Integrand size = 30, antiderivative size = 128 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=-\frac {c}{9 a x^9}+\frac {b c-a d}{6 a^2 x^6}-\frac {b^2 c-a b d+a^2 e}{3 a^3 x^3}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^4}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^4} \] Output:
-1/9*c/a/x^9+1/6*(-a*d+b*c)/a^2/x^6-1/3*(a^2*e-a*b*d+b^2*c)/a^3/x^3-(-a^3* f+a^2*b*e-a*b^2*d+b^3*c)*ln(x)/a^4+1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(b *x^3+a)/a^4
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=-\frac {c}{9 a x^9}+\frac {b c-a d}{6 a^2 x^6}+\frac {-b^2 c+a b d-a^2 e}{3 a^3 x^3}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log (x)}{a^4}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^4} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x]
Output:
-1/9*c/(a*x^9) + (b*c - a*d)/(6*a^2*x^6) + (-(b^2*c) + a*b*d - a^2*e)/(3*a ^3*x^3) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[x])/a^4 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(3*a^4)
Time = 0.65 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2361, 2123, 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.
\(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx\) |
\(\Big \downarrow \) 2361 |
\(\displaystyle \frac {1}{3} \int \frac {f x^9+e x^6+d x^3+c}{x^{12} \left (b x^3+a\right )}dx^3\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{3} \int \left (\frac {c}{a x^{12}}-\frac {b \left (f a^3-b e a^2+b^2 d a-b^3 c\right )}{a^4 \left (b x^3+a\right )}+\frac {f a^3-b e a^2+b^2 d a-b^3 c}{a^4 x^3}+\frac {e a^2-b d a+b^2 c}{a^3 x^6}+\frac {a d-b c}{a^2 x^9}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {b c-a d}{2 a^2 x^6}-\frac {a^2 e-a b d+b^2 c}{a^3 x^3}-\frac {\log \left (x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^4}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^4}-\frac {c}{3 a x^9}\right )\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x]
Output:
(-1/3*c/(a*x^9) + (b*c - a*d)/(2*a^2*x^6) - (b^2*c - a*b*d + a^2*e)/(a^3*x ^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x^3])/a^4 + ((b^3*c - a*b^2 *d + a^2*b*e - a^3*f)*Log[a + b*x^3])/a^4)/3
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[S implify[(m + 1)/n]]
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{4}}-\frac {a d -c b}{6 a^{2} x^{6}}-\frac {c}{9 a \,x^{9}}-\frac {a^{2} e -d a b +b^{2} c}{3 a^{3} x^{3}}+\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (x \right )}{a^{4}}\) | \(120\) |
norman | \(\frac {-\frac {c}{9 a}-\frac {\left (a d -c b \right ) x^{3}}{6 a^{2}}-\frac {\left (a^{2} e -d a b +b^{2} c \right ) x^{6}}{3 a^{3}}}{x^{9}}+\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (x \right )}{a^{4}}-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{4}}\) | \(122\) |
risch | \(\frac {-\frac {c}{9 a}-\frac {\left (a d -c b \right ) x^{3}}{6 a^{2}}-\frac {\left (a^{2} e -d a b +b^{2} c \right ) x^{6}}{3 a^{3}}}{x^{9}}+\frac {\ln \left (x \right ) f}{a}-\frac {\ln \left (x \right ) e b}{a^{2}}+\frac {\ln \left (x \right ) d \,b^{2}}{a^{3}}-\frac {\ln \left (x \right ) b^{3} c}{a^{4}}-\frac {\ln \left (b \,x^{3}+a \right ) f}{3 a}+\frac {\ln \left (b \,x^{3}+a \right ) e b}{3 a^{2}}-\frac {\ln \left (b \,x^{3}+a \right ) d \,b^{2}}{3 a^{3}}+\frac {\ln \left (b \,x^{3}+a \right ) b^{3} c}{3 a^{4}}\) | \(153\) |
parallelrisch | \(\frac {18 \ln \left (x \right ) x^{9} a^{3} f -18 \ln \left (x \right ) x^{9} a^{2} b e +18 \ln \left (x \right ) x^{9} a \,b^{2} d -18 \ln \left (x \right ) x^{9} b^{3} c -6 \ln \left (b \,x^{3}+a \right ) x^{9} a^{3} f +6 \ln \left (b \,x^{3}+a \right ) x^{9} a^{2} b e -6 \ln \left (b \,x^{3}+a \right ) x^{9} a \,b^{2} d +6 \ln \left (b \,x^{3}+a \right ) x^{9} b^{3} c -6 a^{3} e \,x^{6}+6 a^{2} b d \,x^{6}-6 a \,b^{2} c \,x^{6}-3 a^{3} d \,x^{3}+3 a^{2} b c \,x^{3}-2 c \,a^{3}}{18 a^{4} x^{9}}\) | \(180\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/3*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^4*ln(b*x^3+a)-1/6/a^2*(a*d-b*c)/x^6-1 /9*c/a/x^9-1/3*(a^2*e-a*b*d+b^2*c)/a^3/x^3+(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a ^4*ln(x)
Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=\frac {6 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} \log \left (b x^{3} + a\right ) - 18 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} \log \left (x\right ) - 6 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 2 \, a^{3} c + 3 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{18 \, a^{4} x^{9}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x, algorithm="fricas")
Output:
1/18*(6*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^9*log(b*x^3 + a) - 18*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^9*log(x) - 6*(a*b^2*c - a^2*b*d + a^3*e)*x ^6 - 2*a^3*c + 3*(a^2*b*c - a^3*d)*x^3)/(a^4*x^9)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**10/(b*x**3+a),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{4}} - \frac {6 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{6} - 3 \, {\left (a b c - a^{2} d\right )} x^{3} + 2 \, a^{2} c}{18 \, a^{3} x^{9}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x, algorithm="maxima")
Output:
1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(b*x^3 + a)/a^4 - 1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(x^3)/a^4 - 1/18*(6*(b^2*c - a*b*d + a^2*e)* x^6 - 3*(a*b*c - a^2*d)*x^3 + 2*a^2*c)/(a^3*x^9)
Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.41 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=-\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac {11 \, b^{3} c x^{9} - 11 \, a b^{2} d x^{9} + 11 \, a^{2} b e x^{9} - 11 \, a^{3} f x^{9} - 6 \, a b^{2} c x^{6} + 6 \, a^{2} b d x^{6} - 6 \, a^{3} e x^{6} + 3 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{4} x^{9}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x, algorithm="giac")
Output:
-(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*log(abs(x))/a^4 + 1/3*(b^4*c - a*b^3* d + a^2*b^2*e - a^3*b*f)*log(abs(b*x^3 + a))/(a^4*b) + 1/18*(11*b^3*c*x^9 - 11*a*b^2*d*x^9 + 11*a^2*b*e*x^9 - 11*a^3*f*x^9 - 6*a*b^2*c*x^6 + 6*a^2*b *d*x^6 - 6*a^3*e*x^6 + 3*a^2*b*c*x^3 - 3*a^3*d*x^3 - 2*a^3*c)/(a^4*x^9)
Time = 6.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}-\frac {\frac {c}{9\,a}+\frac {x^3\,\left (a\,d-b\,c\right )}{6\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{3\,a^3}}{x^9}-\frac {\ln \left (x\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^4} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)),x)
Output:
(log(a + b*x^3)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^4) - (c/(9*a) + (x^3*(a*d - b*c))/(6*a^2) + (x^6*(b^2*c + a^2*e - a*b*d))/(3*a^3))/x^9 - ( log(x)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/a^4
Time = 0.16 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.41 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )} \, dx=\frac {-6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} f \,x^{9}+6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b e \,x^{9}-6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{2} d \,x^{9}+6 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{3} c \,x^{9}-6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} f \,x^{9}+6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b e \,x^{9}-6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{2} d \,x^{9}+6 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{3} c \,x^{9}+18 \,\mathrm {log}\left (x \right ) a^{3} f \,x^{9}-18 \,\mathrm {log}\left (x \right ) a^{2} b e \,x^{9}+18 \,\mathrm {log}\left (x \right ) a \,b^{2} d \,x^{9}-18 \,\mathrm {log}\left (x \right ) b^{3} c \,x^{9}-2 a^{3} c -3 a^{3} d \,x^{3}-6 a^{3} e \,x^{6}+3 a^{2} b c \,x^{3}+6 a^{2} b d \,x^{6}-6 a \,b^{2} c \,x^{6}}{18 a^{4} x^{9}} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a),x)
Output:
( - 6*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*f*x**9 + 6* log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*e*x**9 - 6*log( a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*d*x**9 + 6*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**3*c*x**9 - 6*log(a**(1/3) + b**(1/3)*x)*a**3*f*x**9 + 6*log(a**(1/3) + b**(1/3)*x)*a**2*b*e*x**9 - 6* log(a**(1/3) + b**(1/3)*x)*a*b**2*d*x**9 + 6*log(a**(1/3) + b**(1/3)*x)*b* *3*c*x**9 + 18*log(x)*a**3*f*x**9 - 18*log(x)*a**2*b*e*x**9 + 18*log(x)*a* b**2*d*x**9 - 18*log(x)*b**3*c*x**9 - 2*a**3*c - 3*a**3*d*x**3 - 6*a**3*e* x**6 + 3*a**2*b*c*x**3 + 6*a**2*b*d*x**6 - 6*a*b**2*c*x**6)/(18*a**4*x**9)