Integrand size = 30, antiderivative size = 100 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {f x^3}{3 b^2}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a b^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^2}-\frac {\left (b^3 c-a^2 b e+2 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^2 b^3} \] Output:
1/3*f*x^3/b^2+1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/a/b^3/(b*x^3+a)+c*ln(x)/a ^2-1/3*(2*a^3*f-a^2*b*e+b^3*c)*ln(b*x^3+a)/a^2/b^3
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {3 c \log (x)+\frac {\frac {a \left (b^3 c-a^3 f+a^2 b \left (e+f x^3\right )+a b^2 \left (-d+f x^6\right )\right )}{a+b x^3}+\left (-b^3 c+a^2 b e-2 a^3 f\right ) \log \left (a+b x^3\right )}{b^3}}{3 a^2} \] Input:
Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x]
Output:
(3*c*Log[x] + ((a*(b^3*c - a^3*f + a^2*b*(e + f*x^3) + a*b^2*(-d + f*x^6)) )/(a + b*x^3) + (-(b^3*c) + a^2*b*e - 2*a^3*f)*Log[a + b*x^3])/b^3)/(3*a^2 )
Time = 0.57 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 2361 |
\(\displaystyle \frac {1}{3} \int \frac {f x^9+e x^6+d x^3+c}{x^3 \left (b x^3+a\right )^2}dx^3\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{3} \int \left (\frac {c}{a^2 x^3}+\frac {f}{b^2}+\frac {-2 f a^3+b e a^2-b^3 c}{a^2 b^2 \left (b x^3+a\right )}+\frac {f a^3-b e a^2+b^2 d a-b^3 c}{a b^2 \left (b x^3+a\right )^2}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {c \log \left (x^3\right )}{a^2}-\frac {\log \left (a+b x^3\right ) \left (2 a^3 f-a^2 b e+b^3 c\right )}{a^2 b^3}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a b^3 \left (a+b x^3\right )}+\frac {f x^3}{b^2}\right )\) |
Input:
Int[(c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x]
Output:
((f*x^3)/b^2 + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a*b^3*(a + b*x^3)) + ( c*Log[x^3])/a^2 - ((b^3*c - a^2*b*e + 2*a^3*f)*Log[a + b*x^3])/(a^2*b^3))/ 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 = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {f \,x^{3}}{3 b^{2}}-\frac {\frac {a \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right )}{b \left (b \,x^{3}+a \right )}+\frac {\left (2 f \,a^{3}-e \,a^{2} b +b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}}{3 a^{2} b^{2}}+\frac {c \ln \left (x \right )}{a^{2}}\) | \(97\) |
norman | \(\frac {\frac {f \,x^{6}}{3 b}-\frac {2 f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c}{3 a \,b^{3}}}{b \,x^{3}+a}+\frac {c \ln \left (x \right )}{a^{2}}-\frac {\left (2 f \,a^{3}-e \,a^{2} b +b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{2} b^{3}}\) | \(98\) |
risch | \(\frac {f \,x^{3}}{3 b^{2}}-\frac {a^{2} f}{3 b^{3} \left (b \,x^{3}+a \right )}+\frac {a e}{3 b^{2} \left (b \,x^{3}+a \right )}-\frac {d}{3 b \left (b \,x^{3}+a \right )}+\frac {c}{3 a \left (b \,x^{3}+a \right )}+\frac {c \ln \left (x \right )}{a^{2}}-\frac {2 \ln \left (b \,x^{3}+a \right ) a f}{3 b^{3}}+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b^{2}}-\frac {c \ln \left (b \,x^{3}+a \right )}{3 a^{2}}\) | \(125\) |
parallelrisch | \(\frac {x^{6} a^{2} b^{2} f +3 \ln \left (x \right ) x^{3} b^{4} c -2 \ln \left (b \,x^{3}+a \right ) x^{3} a^{3} b f +\ln \left (b \,x^{3}+a \right ) x^{3} a^{2} b^{2} e -\ln \left (b \,x^{3}+a \right ) x^{3} b^{4} c +3 \ln \left (x \right ) a \,b^{3} c -2 \ln \left (b \,x^{3}+a \right ) a^{4} f +\ln \left (b \,x^{3}+a \right ) a^{3} b e -\ln \left (b \,x^{3}+a \right ) a \,b^{3} c -2 a^{4} f +a^{3} e b -a^{2} d \,b^{2}+a \,b^{3} c}{3 a^{2} b^{3} \left (b \,x^{3}+a \right )}\) | \(174\) |
Input:
int((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
1/3*f*x^3/b^2-1/3/a^2/b^2*(a*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/b/(b*x^3+a)+(2* a^3*f-a^2*b*e+b^3*c)/b*ln(b*x^3+a))+c*ln(x)/a^2
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.45 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {a^{2} b^{2} f x^{6} + a^{3} b f x^{3} + a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f - {\left (a b^{3} c - a^{3} b e + 2 \, a^{4} f + {\left (b^{4} c - a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \, {\left (b^{4} c x^{3} + a b^{3} c\right )} \log \left (x\right )}{3 \, {\left (a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/3*(a^2*b^2*f*x^6 + a^3*b*f*x^3 + a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f - (a*b^3*c - a^3*b*e + 2*a^4*f + (b^4*c - a^2*b^2*e + 2*a^3*b*f)*x^3)*log(b *x^3 + a) + 3*(b^4*c*x^3 + a*b^3*c)*log(x))/(a^2*b^4*x^3 + a^3*b^3)
