^(1/3)*x^2 + a^(1/3))/(a^(2/3)*c^(1/3)) + 4*e*arctan(c^( 1/3)*x/sqrt(a^(1/3)*c^(1/3)))/(c^(2/3)*sqrt(a^(1/3)*c^(1/3))) - (sqrt(3)*s qrt(a)*c^(1/6)*e + c^(2/3)*d)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a^(2/3)*c) + (sqrt(3)*sqrt(a)*c^(1/6)*e - c^(2/3)*d)*log(c^(1/ 3)*x^2 - sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a^(2/3)*c) - 2*(sqrt(3)*a^( 1/6)*c^(5/6)*d - a^(2/3)*c^(1/3)*e)*arctan((2*c^(1/3)*x + sqrt(3)*a^(1/6)* c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*c*sqrt(a^(1/3)*c^(1/3))) + 2*(sqr t(3)*a^(1/6)*c^(5/6)*d + a^(2/3)*c^(1/3)*e)*arctan((2*c^(1/3)*x - sqrt(3)* a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*c*sqrt(a^(1/3)*c^(1/3)))) /c + 1/10*(2*e*x^5 + 5*d*x^2)/c
Time = 0.41 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.97 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=-\frac {e \left (\frac {a}{c}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c} - \frac {\left (a c^{5}\right )^{\frac {1}{3}} d \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, c^{3}} + \frac {2 \, c^{4} e x^{5} + 5 \, c^{4} d x^{2}}{10 \, c^{5}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{3}} c^{3} d - \left (a c^{5}\right )^{\frac {5}{6}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, c^{6}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{3}} c^{3} d + \left (a c^{5}\right )^{\frac {5}{6}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, c^{6}} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{3}} c^{3} d + \sqrt {3} \left (a c^{5}\right )^{\frac {5}{6}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, c^{6}} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{3}} c^{3} d - \sqrt {3} \left (a c^{5}\right )^{\frac {5}{6}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, c^{6}} \] Input:
integrate(x^7*(e*x^3+d)/(c*x^6+a),x, algorithm="giac")
Output:
-1/3*e*(a/c)^(5/6)*arctan(x/(a/c)^(1/6))/c - 1/6*(a*c^5)^(1/3)*d*log(x^2 + (a/c)^(1/3))/c^3 + 1/10*(2*c^4*e*x^5 + 5*c^4*d*x^2)/c^5 + 1/6*(sqrt(3)*(a *c^5)^(1/3)*c^3*d - (a*c^5)^(5/6)*e)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a /c)^(1/6))/c^6 - 1/6*(sqrt(3)*(a*c^5)^(1/3)*c^3*d + (a*c^5)^(5/6)*e)*arcta n((2*x - sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/c^6 + 1/12*((a*c^5)^(1/3)*c^3*d + sqrt(3)*(a*c^5)^(5/6)*e)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3)) /c^6 + 1/12*((a*c^5)^(1/3)*c^3*d - sqrt(3)*(a*c^5)^(5/6)*e)*log(x^2 - sqrt (3)*x*(a/c)^(1/6) + (a/c)^(1/3))/c^6
Time = 23.45 (sec) , antiderivative size = 1461, normalized size of antiderivative = 4.85 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\text {Too large to display} \] Input:
int((x^7*(d + e*x^3))/(a + c*x^6),x)
Output:
log(a*c^9*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 - 3*c*d ^2*e*(-a^3*c^11)^(1/2))/c^11)^(2/3) - a*e^2*x*(-a^3*c^11)^(1/2) + c*d^2*x* (-a^3*c^11)^(1/2) + 2*a^2*c^6*d*e*x)*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2 ) - 3*a^2*c^6*d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/(216*c^11))^(1/3) + log (a*c^9*(-(a*c^7*d^3 - a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 + 3*c*d^2* e*(-a^3*c^11)^(1/2))/c^11)^(2/3) + a*e^2*x*(-a^3*c^11)^(1/2) - c*d^2*x*(-a ^3*c^11)^(1/2) + 2*a^2*c^6*d*e*x)*(-(a*c^7*d^3 - a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 + 3*c*d^2*e*(-a^3*c^11)^(1/2))/(216*c^11))^(1/3) + (d*x^2 )/(2*c) + (e*x^5)/(5*c) - log(- (((3^(1/2)*1i)/2 - 1/2)*(-(a*c^7*d^3 + a*e ^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/c^11 )^(2/3)*(108*a^5*c*d^2*e - 36*a^6*e^3 + 36*a^4*c^2*x*((3^(1/2)*1i)/2 + 1/2 )*(a*e^2 - c*d^2)*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/c^11)^(1/3)))/36 - (a^5*d*x*(a*e^2 + c*d^2 )^2)/c^3)*((3^(1/2)*1i)/2 + 1/2)*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/(216*c^11))^(1/3) + log(- ( ((3^(1/2)*1i)/2 + 1/2)*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6* d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/c^11)^(2/3)*(36*a^6*e^3 - 108*a^5*c*d ^2*e + 36*a^4*c^2*x*((3^(1/2)*1i)/2 - 1/2)*(a*e^2 - c*d^2)*(-(a*c^7*d^3 + a*e^3*(-a^3*c^11)^(1/2) - 3*a^2*c^6*d*e^2 - 3*c*d^2*e*(-a^3*c^11)^(1/2))/c ^11)^(1/3)))/36 - (a^5*d*x*(a*e^2 + c*d^2)^2)/c^3)*((3^(1/2)*1i)/2 - 1/...
Time = 0.18 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.11 \[ \int \frac {x^7 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {10 c^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e +10 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -10 c^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e +10 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d -20 c^{\frac {1}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) e -5 c^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +5 c^{\frac {1}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -10 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +5 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +5 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +30 a^{\frac {1}{3}} c d \,x^{2}+12 a^{\frac {1}{3}} c e \,x^{5}}{60 a^{\frac {1}{3}} c^{2}} \] Input:
int(x^7*(e*x^3+d)/(c*x^6+a),x)
Output:
(10*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c** (1/6)*a**(1/6)))*a*e + 10*c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a**(1/6 )*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*d - 10*c**(1/6)*a**(1/6)*at an((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*a*e + 1 0*c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x )/(c**(1/6)*a**(1/6)))*d - 20*c**(1/6)*a**(1/6)*atan((c**(1/3)*x)/(c**(1/6 )*a**(1/6)))*a*e - 5*c**(1/6)*a**(1/6)*sqrt(3)*log( - c**(1/6)*a**(1/6)*sq rt(3)*x + a**(1/3) + c**(1/3)*x**2)*a*e + 5*c**(1/6)*a**(1/6)*sqrt(3)*log( c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*a*e - 10*c**(2/3)* a**(2/3)*log(a**(1/3) + c**(1/3)*x**2)*d + 5*c**(2/3)*a**(2/3)*log( - c**( 1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*d + 5*c**(2/3)*a**(2/3 )*log(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*d + 30*a**(1 /3)*c*d*x**2 + 12*a**(1/3)*c*e*x**5)/(60*a**(1/3)*c**2)