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

Optimal. Leaf size=218 \[ -\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 \left (a+b x^3\right )}-\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^4 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c\right )}{3 a^6}-\frac{\log (x) \left (3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c\right )}{a^6}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{3 b c-a d}{6 a^4 x^6}-\frac{c}{9 a^3 x^9} \]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(6*a^4*x^6) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) - (b^3*c - a*b^2*d + a^2*
b*e - a^3*f)/(6*a^4*(a + b*x^3)^2) - (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*(a + b*x^3)) - ((10*b^3*
c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[x])/a^6 + ((10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])/(
3*a^6)

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Rubi [A]  time = 0.263711, antiderivative size = 218, 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} \[ -\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 \left (a+b x^3\right )}-\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^4 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c\right )}{3 a^6}-\frac{\log (x) \left (3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c\right )}{a^6}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{3 b c-a d}{6 a^4 x^6}-\frac{c}{9 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(6*a^4*x^6) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) - (b^3*c - a*b^2*d + a^2*
b*e - a^3*f)/(6*a^4*(a + b*x^3)^2) - (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*(a + b*x^3)) - ((10*b^3*
c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[x])/a^6 + ((10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])/(
3*a^6)

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

Mathematica [A]  time = 0.137317, size = 200, normalized size = 0.92 \[ \frac{\frac{3 a^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{6 a \left (-2 a^2 b e+a^3 f+3 a b^2 d-4 b^3 c\right )}{a+b x^3}+6 \log \left (a+b x^3\right ) \left (3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c\right )+18 \log (x) \left (-3 a^2 b e+a^3 f+6 a b^2 d-10 b^3 c\right )-\frac{6 a \left (a^2 e-3 a b d+6 b^2 c\right )}{x^3}-\frac{3 a^2 (a d-3 b c)}{x^6}-\frac{2 a^3 c}{x^9}}{18 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a^3*c)/x^9 - (3*a^2*(-3*b*c + a*d))/x^6 - (6*a*(6*b^2*c - 3*a*b*d + a^2*e))/x^3 + (3*a^2*(-(b^3*c) + a*b^
2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 + (6*a*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/(a + b*x^3) + 18*(-10
*b^3*c + 6*a*b^2*d - 3*a^2*b*e + a^3*f)*Log[x] + 6*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])/
(18*a^6)

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Maple [A]  time = 0.02, size = 293, normalized size = 1.3 \begin{align*}{\frac{f}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{be}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{b}^{2}d}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{b}^{3}c}{6\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{3}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) e}{{a}^{4}}}-2\,{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{a}^{5}}}+{\frac{10\,{b}^{3}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{6}}}+{\frac{f}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,be}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}d}{{a}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,{b}^{3}c}{3\,{a}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{9\,{a}^{3}{x}^{9}}}-{\frac{d}{6\,{a}^{3}{x}^{6}}}+{\frac{bc}{2\,{a}^{4}{x}^{6}}}-{\frac{e}{3\,{a}^{3}{x}^{3}}}+{\frac{bd}{{a}^{4}{x}^{3}}}-2\,{\frac{{b}^{2}c}{{a}^{5}{x}^{3}}}+{\frac{\ln \left ( x \right ) f}{{a}^{3}}}-3\,{\frac{\ln \left ( x \right ) be}{{a}^{4}}}+6\,{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{5}}}-10\,{\frac{\ln \left ( x \right ){b}^{3}c}{{a}^{6}}} \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)^3,x)

[Out]

1/6/a/(b*x^3+a)^2*f-1/6*b/a^2/(b*x^3+a)^2*e+1/6*b^2/a^3/(b*x^3+a)^2*d-1/6*b^3/a^4/(b*x^3+a)^2*c-1/3/a^3*ln(b*x
^3+a)*f+b/a^4*ln(b*x^3+a)*e-2*b^2/a^5*ln(b*x^3+a)*d+10/3*b^3/a^6*ln(b*x^3+a)*c+1/3/a^2/(b*x^3+a)*f-2/3*b/a^3/(
b*x^3+a)*e+b^2/a^4/(b*x^3+a)*d-4/3*b^3/a^5/(b*x^3+a)*c-1/9*c/a^3/x^9-1/6/a^3/x^6*d+1/2/a^4/x^6*b*c-1/3/a^3/x^3
*e+1/a^4/x^3*b*d-2/a^5/x^3*b^2*c+1/a^3*ln(x)*f-3/a^4*ln(x)*b*e+6/a^5*ln(x)*b^2*d-10/a^6*ln(x)*b^3*c

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Maxima [A]  time = 0.980614, size = 313, normalized size = 1.44 \begin{align*} -\frac{6 \,{\left (10 \, b^{4} c - 6 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} + 9 \,{\left (10 \, a b^{3} c - 6 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{9} + 2 \,{\left (10 \, a^{2} b^{2} c - 6 \, a^{3} b d + 3 \, a^{4} e\right )} x^{6} + 2 \, a^{4} c -{\left (5 \, a^{3} b c - 3 \, a^{4} d\right )} x^{3}}{18 \,{\left (a^{5} b^{2} x^{15} + 2 \, a^{6} b x^{12} + a^{7} x^{9}\right )}} + \frac{{\left (10 \, b^{3} c - 6 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} - \frac{{\left (10 \, b^{3} c - 6 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} \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)^3,x, algorithm="maxima")

