\(\int \frac {c+d x^3+e x^6+f x^9}{x (a+b x^3)^3} \, dx\) [281]

Optimal result

Integrand size = 30, antiderivative size = 114 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a b^3 \left (a+b x^3\right )^2}+\frac {b^3 c-a^2 b e+2 a^3 f}{3 a^2 b^3 \left (a+b x^3\right )}+\frac {c \log (x)}{a^3}-\frac {1}{3} \left (\frac {c}{a^3}-\frac {f}{b^3}\right ) \log \left (a+b x^3\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 114, 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} \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=-\frac {1}{3} \left (\frac {c}{a^3}-\frac {f}{b^3}\right ) \log \left (a+b x^3\right )+\frac {c \log (x)}{a^3}+\frac {2 a^3 f-a^2 b e+b^3 c}{3 a^2 b^3 \left (a+b x^3\right )}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a b^3 \left (a+b x^3\right )^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {6 c \log (x)+\frac {\frac {a \left (3 a b^3 c+3 a^4 f+2 b^4 c x^3-a^2 b^2 \left (d+2 e x^3\right )-a^3 b \left (e-4 f x^3\right )\right )}{\left (a+b x^3\right )^2}+2 \left (-b^3 c+a^3 f\right ) \log \left (a+b x^3\right )}{b^3}}{6 a^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {\frac {3 f \,a^{3}-a^{2} b e -a \,b^{2} d +3 b^{3} c}{6 a \,b^{3}}+\frac {\left (2 f \,a^{3}-a^{2} b e +b^{3} c \right ) x^{3}}{3 a^{2} b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {c \ln \left (x \right )}{a^{3}}+\frac {\left (f \,a^{3}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{3} b^{3}}\)	\(113\)
default	\(\frac {c \ln \left (x \right )}{a^{3}}+\frac {\frac {\left (f \,a^{3}-b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b^{3}}-\frac {a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {a \left (2 f \,a^{3}-a^{2} b e +b^{3} c \right )}{b^{3} \left (b \,x^{3}+a \right )}}{3 a^{3}}\)	\(114\)
risch	\(\frac {\frac {3 f \,a^{3}-a^{2} b e -a \,b^{2} d +3 b^{3} c}{6 a \,b^{3}}+\frac {\left (2 f \,a^{3}-a^{2} b e +b^{3} c \right ) x^{3}}{3 a^{2} b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {c \ln \left (x \right )}{a^{3}}+\frac {\ln \left (-b \,x^{3}-a \right ) f}{3 b^{3}}-\frac {\ln \left (-b \,x^{3}-a \right ) c}{3 a^{3}}\)	\(119\)
parallelrisch	\(\frac {6 \ln \left (x \right ) x^{6} b^{5} c +2 \ln \left (b \,x^{3}+a \right ) x^{6} a^{3} b^{2} f -2 \ln \left (b \,x^{3}+a \right ) x^{6} b^{5} c +12 \ln \left (x \right ) x^{3} a \,b^{4} c +4 \ln \left (b \,x^{3}+a \right ) x^{3} a^{4} b f -4 \ln \left (b \,x^{3}+a \right ) x^{3} a \,b^{4} c +4 a^{4} b f \,x^{3}-2 a^{3} b^{2} e \,x^{3}+2 a \,b^{4} c \,x^{3}+6 \ln \left (x \right ) a^{2} b^{3} c +2 \ln \left (b \,x^{3}+a \right ) a^{5} f -2 \ln \left (b \,x^{3}+a \right ) a^{2} b^{3} c +3 f \,a^{5}-a^{4} e b -a^{3} d \,b^{2}+3 a^{2} c \,b^{3}}{6 a^{3} b^{3} \left (b \,x^{3}+a \right )^{2}}\)	\(220\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.64 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {3 \, a^{2} b^{3} c - a^{3} b^{2} d - a^{4} b e + 3 \, a^{5} f + 2 \, {\left (a b^{4} c - a^{3} b^{2} e + 2 \, a^{4} b f\right )} x^{3} - 2 \, {\left ({\left (b^{5} c - a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - a^{5} f + 2 \, {\left (a b^{4} c - a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left (b^{5} c x^{6} + 2 \, a b^{4} c x^{3} + a^{2} b^{3} c\right )} \log \left (x\right )}{6 \, {\left (a^{3} b^{5} x^{6} + 2 \, a^{4} b^{4} x^{3} + a^{5} b^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f + 2 \, {\left (b^{4} c - a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{3}}{6 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}} + \frac {c \log \left (x^{3}\right )}{3 \, a^{3}} - \frac {{\left (b^{3} c - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} c - a^{3} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b^{3}} + \frac {3 \, b^{4} c x^{6} - 3 \, a^{3} b f x^{6} + 8 \, a b^{3} c x^{3} - 2 \, a^{3} b e x^{3} - 2 \, a^{4} f x^{3} + 6 \, a^{2} b^{2} c - a^{3} b d - a^{4} e}{6 \, {\left (b x^{3} + a\right )}^{2} a^{3} b^{2}} \]

[In]

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

[Out]

c*log(abs(x))/a^3 - 1/3*(b^3*c - a^3*f)*log(abs(b*x^3 + a))/(a^3*b^3) + 1/6*(3*b^4*c*x^6 - 3*a^3*b*f*x^6 + 8*a
*b^3*c*x^3 - 2*a^3*b*e*x^3 - 2*a^4*f*x^3 + 6*a^2*b^2*c - a^3*b*d - a^4*e)/((b*x^3 + a)^2*a^3*b^2)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x^3+e x^6+f x^9}{x \left (a+b x^3\right )^3} \, dx=\frac {\frac {3\,f\,a^3-e\,a^2\,b-d\,a\,b^2+3\,c\,b^3}{6\,a\,b^3}+\frac {x^3\,\left (2\,f\,a^3-e\,a^2\,b+c\,b^3\right )}{3\,a^2\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {c\,\ln \left (x\right )}{a^3}-\frac {\ln \left (b\,x^3+a\right )\,\left (b^3\,c-a^3\,f\right )}{3\,a^3\,b^3} \]

[In]

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

[Out]

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