Integrand size = 21, antiderivative size = 71 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {d x^3}{3 c}-\frac {e x^6}{6 c}+\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{3 c^{3/2}}-\frac {a e \log \left (a-c x^6\right )}{6 c^2} \] Output:
-1/3*d*x^3/c-1/6*e*x^6/c+1/3*a^(1/2)*d*arctanh(c^(1/2)*x^3/a^(1/2))/c^(3/2 )-1/6*a*e*ln(-c*x^6+a)/c^2
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {2 c d x^3+c e x^6-2 \sqrt {a} \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )+a e \log \left (a-c x^6\right )}{6 c^2} \] Input:
Integrate[(x^8*(d + e*x^3))/(a - c*x^6),x]
Output:
-1/6*(2*c*d*x^3 + c*e*x^6 - 2*Sqrt[a]*Sqrt[c]*d*ArcTanh[(Sqrt[c]*x^3)/Sqrt [a]] + a*e*Log[a - c*x^6])/c^2
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1803, 523, 2009}
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.
\(\displaystyle \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx\) |
\(\Big \downarrow \) 1803 |
\(\displaystyle \frac {1}{3} \int \frac {x^6 \left (e x^3+d\right )}{a-c x^6}dx^3\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {1}{3} \int \left (-\frac {e x^3}{c}-\frac {d}{c}+\frac {a e x^3+a d}{c \left (a-c x^6\right )}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{c^{3/2}}-\frac {a e \log \left (a-c x^6\right )}{2 c^2}-\frac {d x^3}{c}-\frac {e x^6}{2 c}\right )\) |
Input:
Int[(x^8*(d + e*x^3))/(a - c*x^6),x]
Output:
(-((d*x^3)/c) - (e*x^6)/(2*c) + (Sqrt[a]*d*ArcTanh[(Sqrt[c]*x^3)/Sqrt[a]]) /c^(3/2) - (a*e*Log[a - c*x^6])/(2*c^2))/3
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x )^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\frac {1}{2} e \,x^{6}+d \,x^{3}}{3 c}-\frac {a \left (\frac {e \ln \left (-c \,x^{6}+a \right )}{2 c}-\frac {d \,\operatorname {arctanh}\left (\frac {c \,x^{3}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{3 c}\) | \(60\) |
risch | \(-\frac {e \,x^{6}}{6 c}-\frac {d \,x^{3}}{3 c}-\frac {d^{2}}{6 e c}+\frac {\ln \left (c \,x^{3}+\sqrt {a c}\right ) d \sqrt {a c}}{6 c^{2}}-\frac {\ln \left (c \,x^{3}+\sqrt {a c}\right ) a e}{6 c^{2}}-\frac {\ln \left (c \,x^{3}-\sqrt {a c}\right ) d \sqrt {a c}}{6 c^{2}}-\frac {\ln \left (c \,x^{3}-\sqrt {a c}\right ) a e}{6 c^{2}}\) | \(119\) |
Input:
int(x^8*(e*x^3+d)/(-c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-1/3/c*(1/2*e*x^6+d*x^3)-1/3*a/c*(1/2*e*ln(-c*x^6+a)/c-d/(a*c)^(1/2)*arcta nh(c*x^3/(a*c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.94 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=\left [-\frac {c e x^{6} + 2 \, c d x^{3} - c d \sqrt {\frac {a}{c}} \log \left (\frac {c x^{6} + 2 \, c x^{3} \sqrt {\frac {a}{c}} + a}{c x^{6} - a}\right ) + a e \log \left (c x^{6} - a\right )}{6 \, c^{2}}, -\frac {c e x^{6} + 2 \, c d x^{3} + 2 \, c d \sqrt {-\frac {a}{c}} \arctan \left (\frac {c x^{3} \sqrt {-\frac {a}{c}}}{a}\right ) + a e \log \left (c x^{6} - a\right )}{6 \, c^{2}}\right ] \] Input:
integrate(x^8*(e*x^3+d)/(-c*x^6+a),x, algorithm="fricas")
Output:
[-1/6*(c*e*x^6 + 2*c*d*x^3 - c*d*sqrt(a/c)*log((c*x^6 + 2*c*x^3*sqrt(a/c) + a)/(c*x^6 - a)) + a*e*log(c*x^6 - a))/c^2, -1/6*(c*e*x^6 + 2*c*d*x^3 + 2 *c*d*sqrt(-a/c)*arctan(c*x^3*sqrt(-a/c)/a) + a*e*log(c*x^6 - a))/c^2]
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (61) = 122\).