[Out]

-1/18*(6*(10*b^4*c - 6*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^12 + 9*(10*a*b^3*c - 6*a^2*b^2*d + 3*a^3*b*e - a^4*f
)*x^9 + 2*(10*a^2*b^2*c - 6*a^3*b*d + 3*a^4*e)*x^6 + 2*a^4*c - (5*a^3*b*c - 3*a^4*d)*x^3)/(a^5*b^2*x^15 + 2*a^
6*b*x^12 + a^7*x^9) + 1/3*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*log(b*x^3 + a)/a^6 - 1/3*(10*b^3*c - 6*a*
b^2*d + 3*a^2*b*e - a^3*f)*log(x^3)/a^6

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Fricas [A]  time = 1.44256, size = 837, normalized size = 3.84 \begin{align*} -\frac{6 \,{\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 9 \,{\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9} + 2 \,{\left (10 \, a^{3} b^{2} c - 6 \, a^{4} b d + 3 \, a^{5} e\right )} x^{6} + 2 \, a^{5} c -{\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} x^{3} - 6 \,{\left ({\left (10 \, b^{5} c - 6 \, a b^{4} d + 3 \, a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} + 2 \,{\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} +{\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left ({\left (10 \, b^{5} c - 6 \, a b^{4} d + 3 \, a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} + 2 \,{\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} +{\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9}\right )} \log \left (x\right )}{18 \,{\left (a^{6} b^{2} x^{15} + 2 \, a^{7} b x^{12} + a^{8} 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)^3,x, algorithm="fricas")

[Out]

-1/18*(6*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + 9*(10*a^2*b^3*c - 6*a^3*b^2*d + 3*a^4*b*e -
a^5*f)*x^9 + 2*(10*a^3*b^2*c - 6*a^4*b*d + 3*a^5*e)*x^6 + 2*a^5*c - (5*a^4*b*c - 3*a^5*d)*x^3 - 6*((10*b^5*c
- 6*a*b^4*d + 3*a^2*b^3*e - a^3*b^2*f)*x^15 + 2*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + (10*
a^2*b^3*c - 6*a^3*b^2*d + 3*a^4*b*e - a^5*f)*x^9)*log(b*x^3 + a) + 18*((10*b^5*c - 6*a*b^4*d + 3*a^2*b^3*e - a
^3*b^2*f)*x^15 + 2*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + (10*a^2*b^3*c - 6*a^3*b^2*d + 3*a
^4*b*e - a^5*f)*x^9)*log(x))/(a^6*b^2*x^15 + 2*a^7*b*x^12 + a^8*x^9)

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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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.1005, size = 437, normalized size = 2. \begin{align*} -\frac{{\left (10 \, b^{3} c - 6 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (10 \, b^{4} c - 6 \, a b^{3} d - a^{3} b f + 3 \, a^{2} b^{2} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} - \frac{30 \, b^{5} c x^{6} - 18 \, a b^{4} d x^{6} - 3 \, a^{3} b^{2} f x^{6} + 9 \, a^{2} b^{3} x^{6} e + 68 \, a b^{4} c x^{3} - 42 \, a^{2} b^{3} d x^{3} - 8 \, a^{4} b f x^{3} + 22 \, a^{3} b^{2} x^{3} e + 39 \, a^{2} b^{3} c - 25 \, a^{3} b^{2} d - 6 \, a^{5} f + 14 \, a^{4} b e}{6 \,{\left (b x^{3} + a\right )}^{2} a^{6}} + \frac{110 \, b^{3} c x^{9} - 66 \, a b^{2} d x^{9} - 11 \, a^{3} f x^{9} + 33 \, a^{2} b x^{9} e - 36 \, a b^{2} c x^{6} + 18 \, a^{2} b d x^{6} - 6 \, a^{3} x^{6} e + 9 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{6} 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)^3,x, algorithm="giac")

[Out]

-(10*b^3*c - 6*a*b^2*d - a^3*f + 3*a^2*b*e)*log(abs(x))/a^6 + 1/3*(10*b^4*c - 6*a*b^3*d - a^3*b*f + 3*a^2*b^2*
e)*log(abs(b*x^3 + a))/(a^6*b) - 1/6*(30*b^5*c*x^6 - 18*a*b^4*d*x^6 - 3*a^3*b^2*f*x^6 + 9*a^2*b^3*x^6*e + 68*a
*b^4*c*x^3 - 42*a^2*b^3*d*x^3 - 8*a^4*b*f*x^3 + 22*a^3*b^2*x^3*e + 39*a^2*b^3*c - 25*a^3*b^2*d - 6*a^5*f + 14*
a^4*b*e)/((b*x^3 + a)^2*a^6) + 1/18*(110*b^3*c*x^9 - 66*a*b^2*d*x^9 - 11*a^3*f*x^9 + 33*a^2*b*x^9*e - 36*a*b^2
*c*x^6 + 18*a^2*b*d*x^6 - 6*a^3*x^6*e + 9*a^2*b*c*x^3 - 3*a^3*d*x^3 - 2*a^3*c)/(a^6*x^9)