Integrand size = 30, antiderivative size = 164 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=-\frac {c}{12 a x^{12}}+\frac {b c-a d}{9 a^2 x^9}-\frac {b^2 c-a b d+a^2 e}{6 a^3 x^6}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^5}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^5} \] Output:
-1/12*c/a/x^12+1/9*(-a*d+b*c)/a^2/x^9-1/6*(a^2*e-a*b*d+b^2*c)/a^3/x^6+1/3* (-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a^4/x^3+b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln (x)/a^5-1/3*b*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(b*x^3+a)/a^5
Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=\frac {12 a b^3 c x^9-6 a^2 b^2 x^6 \left (c+2 d x^3\right )+2 a^3 b x^3 \left (2 c+3 d x^3+6 e x^6\right )-a^4 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+36 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12} \log (x)-12 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12} \log \left (a+b x^3\right )}{36 a^5 x^{12}} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]
Output:
(12*a*b^3*c*x^9 - 6*a^2*b^2*x^6*(c + 2*d*x^3) + 2*a^3*b*x^3*(2*c + 3*d*x^3 + 6*e*x^6) - a^4*(3*c + 4*d*x^3 + 6*e*x^6 + 12*f*x^9) + 36*b*(b^3*c - a*b ^2*d + a^2*b*e - a^3*f)*x^12*Log[x] - 12*b*(b^3*c - a*b^2*d + a^2*b*e - a^ 3*f)*x^12*Log[a + b*x^3])/(36*a^5*x^12)
Time = 0.69 (sec) , antiderivative size = 165, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2361 |
| \(\displaystyle \frac {1}{3} \int \frac {f x^9+e x^6+d x^3+c}{x^{15} \left (b x^3+a\right )}dx^3\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {\left (f a^3-b e a^2+b^2 d a-b^3 c\right ) b^2}{a^5 \left (b x^3+a\right )}-\frac {\left (f a^3-b e a^2+b^2 d a-b^3 c\right ) b}{a^5 x^3}+\frac {f a^3-b e a^2+b^2 d a-b^3 c}{a^4 x^6}+\frac {e a^2-b d a+b^2 c}{a^3 x^9}+\frac {a d-b c}{a^2 x^{12}}+\frac {c}{a x^{15}}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {b c-a d}{3 a^2 x^9}-\frac {a^2 e-a b d+b^2 c}{2 a^3 x^6}+\frac {b \log \left (x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5}-\frac {b \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^5}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x^3}-\frac {c}{4 a x^{12}}\right )\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]
Output:
(-1/4*c/(a*x^12) + (b*c - a*d)/(3*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(2*a^ 3*x^6) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a^4*x^3) + (b*(b^3*c - a*b^2 *d + a^2*b*e - a^3*f)*Log[x^3])/a^5 - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3* f)*Log[a + b*x^3])/a^5)/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 = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {b \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{5}}-\frac {a d -c b}{9 a^{2} x^{9}}-\frac {c}{12 a \,x^{12}}-\frac {a^{2} e -d a b +b^{2} c}{6 a^{3} x^{6}}-\frac {f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c}{3 a^{4} x^{3}}-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) b \ln \left (x \right )}{a^{5}}\) | \(156\) |
| norman | \(\frac {-\frac {c}{12 a}-\frac {\left (a d -c b \right ) x^{3}}{9 a^{2}}-\frac {\left (a^{2} e -d a b +b^{2} c \right ) x^{6}}{6 a^{3}}-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) x^{9}}{3 a^{4}}}{x^{12}}-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) b \ln \left (x \right )}{a^{5}}+\frac {b \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{5}}\) | \(158\) |
