Integrand size = 30, antiderivative size = 132 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^6}{6 b^3}+\frac {(b e-a f) x^9}{9 b^2}+\frac {f x^{12}}{12 b}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 b^5} \] Output:
1/3*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^3/b^4+1/6*(a^2*f-a*b*e+b^2*d)*x^6/b^3 +1/9*(-a*f+b*e)*x^9/b^2+1/12*f*x^12/b-1/3*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c) *ln(b*x^3+a)/b^5
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.90 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {b x^3 \left (-12 a^3 f+6 a^2 b \left (2 e+f x^3\right )-2 a b^2 \left (6 d+3 e x^3+2 f x^6\right )+b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )\right )+12 a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \log \left (a+b x^3\right )}{36 b^5} \] Input:
Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
Output:
(b*x^3*(-12*a^3*f + 6*a^2*b*(2*e + f*x^3) - 2*a*b^2*(6*d + 3*e*x^3 + 2*f*x ^6) + b^3*(12*c + 6*d*x^3 + 4*e*x^6 + 3*f*x^9)) + 12*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(36*b^5)
Time = 0.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
\(\displaystyle \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 2361 |
\(\displaystyle \frac {1}{3} \int \frac {x^3 \left (f x^9+e x^6+d x^3+c\right )}{b x^3+a}dx^3\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{3} \int \left (\frac {f x^9}{b}+\frac {(b e-a f) x^6}{b^2}+\frac {\left (f a^2-b e a+b^2 d\right ) x^3}{b^3}+\frac {-f a^3+b e a^2-b^2 d a+b^3 c}{b^4}+\frac {a \left (f a^3-b e a^2+b^2 d a-b^3 c\right )}{b^4 \left (b x^3+a\right )}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {x^6 \left (a^2 f-a b e+b^2 d\right )}{2 b^3}-\frac {a \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac {x^9 (b e-a f)}{3 b^2}+\frac {f x^{12}}{4 b}\right )\) |
Input:
Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]
Output:
(((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/b^4 + ((b^2*d - a*b*e + a^2*f)* x^6)/(2*b^3) + ((b*e - a*f)*x^9)/(3*b^2) + (f*x^12)/(4*b) - (a*(b^3*c - a* b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/b^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 = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.93
method | result | size |
norman | \(-\frac {\left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) x^{3}}{3 b^{4}}-\frac {\left (a f -b e \right ) x^{9}}{9 b^{2}}+\frac {f \,x^{12}}{12 b}+\frac {\left (f \,a^{2}-a b e +b^{2} d \right ) x^{6}}{6 b^{3}}+\frac {a \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{5}}\) | \(123\) |
default | \(-\frac {-\frac {1}{4} f \,x^{12} b^{3}+\frac {1}{3} a \,b^{2} f \,x^{9}-\frac {1}{3} b^{3} e \,x^{9}-\frac {1}{2} a^{2} b f \,x^{6}+\frac {1}{2} a \,b^{2} e \,x^{6}-\frac {1}{2} b^{3} d \,x^{6}+f \,a^{3} x^{3}-a^{2} b e \,x^{3}+a \,b^{2} d \,x^{3}-b^{3} c \,x^{3}}{3 b^{4}}+\frac {a \left (f \,a^{3}-e \,a^{2} b +d a \,b^{2}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{5}}\) | \(140\) |
parallelrisch | \(\frac {3 f \,x^{12} b^{4}-4 x^{9} a \,b^{3} f +4 x^{9} b^{4} e +6 x^{6} a^{2} b^{2} f -6 x^{6} a \,b^{3} e +6 x^{6} b^{4} d -12 x^{3} a^{3} b f +12 x^{3} a^{2} b^{2} e -12 x^{3} a \,b^{3} d +12 x^{3} b^{4} c +12 \ln \left (b \,x^{3}+a \right ) a^{4} f -12 \ln \left (b \,x^{3}+a \right ) a^{3} b e +12 \ln \left (b \,x^{3}+a \right ) a^{2} b^{2} d -12 \ln \left (b \,x^{3}+a \right ) a \,b^{3} c}{36 b^{5}}\) | \(168\) |
risch | \(\frac {f \,x^{12}}{12 b}-\frac {a f \,x^{9}}{9 b^{2}}+\frac {e \,x^{9}}{9 b}+\frac {a^{2} f \,x^{6}}{6 b^{3}}-\frac {a e \,x^{6}}{6 b^{2}}+\frac {d \,x^{6}}{6 b}-\frac {f \,a^{3} x^{3}}{3 b^{4}}+\frac {a^{2} e \,x^{3}}{3 b^{3}}-\frac {a d \,x^{3}}{3 b^{2}}+\frac {c \,x^{3}}{3 b}+\frac {a^{4} \ln \left (b \,x^{3}+a \right ) f}{3 b^{5}}-\frac {a^{3} \ln \left (b \,x^{3}+a \right ) e}{3 b^{4}}+\frac {a^{2} \ln \left (b \,x^{3}+a \right ) d}{3 b^{3}}-\frac {a \ln \left (b \,x^{3}+a \right ) c}{3 b^{2}}\) | \(170\) |
Input:
int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-1/3/b^4*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)*x^3-1/9/b^2*(a*f-b*e)*x^9+1/12*f*x^ 12/b+1/6*(a^2*f-a*b*e+b^2*d)*x^6/b^3+1/3*a/b^5*(a^3*f-a^2*b*e+a*b^2*d-b^3* c)*ln(b*x^3+a)
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {3 \, b^{4} f x^{12} + 4 \, {\left (b^{4} e - a b^{3} f\right )} x^{9} + 6 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{6} + 12 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 12 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{36 \, b^{5}} \] Input:
integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")
Output:
1/36*(3*b^4*f*x^12 + 4*(b^4*e - a*b^3*f)*x^9 + 6*(b^4*d - a*b^3*e + a^2*b^ 2*f)*x^6 + 12*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^3 - 12*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a))/b^5
Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{5}} + x^{9} \left (- \frac {a f}{9 b^{2}} + \frac {e}{9 b}\right ) + x^{6} \left (\frac {a^{2} f}{6 b^{3}} - \frac {a e}{6 b^{2}} + \frac {d}{6 b}\right ) + x^{3} \left (- \frac {a^{3} f}{3 b^{4}} + \frac {a^{2} e}{3 b^{3}} - \frac {a d}{3 b^{2}} + \frac {c}{3 b}\right ) + \frac {f x^{12}}{12 b} \] Input:
integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)
Output:
a*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a + b*x**3)/(3*b**5) + x**9* (-a*f/(9*b**2) + e/(9*b)) + x**6*(a**2*f/(6*b**3) - a*e/(6*b**2) + d/(6*b) ) + x**3*(-a**3*f/(3*b**4) + a**2*e/(3*b**3) - a*d/(3*b**2) + c/(3*b)) + f *x**12/(12*b)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {3 \, b^{3} f x^{12} + 4 \, {\left (b^{3} e - a b^{2} f\right )} x^{9} + 6 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{6} + 12 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{36 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \] Input:
integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")
Output:
1/36*(3*b^3*f*x^12 + 4*(b^3*e - a*b^2*f)*x^9 + 6*(b^3*d - a*b^2*e + a^2*b* f)*x^6 + 12*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a)/b^5
Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {3 \, b^{3} f x^{12} + 4 \, b^{3} e x^{9} - 4 \, a b^{2} f x^{9} + 6 \, b^{3} d x^{6} - 6 \, a b^{2} e x^{6} + 6 \, a^{2} b f x^{6} + 12 \, b^{3} c x^{3} - 12 \, a b^{2} d x^{3} + 12 \, a^{2} b e x^{3} - 12 \, a^{3} f x^{3}}{36 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} \] Input:
integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="giac")
Output:
1/36*(3*b^3*f*x^12 + 4*b^3*e*x^9 - 4*a*b^2*f*x^9 + 6*b^3*d*x^6 - 6*a*b^2*e *x^6 + 6*a^2*b*f*x^6 + 12*b^3*c*x^3 - 12*a*b^2*d*x^3 + 12*a^2*b*e*x^3 - 12 *a^3*f*x^3)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(abs(b*x^ 3 + a))/b^5
Time = 5.88 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.07 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=x^9\,\left (\frac {e}{9\,b}-\frac {a\,f}{9\,b^2}\right )+x^6\,\left (\frac {d}{6\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{6\,b}\right )+x^3\,\left (\frac {c}{3\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^{12}}{12\,b}+\frac {\ln \left (b\,x^3+a\right )\,\left (f\,a^4-e\,a^3\,b+d\,a^2\,b^2-c\,a\,b^3\right )}{3\,b^5} \] Input:
int((x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x)
Output:
x^9*(e/(9*b) - (a*f)/(9*b^2)) + x^6*(d/(6*b) - (a*(e/b - (a*f)/b^2))/(6*b) ) + x^3*(c/(3*b) - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/(3*b)) + (f*x^12)/( 12*b) + (log(a + b*x^3)*(a^4*f + a^2*b^2*d - a*b^3*c - a^3*b*e))/(3*b^5)
Time = 0.15 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.18 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx=\frac {12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{4} f -12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b e +12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b^{2} d -12 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{3} c +12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{4} f -12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b e +12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b^{2} d -12 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{3} c -12 a^{3} b f \,x^{3}+12 a^{2} b^{2} e \,x^{3}+6 a^{2} b^{2} f \,x^{6}-12 a \,b^{3} d \,x^{3}-6 a \,b^{3} e \,x^{6}-4 a \,b^{3} f \,x^{9}+12 b^{4} c \,x^{3}+6 b^{4} d \,x^{6}+4 b^{4} e \,x^{9}+3 b^{4} f \,x^{12}}{36 b^{5}} \] Input:
int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)
Output:
(12*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*f - 12*log(a* *(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*e + 12*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*d - 12*log(a**(2/3) - b**( 1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**3*c + 12*log(a**(1/3) + b**(1/3)*x)* a**4*f - 12*log(a**(1/3) + b**(1/3)*x)*a**3*b*e + 12*log(a**(1/3) + b**(1/ 3)*x)*a**2*b**2*d - 12*log(a**(1/3) + b**(1/3)*x)*a*b**3*c - 12*a**3*b*f*x **3 + 12*a**2*b**2*e*x**3 + 6*a**2*b**2*f*x**6 - 12*a*b**3*d*x**3 - 6*a*b* *3*e*x**6 - 4*a*b**3*f*x**9 + 12*b**4*c*x**3 + 6*b**4*d*x**6 + 4*b**4*e*x* *9 + 3*b**4*f*x**12)/(36*b**5)