\(\int \frac {x^5 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\) [279]

Optimal result

Integrand size = 30, antiderivative size = 146 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(b e-3 a f) x^3}{3 b^4}+\frac {f x^6}{6 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{3 b^5 \left (a+b x^3\right )}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^5} \]

[Out]

1/3*(-3*a*f+b*e)*x^3/b^4+1/6*f*x^6/b^3+1/6*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)/b^5/(b*x^3+a)^2+1/3*(4*a^3*f-3*a^2
*b*e+2*a*b^2*d-b^3*c)/b^5/(b*x^3+a)+1/3*(6*a^2*f-3*a*b*e+b^2*d)*ln(b*x^3+a)/b^5

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1835, 1634} \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}-\frac {-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac {a \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac {x^3 (b e-3 a f)}{3 b^4}+\frac {f x^6}{6 b^3} \]

[In]

Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b*e - 3*a*f)*x^3)/(3*b^4) + (f*x^6)/(6*b^3) + (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^5*(a + b*x^3)^2)
- (b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)/(3*b^5*(a + b*x^3)) + ((b^2*d - 3*a*b*e + 6*a^2*f)*Log[a + b*x^3])
/(3*b^5)

Rule 1634

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]) &&
GtQ[Expon[Px, x], 2]

Rule 1835

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[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] && Intege
rQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {b e-3 a f}{b^4}+\frac {f x}{b^3}+\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^4 (a+b x)^3}+\frac {b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{b^4 (a+b x)^2}+\frac {b^2 d-3 a b e+6 a^2 f}{b^4 (a+b x)}\right ) \, dx,x,x^3\right ) \\ & = \frac {(b e-3 a f) x^3}{3 b^4}+\frac {f x^6}{6 b^3}+\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^5 \left (a+b x^3\right )^2}-\frac {b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{3 b^5 \left (a+b x^3\right )}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) \log \left (a+b x^3\right )}{3 b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {7 a^4 f+a^3 b \left (-5 e+2 f x^3\right )+a^2 b^2 \left (3 d-4 e x^3-11 f x^6\right )+b^4 x^3 \left (-2 c+2 e x^6+f x^9\right )-a b^3 \left (c-4 x^3 \left (d+e x^3-f x^6\right )\right )+2 \left (b^2 d-3 a b e+6 a^2 f\right ) \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 b^5 \left (a+b x^3\right )^2} \]

[In]

Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(7*a^4*f + a^3*b*(-5*e + 2*f*x^3) + a^2*b^2*(3*d - 4*e*x^3 - 11*f*x^6) + b^4*x^3*(-2*c + 2*e*x^6 + f*x^9) - a*
b^3*(c - 4*x^3*(d + e*x^3 - f*x^6)) + 2*(b^2*d - 3*a*b*e + 6*a^2*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^5*(a +
b*x^3)^2)

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94

method	result	size
norman	\(\frac {\frac {a \left (18 f \,a^{3}-9 a^{2} b e +3 a \,b^{2} d -b^{3} c \right )}{6 b^{5}}+\frac {f \,x^{12}}{6 b}-\frac {\left (2 a f -b e \right ) x^{9}}{3 b^{2}}+\frac {\left (12 f \,a^{3}-6 a^{2} b e +2 a \,b^{2} d -b^{3} c \right ) x^{3}}{3 b^{4}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (6 a^{2} f -3 a e b +b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{3 b^{5}}\)	\(137\)
default	\(\frac {\left (-f \,x^{3} b +3 a f -b e \right )^{2}}{6 b^{5} f}+\frac {\frac {\left (6 a^{2} f -3 a e b +b^{2} d \right ) \ln \left (b \,x^{3}+a \right )}{b}-\frac {a \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b \left (b \,x^{3}+a \right )^{2}}-\frac {-4 f \,a^{3}+3 a^{2} b e -2 a \,b^{2} d +b^{3} c}{b \left (b \,x^{3}+a \right )}}{3 b^{4}}\)	\(143\)
risch	\(\frac {f \,x^{6}}{6 b^{3}}-\frac {f a \,x^{3}}{b^{4}}+\frac {e \,x^{3}}{3 b^{3}}+\frac {3 f \,a^{2}}{2 b^{5}}-\frac {a e}{b^{4}}+\frac {e^{2}}{6 b^{3} f}+\frac {\left (\frac {4}{3} f \,a^{3}-a^{2} b e +\frac {2}{3} a \,b^{2} d -\frac {1}{3} b^{3} c \right ) x^{3}+\frac {a \left (7 f \,a^{3}-5 a^{2} b e +3 a \,b^{2} d -b^{3} c \right )}{6 b}}{b^{4} \left (b \,x^{3}+a \right )^{2}}+\frac {2 \ln \left (b \,x^{3}+a \right ) a^{2} f}{b^{5}}-\frac {\ln \left (b \,x^{3}+a \right ) a e}{b^{4}}+\frac {\ln \left (b \,x^{3}+a \right ) d}{3 b^{3}}\)	\(181\)
parallelrisch	\(\frac {f \,x^{12} b^{4}-4 x^{9} a \,b^{3} f +2 x^{9} b^{4} e +12 \ln \left (b \,x^{3}+a \right ) x^{6} a^{2} b^{2} f -6 \ln \left (b \,x^{3}+a \right ) x^{6} a \,b^{3} e +2 \ln \left (b \,x^{3}+a \right ) x^{6} b^{4} d +24 \ln \left (b \,x^{3}+a \right ) x^{3} a^{3} b f -12 \ln \left (b \,x^{3}+a \right ) x^{3} a^{2} b^{2} e +4 \ln \left (b \,x^{3}+a \right ) x^{3} a \,b^{3} d +24 a^{3} b f \,x^{3}-12 a^{2} b^{2} e \,x^{3}+4 a \,b^{3} d \,x^{3}-2 b^{4} c \,x^{3}+12 \ln \left (b \,x^{3}+a \right ) a^{4} f -6 \ln \left (b \,x^{3}+a \right ) a^{3} b e +2 \ln \left (b \,x^{3}+a \right ) a^{2} b^{2} d +18 a^{4} f -9 a^{3} b e +3 a^{2} b^{2} d -a \,b^{3} c}{6 b^{5} \left (b \,x^{3}+a \right )^{2}}\)	\(270\)