| risch | \(\frac {-\frac {c}{12 a}-\frac {\left (a d -c b \right ) x^{3}}{9 a^{2}}-\frac {\left (a^{2} e -d a b +b^{2} c \right ) x^{6}}{6 a^{3}}-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) x^{9}}{3 a^{4}}}{x^{12}}-\frac {b \ln \left (x \right ) f}{a^{2}}+\frac {b^{2} \ln \left (x \right ) e}{a^{3}}-\frac {b^{3} \ln \left (x \right ) d}{a^{4}}+\frac {b^{4} \ln \left (x \right ) c}{a^{5}}+\frac {b \ln \left (-b \,x^{3}-a \right ) f}{3 a^{2}}-\frac {b^{2} \ln \left (-b \,x^{3}-a \right ) e}{3 a^{3}}+\frac {b^{3} \ln \left (-b \,x^{3}-a \right ) d}{3 a^{4}}-\frac {b^{4} \ln \left (-b \,x^{3}-a \right ) c}{3 a^{5}}\) | \(204\) |
| parallelrisch | \(-\frac {36 \ln \left (x \right ) x^{12} a^{3} b f -36 \ln \left (x \right ) x^{12} a^{2} b^{2} e +36 \ln \left (x \right ) x^{12} a \,b^{3} d -36 \ln \left (x \right ) x^{12} b^{4} c -12 \ln \left (b \,x^{3}+a \right ) x^{12} a^{3} b f +12 \ln \left (b \,x^{3}+a \right ) x^{12} a^{2} b^{2} e -12 \ln \left (b \,x^{3}+a \right ) x^{12} a \,b^{3} d +12 \ln \left (b \,x^{3}+a \right ) x^{12} b^{4} c +12 a^{4} f \,x^{9}-12 a^{3} b e \,x^{9}+12 a^{2} b^{2} d \,x^{9}-12 a \,b^{3} c \,x^{9}+6 a^{4} e \,x^{6}-6 a^{3} b d \,x^{6}+6 a^{2} b^{2} c \,x^{6}+4 a^{4} d \,x^{3}-4 a^{3} b c \,x^{3}+3 c \,a^{4}}{36 a^{5} x^{12}}\) | \(229\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/3*b*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5*ln(b*x^3+a)-1/9/a^2*(a*d-b*c)/x^9- 1/12*c/a/x^12-1/6*(a^2*e-a*b*d+b^2*c)/a^3/x^6-1/3*(a^3*f-a^2*b*e+a*b^2*d-b ^3*c)/a^4/x^3-(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^5*b*ln(x)
Time = 0.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=-\frac {12 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (b x^{3} + a\right ) - 36 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (x\right ) - 12 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 6 \, {\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 3 \, a^{4} c - 4 \, {\left (a^{3} b c - a^{4} d\right )} x^{3}}{36 \, a^{5} x^{12}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x, algorithm="fricas")
Output:
-1/36*(12*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^12*log(b*x^3 + a) - 36 *(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^12*log(x) - 12*(a*b^3*c - a^2*b ^2*d + a^3*b*e - a^4*f)*x^9 + 6*(a^2*b^2*c - a^3*b*d + a^4*e)*x^6 + 3*a^4* c - 4*(a^3*b*c - a^4*d)*x^3)/(a^5*x^12)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x**13/(b*x**3+a),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=-\frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} + \frac {12 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 6 \, {\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 3 \, a^{3} c + 4 \, {\left (a^{2} b c - a^{3} d\right )} x^{3}}{36 \, a^{4} x^{12}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x, algorithm="maxima")
Output:
-1/3*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(b*x^3 + a)/a^5 + 1/3*(b^4 *c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(x^3)/a^5 + 1/36*(12*(b^3*c - a*b^2 *d + a^2*b*e - a^3*f)*x^9 - 6*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 3*a^3*c + 4*(a^2*b*c - a^3*d)*x^3)/(a^4*x^12)
Time = 0.13 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.40 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=\frac {{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac {25 \, b^{4} c x^{12} - 25 \, a b^{3} d x^{12} + 25 \, a^{2} b^{2} e x^{12} - 25 \, a^{3} b f x^{12} - 12 \, a b^{3} c x^{9} + 12 \, a^{2} b^{2} d x^{9} - 12 \, a^{3} b e x^{9} + 12 \, a^{4} f x^{9} + 6 \, a^{2} b^{2} c x^{6} - 6 \, a^{3} b d x^{6} + 6 \, a^{4} e x^{6} - 4 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{5} x^{12}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x, algorithm="giac")