Time = 18.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \frac {f x^{3}}{3 b^{2}} + \frac {c \log {\left (x \right )}}{a^{2}} - \frac {\left (2 a^{3} f - a^{2} b e + b^{3} c\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{2} b^{3}} \] Input:
integrate((f*x**9+e*x**6+d*x**3+c)/x/(b*x**3+a)**2,x)
Output:
(-a**3*f + a**2*b*e - a*b**2*d + b**3*c)/(3*a**2*b**3 + 3*a*b**4*x**3) + f *x**3/(3*b**2) + c*log(x)/a**2 - (2*a**3*f - a**2*b*e + b**3*c)*log(a/b + x**3)/(3*a**2*b**3)
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {f x^{3}}{3 \, b^{2}} + \frac {b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {c \log \left (x^{3}\right )}{3 \, a^{2}} - \frac {{\left (b^{3} c - a^{2} b e + 2 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{3}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/3*f*x^3/b^2 + 1/3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a*b^4*x^3 + a^2*b ^3) + 1/3*c*log(x^3)/a^2 - 1/3*(b^3*c - a^2*b*e + 2*a^3*f)*log(b*x^3 + a)/ (a^2*b^3)
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.23 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {f x^{3}}{3 \, b^{2}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {{\left (b^{3} c - a^{2} b e + 2 \, a^{3} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{3}} + \frac {b^{4} c x^{3} - a^{2} b^{2} e x^{3} + 2 \, a^{3} b f x^{3} + 2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f}{3 \, {\left (b x^{3} + a\right )} a^{2} b^{3}} \] Input:
integrate((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^2,x, algorithm="giac")
Output:
1/3*f*x^3/b^2 + c*log(abs(x))/a^2 - 1/3*(b^3*c - a^2*b*e + 2*a^3*f)*log(ab s(b*x^3 + a))/(a^2*b^3) + 1/3*(b^4*c*x^3 - a^2*b^2*e*x^3 + 2*a^3*b*f*x^3 + 2*a*b^3*c - a^2*b^2*d + a^4*f)/((b*x^3 + a)*a^2*b^3)
Time = 6.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {f\,x^3}{3\,b^2}+\frac {c\,\ln \left (x\right )}{a^2}+\frac {-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3}{3\,a\,b\,\left (b^3\,x^3+a\,b^2\right )}-\frac {\ln \left (b\,x^3+a\right )\,\left (2\,f\,a^3-e\,a^2\,b+c\,b^3\right )}{3\,a^2\,b^3} \] Input:
int((c + d*x^3 + e*x^6 + f*x^9)/(x*(a + b*x^3)^2),x)
Output:
(f*x^3)/(3*b^2) + (c*log(x))/a^2 + (b^3*c - a^3*f - a*b^2*d + a^2*b*e)/(3* a*b*(a*b^2 + b^3*x^3)) - (log(a + b*x^3)*(b^3*c + 2*a^3*f - a^2*b*e))/(3*a ^2*b^3)
Time = 0.15 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.73 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^2} \, dx=\frac {-2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{4} f +\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b e -2 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b f \,x^{3}+\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b^{2} e \,x^{3}-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{3} c -\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{4} c \,x^{3}-2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{4} f +\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b e -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b f \,x^{3}+\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b^{2} e \,x^{3}-\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{3} c -\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{4} c \,x^{3}+3 \,\mathrm {log}\left (x \right ) a \,b^{3} c +3 \,\mathrm {log}\left (x \right ) b^{4} c \,x^{3}+2 a^{3} b f \,x^{3}-a^{2} b^{2} e \,x^{3}+a^{2} b^{2} f \,x^{6}+a \,b^{3} d \,x^{3}-b^{4} c \,x^{3}}{3 a^{2} b^{3} \left (b \,x^{3}+a \right )} \] Input:
int((f*x^9+e*x^6+d*x^3+c)/x/(b*x^3+a)^2,x)
Output:
( - 2*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*f + log(a** (2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*e - 2*log(a**(2/3) - b **(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*f*x**3 + log(a**(2/3) - b**(1/3 )*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*e*x**3 - log(a**(2/3) - b**(1/3)*a **(1/3)*x + b**(2/3)*x**2)*a*b**3*c - log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**4*c*x**3 - 2*log(a**(1/3) + b**(1/3)*x)*a**4*f + log(a* *(1/3) + b**(1/3)*x)*a**3*b*e - 2*log(a**(1/3) + b**(1/3)*x)*a**3*b*f*x**3 + log(a**(1/3) + b**(1/3)*x)*a**2*b**2*e*x**3 - log(a**(1/3) + b**(1/3)*x )*a*b**3*c - log(a**(1/3) + b**(1/3)*x)*b**4*c*x**3 + 3*log(x)*a*b**3*c + 3*log(x)*b**4*c*x**3 + 2*a**3*b*f*x**3 - a**2*b**2*e*x**3 + a**2*b**2*f*x* *6 + a*b**3*d*x**3 - b**4*c*x**3)/(3*a**2*b**3*(a + b*x**3))