[In]

int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

(1/6*a*(18*a^3*f-9*a^2*b*e+3*a*b^2*d-b^3*c)/b^5+1/6*f*x^12/b-1/3*(2*a*f-b*e)/b^2*x^9+1/3*(12*a^3*f-6*a^2*b*e+2
*a*b^2*d-b^3*c)/b^4*x^3)/(b*x^3+a)^2+1/3*(6*a^2*f-3*a*b*e+b^2*d)*ln(b*x^3+a)/b^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.54 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {b^{4} f x^{12} + 2 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{9} + {\left (4 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{6} - a b^{3} c + 3 \, a^{2} b^{2} d - 5 \, a^{3} b e + 7 \, a^{4} f - 2 \, {\left (b^{4} c - 2 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 2 \, {\left ({\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{6} + a^{2} b^{2} d - 3 \, a^{3} b e + 6 \, a^{4} f + 2 \, {\left (a b^{3} d - 3 \, a^{2} b^{2} e + 6 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \]

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(b^4*f*x^12 + 2*(b^4*e - 2*a*b^3*f)*x^9 + (4*a*b^3*e - 11*a^2*b^2*f)*x^6 - a*b^3*c + 3*a^2*b^2*d - 5*a^3*b
*e + 7*a^4*f - 2*(b^4*c - 2*a*b^3*d + 2*a^2*b^2*e - a^3*b*f)*x^3 + 2*((b^4*d - 3*a*b^3*e + 6*a^2*b^2*f)*x^6 +
a^2*b^2*d - 3*a^3*b*e + 6*a^4*f + 2*(a*b^3*d - 3*a^2*b^2*e + 6*a^3*b*f)*x^3)*log(b*x^3 + a))/(b^7*x^6 + 2*a*b^
6*x^3 + a^2*b^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {b f x^{6} + 2 \, {\left (b e - 3 \, a f\right )} x^{3}}{6 \, b^{4}} + \frac {{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/6*(a*b^3*c - 3*a^2*b^2*d + 5*a^3*b*e - 7*a^4*f + 2*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^3)/(b^7*
x^6 + 2*a*b^6*x^3 + a^2*b^5) + 1/6*(b*f*x^6 + 2*(b*e - 3*a*f)*x^3)/b^4 + 1/3*(b^2*d - 3*a*b*e + 6*a^2*f)*log(b
*x^3 + a)/b^5

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=\frac {{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} + \frac {b^{3} f x^{6} + 2 \, b^{3} e x^{3} - 6 \, a b^{2} f x^{3}}{6 \, b^{6}} - \frac {a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{5}} \]

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*(b^2*d - 3*a*b*e + 6*a^2*f)*log(abs(b*x^3 + a))/b^5 + 1/6*(b^3*f*x^6 + 2*b^3*e*x^3 - 6*a*b^2*f*x^3)/b^6 -
1/6*(a*b^3*c - 3*a^2*b^2*d + 5*a^3*b*e - 7*a^4*f + 2*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^3)/((b*x^
3 + a)^2*b^5)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx=x^3\,\left (\frac {e}{3\,b^3}-\frac {a\,f}{b^4}\right )+\frac {\frac {7\,f\,a^4-5\,e\,a^3\,b+3\,d\,a^2\,b^2-c\,a\,b^3}{6\,b}-x^3\,\left (-\frac {4\,f\,a^3}{3}+e\,a^2\,b-\frac {2\,d\,a\,b^2}{3}+\frac {c\,b^3}{3}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+\frac {f\,x^6}{6\,b^3}+\frac {\ln \left (b\,x^3+a\right )\,\left (6\,f\,a^2-3\,e\,a\,b+d\,b^2\right )}{3\,b^5} \]

[In]

int((x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x)

[Out]

x^3*(e/(3*b^3) - (a*f)/b^4) + ((7*a^4*f + 3*a^2*b^2*d - a*b^3*c - 5*a^3*b*e)/(6*b) - x^3*((b^3*c)/3 - (4*a^3*f
)/3 - (2*a*b^2*d)/3 + a^2*b*e))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3) + (f*x^6)/(6*b^3) + (log(a + b*x^3)*(b^2*d +
6*a^2*f - 3*a*b*e))/(3*b^5)