3.258 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{10} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=175 \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{3 a^5}-\frac{\log (x) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{a^5}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{6 a^3 x^6}-\frac{c}{9 a^2 x^9} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.202541, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1821, 1620} \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{3 a^5}-\frac{\log (x) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )}{a^5}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{6 a^3 x^6}-\frac{c}{9 a^2 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^2),x]

[Out]

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

Rule 1821

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]]

Rule 1620

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]

Rubi steps

\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^4 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a^2 x^4}+\frac{-2 b c+a d}{a^3 x^3}+\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x^2}+\frac{-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f}{a^5 x}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 (a+b x)^2}-\frac{b \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{9 a^2 x^9}+\frac{2 b c-a d}{6 a^3 x^6}-\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}-\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 \left (a+b x^3\right )}-\frac{\left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) \log (x)}{a^5}+\frac{\left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^5}\\ \end{align*}

Mathematica [A]  time = 0.10454, size = 160, normalized size = 0.91 \[ \frac{\frac{6 a \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+6 \log \left (a+b x^3\right ) \left (2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c\right )+18 \log (x) \left (-2 a^2 b e+a^3 f+3 a b^2 d-4 b^3 c\right )-\frac{6 a \left (a^2 e-2 a b d+3 b^2 c\right )}{x^3}-\frac{3 a^2 (a d-2 b c)}{x^6}-\frac{2 a^3 c}{x^9}}{18 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^2),x]

[Out]

((-2*a^3*c)/x^9 - (3*a^2*(-2*b*c + a*d))/x^6 - (6*a*(3*b^2*c - 2*a*b*d + a^2*e))/x^3 + (6*a*(-(b^3*c) + a*b^2*
d - a^2*b*e + a^3*f))/(a + b*x^3) + 18*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f)*Log[x] + 6*(4*b^3*c - 3*a*b^
2*d + 2*a^2*b*e - a^3*f)*Log[a + b*x^3])/(18*a^5)

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 229, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{2}}}+{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) e}{3\,{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{a}^{4}}}+{\frac{4\,{b}^{3}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{5}}}+{\frac{f}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{be}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}d}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{{b}^{3}c}{3\,{a}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{9\,{a}^{2}{x}^{9}}}-{\frac{d}{6\,{a}^{2}{x}^{6}}}+{\frac{bc}{3\,{a}^{3}{x}^{6}}}-{\frac{e}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bd}{3\,{a}^{3}{x}^{3}}}-{\frac{{b}^{2}c}{{a}^{4}{x}^{3}}}+{\frac{\ln \left ( x \right ) f}{{a}^{2}}}-2\,{\frac{\ln \left ( x \right ) be}{{a}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{4}}}-4\,{\frac{\ln \left ( x \right ){b}^{3}c}{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04895, size = 244, normalized size = 1.39 \begin{align*} -\frac{6 \,{\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} x^{9} + 3 \,{\left (4 \, a b^{2} c - 3 \, a^{2} b d + 2 \, a^{3} e\right )} x^{6} + 2 \, a^{3} c -{\left (4 \, a^{2} b c - 3 \, a^{3} d\right )} x^{3}}{18 \,{\left (a^{4} b x^{12} + a^{5} x^{9}\right )}} + \frac{{\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} - \frac{{\left (4 \, b^{3} c - 3 \, a b^{2} d + 2 \, a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.6199, size = 549, normalized size = 3.14 \begin{align*} -\frac{6 \,{\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9} + 3 \,{\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d + 2 \, a^{4} e\right )} x^{6} + 2 \, a^{4} c -{\left (4 \, a^{3} b c - 3 \, a^{4} d\right )} x^{3} - 6 \,{\left ({\left (4 \, b^{4} c - 3 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} +{\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left ({\left (4 \, b^{4} c - 3 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} +{\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d + 2 \, a^{3} b e - a^{4} f\right )} x^{9}\right )} \log \left (x\right )}{18 \,{\left (a^{5} b x^{12} + a^{6} x^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**10/(b*x**3+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.06707, size = 371, normalized size = 2.12 \begin{align*} -\frac{{\left (4 \, b^{3} c - 3 \, a b^{2} d - a^{3} f + 2 \, a^{2} b e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (4 \, b^{4} c - 3 \, a b^{3} d - a^{3} b f + 2 \, a^{2} b^{2} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac{4 \, b^{4} c x^{3} - 3 \, a b^{3} d x^{3} - a^{3} b f x^{3} + 2 \, a^{2} b^{2} x^{3} e + 5 \, a b^{3} c - 4 \, a^{2} b^{2} d - 2 \, a^{4} f + 3 \, a^{3} b e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{44 \, b^{3} c x^{9} - 33 \, a b^{2} d x^{9} - 11 \, a^{3} f x^{9} + 22 \, a^{2} b x^{9} e - 18 \, a b^{2} c x^{6} + 12 \, a^{2} b d x^{6} - 6 \, a^{3} x^{6} e + 6 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{5} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-(4*b^3*c - 3*a*b^2*d - a^3*f + 2*a^2*b*e)*log(abs(x))/a^5 + 1/3*(4*b^4*c - 3*a*b^3*d - a^3*b*f + 2*a^2*b^2*e)
*log(abs(b*x^3 + a))/(a^5*b) - 1/3*(4*b^4*c*x^3 - 3*a*b^3*d*x^3 - a^3*b*f*x^3 + 2*a^2*b^2*x^3*e + 5*a*b^3*c -
4*a^2*b^2*d - 2*a^4*f + 3*a^3*b*e)/((b*x^3 + a)*a^5) + 1/18*(44*b^3*c*x^9 - 33*a*b^2*d*x^9 - 11*a^3*f*x^9 + 22
*a^2*b*x^9*e - 18*a*b^2*c*x^6 + 12*a^2*b*d*x^6 - 6*a^3*x^6*e + 6*a^2*b*c*x^3 - 3*a^3*d*x^3 - 2*a^3*c)/(a^5*x^9
)