Time = 0.52 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=- \left (\frac {a e}{6 c^{2}} - \frac {d \sqrt {a c^{5}}}{6 c^{4}}\right ) \log {\left (x^{3} + \frac {a e - 6 c^{2} \left (\frac {a e}{6 c^{2}} - \frac {d \sqrt {a c^{5}}}{6 c^{4}}\right )}{c d} \right )} - \left (\frac {a e}{6 c^{2}} + \frac {d \sqrt {a c^{5}}}{6 c^{4}}\right ) \log {\left (x^{3} + \frac {a e - 6 c^{2} \left (\frac {a e}{6 c^{2}} + \frac {d \sqrt {a c^{5}}}{6 c^{4}}\right )}{c d} \right )} - \frac {d x^{3}}{3 c} - \frac {e x^{6}}{6 c} \] Input:
integrate(x**8*(e*x**3+d)/(-c*x**6+a),x)
Output:
-(a*e/(6*c**2) - d*sqrt(a*c**5)/(6*c**4))*log(x**3 + (a*e - 6*c**2*(a*e/(6 *c**2) - d*sqrt(a*c**5)/(6*c**4)))/(c*d)) - (a*e/(6*c**2) + d*sqrt(a*c**5) /(6*c**4))*log(x**3 + (a*e - 6*c**2*(a*e/(6*c**2) + d*sqrt(a*c**5)/(6*c**4 )))/(c*d)) - d*x**3/(3*c) - e*x**6/(6*c)
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {a d \log \left (\frac {c x^{3} - \sqrt {a c}}{c x^{3} + \sqrt {a c}}\right )}{6 \, \sqrt {a c} c} - \frac {a e \log \left (c x^{6} - a\right )}{6 \, c^{2}} - \frac {e x^{6} + 2 \, d x^{3}}{6 \, c} \] Input:
integrate(x^8*(e*x^3+d)/(-c*x^6+a),x, algorithm="maxima")
Output:
-1/6*a*d*log((c*x^3 - sqrt(a*c))/(c*x^3 + sqrt(a*c)))/(sqrt(a*c)*c) - 1/6* a*e*log(c*x^6 - a)/c^2 - 1/6*(e*x^6 + 2*d*x^3)/c
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {a d \arctan \left (\frac {c x^{3}}{\sqrt {-a c}}\right )}{3 \, \sqrt {-a c} c} - \frac {a e \log \left (c x^{6} - a\right )}{6 \, c^{2}} - \frac {c e x^{6} + 2 \, c d x^{3}}{6 \, c^{2}} \] Input:
integrate(x^8*(e*x^3+d)/(-c*x^6+a),x, algorithm="giac")
Output:
-1/3*a*d*arctan(c*x^3/sqrt(-a*c))/(sqrt(-a*c)*c) - 1/6*a*e*log(c*x^6 - a)/ c^2 - 1/6*(c*e*x^6 + 2*c*d*x^3)/c^2
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.34 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {\ln \left (\sqrt {a\,c^5}+c^3\,x^3\right )\,\left (d\,\sqrt {a\,c^5}-a\,c^2\,e\right )}{6\,c^4}-\frac {e\,x^6}{6\,c}-\frac {d\,x^3}{3\,c}-\frac {\ln \left (\sqrt {a\,c^5}-c^3\,x^3\right )\,\left (d\,\sqrt {a\,c^5}+a\,c^2\,e\right )}{6\,c^4} \] Input:
int((x^8*(d + e*x^3))/(a - c*x^6),x)
Output:
(log((a*c^5)^(1/2) + c^3*x^3)*(d*(a*c^5)^(1/2) - a*c^2*e))/(6*c^4) - (e*x^ 6)/(6*c) - (d*x^3)/(3*c) - (log((a*c^5)^(1/2) - c^3*x^3)*(d*(a*c^5)^(1/2) + a*c^2*e))/(6*c^4)
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.48 \[ \int \frac {x^8 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {c^{\frac {7}{6}} a^{\frac {7}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +c^{\frac {7}{6}} a^{\frac {7}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d -c^{\frac {7}{6}} a^{\frac {7}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -c^{\frac {7}{6}} a^{\frac {7}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d -c^{\frac {2}{3}} a^{\frac {5}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -c^{\frac {2}{3}} a^{\frac {5}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -c^{\frac {2}{3}} a^{\frac {5}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -c^{\frac {2}{3}} a^{\frac {5}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -2 c^{\frac {5}{3}} a^{\frac {2}{3}} d \,x^{3}-c^{\frac {5}{3}} a^{\frac {2}{3}} e \,x^{6}}{6 c^{\frac {8}{3}} a^{\frac {2}{3}}} \] Input:
int(x^8*(e*x^3+d)/(-c*x^6+a),x)
Output:
(c**(1/6)*a**(1/6)*log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)* a*c*d + c**(1/6)*a**(1/6)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*a*c*d - c **(1/6)*a**(1/6)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*c*d - c**(1/6)*a**(1/6)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*a*c*d - c**(2/3)* a**(2/3)*log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*e - c**( 2/3)*a**(2/3)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*a*e - c**(2/3)*a**(2/ 3)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*e - c**(2/3)*a**( 2/3)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*a*e - 2*c**(2/3)*a**(2/3)*c*d*x** 3 - c**(2/3)*a**(2/3)*c*e*x**6)/(6*c**(2/3)*a**(2/3)*c**2)