Output:
(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(abs(x))/a^5 - 1/3*(b^5*c - a*b ^4*d + a^2*b^3*e - a^3*b^2*f)*log(abs(b*x^3 + a))/(a^5*b) - 1/36*(25*b^4*c *x^12 - 25*a*b^3*d*x^12 + 25*a^2*b^2*e*x^12 - 25*a^3*b*f*x^12 - 12*a*b^3*c *x^9 + 12*a^2*b^2*d*x^9 - 12*a^3*b*e*x^9 + 12*a^4*f*x^9 + 6*a^2*b^2*c*x^6 - 6*a^3*b*d*x^6 + 6*a^4*e*x^6 - 4*a^3*b*c*x^3 + 4*a^4*d*x^3 + 3*a^4*c)/(a^ 5*x^12)
Time = 6.68 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=\frac {\ln \left (x\right )\,\left (-f\,a^3\,b+e\,a^2\,b^2-d\,a\,b^3+c\,b^4\right )}{a^5}-\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3\,b+e\,a^2\,b^2-d\,a\,b^3+c\,b^4\right )}{3\,a^5}-\frac {\frac {c}{12\,a}-\frac {x^9\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^4}+\frac {x^3\,\left (a\,d-b\,c\right )}{9\,a^2}+\frac {x^6\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{6\,a^3}}{x^{12}} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x)
Output:
(log(x)*(b^4*c + a^2*b^2*e - a*b^3*d - a^3*b*f))/a^5 - (log(a + b*x^3)*(b^ 4*c + a^2*b^2*e - a*b^3*d - a^3*b*f))/(3*a^5) - (c/(12*a) - (x^9*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*a^4) + (x^3*(a*d - b*c))/(9*a^2) + (x^6*(b^ 2*c + a^2*e - a*b*d))/(6*a^3))/x^12
Time = 0.16 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.20 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )} \, dx=\frac {12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b f \,x^{12}-12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b^{2} e \,x^{12}+12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{3} d \,x^{12}-12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{4} c \,x^{12}+12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b f \,x^{12}-12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b^{2} e \,x^{12}+12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{3} d \,x^{12}-12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{4} c \,x^{12}-36 \,\mathrm {log}\left (x \right ) a^{3} b f \,x^{12}+36 \,\mathrm {log}\left (x \right ) a^{2} b^{2} e \,x^{12}-36 \,\mathrm {log}\left (x \right ) a \,b^{3} d \,x^{12}+36 \,\mathrm {log}\left (x \right ) b^{4} c \,x^{12}-3 a^{4} c -4 a^{4} d \,x^{3}-6 a^{4} e \,x^{6}-12 a^{4} f \,x^{9}+4 a^{3} b c \,x^{3}+6 a^{3} b d \,x^{6}+12 a^{3} b e \,x^{9}-6 a^{2} b^{2} c \,x^{6}-12 a^{2} b^{2} d \,x^{9}+12 a \,b^{3} c \,x^{9}}{36 a^{5} x^{12}} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a),x)
Output:
(12*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*f*x**12 - 1 2*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*e*x**12 + 12*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**3*d*x**12 - 12 *log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**4*c*x**12 + 12*log (a**(1/3) + b**(1/3)*x)*a**3*b*f*x**12 - 12*log(a**(1/3) + b**(1/3)*x)*a** 2*b**2*e*x**12 + 12*log(a**(1/3) + b**(1/3)*x)*a*b**3*d*x**12 - 12*log(a** (1/3) + b**(1/3)*x)*b**4*c*x**12 - 36*log(x)*a**3*b*f*x**12 + 36*log(x)*a* *2*b**2*e*x**12 - 36*log(x)*a*b**3*d*x**12 + 36*log(x)*b**4*c*x**12 - 3*a* *4*c - 4*a**4*d*x**3 - 6*a**4*e*x**6 - 12*a**4*f*x**9 + 4*a**3*b*c*x**3 + 6*a**3*b*d*x**6 + 12*a**3*b*e*x**9 - 6*a**2*b**2*c*x**6 - 12*a**2*b**2*d*x **9 + 12*a*b**3*c*x**9)/(36*a**5*x**12)