3.251 \(\int \frac{x^{11} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=220 \[ \frac{x^6 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{6 b^5}-\frac{a x^3 \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^2 \log \left (a+b x^3\right ) \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^9 \left (3 a^2 f-2 a b e+b^2 d\right )}{9 b^4}+\frac{x^{12} (b e-2 a f)}{12 b^3}+\frac{f x^{15}}{15 b^2} \]

[Out]

-(a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x^3)/(3*b^6) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^6)
/(6*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^9)/(9*b^4) + ((b*e - 2*a*f)*x^12)/(12*b^3) + (f*x^15)/(15*b^2) + (a^
3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*b^7*(a + b*x^3)) + (a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f)*
Log[a + b*x^3])/(3*b^7)

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Rubi [A]  time = 0.341178, antiderivative size = 220, 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} \[ \frac{x^6 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{6 b^5}-\frac{a x^3 \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^2 \log \left (a+b x^3\right ) \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^9 \left (3 a^2 f-2 a b e+b^2 d\right )}{9 b^4}+\frac{x^{12} (b e-2 a f)}{12 b^3}+\frac{f x^{15}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

-(a*(2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*x^3)/(3*b^6) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^6)
/(6*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^9)/(9*b^4) + ((b*e - 2*a*f)*x^12)/(12*b^3) + (f*x^15)/(15*b^2) + (a^
3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*b^7*(a + b*x^3)) + (a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f)*
Log[a + b*x^3])/(3*b^7)

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{x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right )}{b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{b^4}+\frac{(b e-2 a f) x^3}{b^3}+\frac{f x^4}{b^2}+\frac{a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6 (a+b x)^2}-\frac{a^2 \left (-3 b^3 c+4 a b^2 d-5 a^2 b e+6 a^3 f\right )}{b^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x^3}{3 b^6}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^6}{6 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^9}{9 b^4}+\frac{(b e-2 a f) x^{12}}{12 b^3}+\frac{f x^{15}}{15 b^2}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7}\\ \end{align*}

Mathematica [A]  time = 0.138523, size = 205, normalized size = 0.93 \[ \frac{30 b^2 x^6 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+60 a b x^3 \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )-\frac{60 a^3 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+60 a^2 \log \left (a+b x^3\right ) \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )+20 b^3 x^9 \left (3 a^2 f-2 a b e+b^2 d\right )+15 b^4 x^{12} (b e-2 a f)+12 b^5 f x^{15}}{180 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(60*a*b*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x^3 + 30*b^2*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^
6 + 20*b^3*(b^2*d - 2*a*b*e + 3*a^2*f)*x^9 + 15*b^4*(b*e - 2*a*f)*x^12 + 12*b^5*f*x^15 - (60*a^3*(-(b^3*c) + a
*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 60*a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f)*Log[a + b*x^3])/(1
80*b^7)

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Maple [A]  time = 0.012, size = 288, normalized size = 1.3 \begin{align*}{\frac{f{x}^{15}}{15\,{b}^{2}}}-{\frac{{x}^{12}af}{6\,{b}^{3}}}+{\frac{{x}^{12}e}{12\,{b}^{2}}}+{\frac{{x}^{9}{a}^{2}f}{3\,{b}^{4}}}-{\frac{2\,{x}^{9}ae}{9\,{b}^{3}}}+{\frac{{x}^{9}d}{9\,{b}^{2}}}-{\frac{2\,{a}^{3}f{x}^{6}}{3\,{b}^{5}}}+{\frac{{a}^{2}e{x}^{6}}{2\,{b}^{4}}}-{\frac{ad{x}^{6}}{3\,{b}^{3}}}+{\frac{{x}^{6}c}{6\,{b}^{2}}}+{\frac{5\,{a}^{4}f{x}^{3}}{3\,{b}^{6}}}-{\frac{4\,{a}^{3}e{x}^{3}}{3\,{b}^{5}}}+{\frac{{a}^{2}d{x}^{3}}{{b}^{4}}}-{\frac{2\,ac{x}^{3}}{3\,{b}^{3}}}-2\,{\frac{{a}^{5}\ln \left ( b{x}^{3}+a \right ) f}{{b}^{7}}}+{\frac{5\,{a}^{4}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{6}}}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{5}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) c}{{b}^{4}}}-{\frac{{a}^{6}f}{3\,{b}^{7} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{5}e}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{4}d}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}c}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.958348, size = 300, normalized size = 1.36 \begin{align*} \frac{a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f}{3 \,{\left (b^{8} x^{3} + a b^{7}\right )}} + \frac{12 \, b^{4} f x^{15} + 15 \,{\left (b^{4} e - 2 \, a b^{3} f\right )} x^{12} + 20 \,{\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{9} + 30 \,{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{6} - 60 \,{\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x^{3}}{180 \, b^{6}} + \frac{{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 5 \, a^{4} b e - 6 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)/(b^8*x^3 + a*b^7) + 1/180*(12*b^4*f*x^15 + 15*(b^4*e - 2*a*b^3*f
)*x^12 + 20*(b^4*d - 2*a*b^3*e + 3*a^2*b^2*f)*x^9 + 30*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^6 - 60*
(2*a*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x^3)/b^6 + 1/3*(3*a^2*b^3*c - 4*a^3*b^2*d + 5*a^4*b*e - 6*a^5*
f)*log(b*x^3 + a)/b^7

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Fricas [A]  time = 1.244, size = 656, normalized size = 2.98 \begin{align*} \frac{12 \, b^{6} f x^{18} + 3 \,{\left (5 \, b^{6} e - 6 \, a b^{5} f\right )} x^{15} + 5 \,{\left (4 \, b^{6} d - 5 \, a b^{5} e + 6 \, a^{2} b^{4} f\right )} x^{12} + 10 \,{\left (3 \, b^{6} c - 4 \, a b^{5} d + 5 \, a^{2} b^{4} e - 6 \, a^{3} b^{3} f\right )} x^{9} + 60 \, a^{3} b^{3} c - 60 \, a^{4} b^{2} d + 60 \, a^{5} b e - 60 \, a^{6} f - 30 \,{\left (3 \, a b^{5} c - 4 \, a^{2} b^{4} d + 5 \, a^{3} b^{3} e - 6 \, a^{4} b^{2} f\right )} x^{6} - 60 \,{\left (2 \, a^{2} b^{4} c - 3 \, a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 5 \, a^{5} b f\right )} x^{3} + 60 \,{\left (3 \, a^{3} b^{3} c - 4 \, a^{4} b^{2} d + 5 \, a^{5} b e - 6 \, a^{6} f +{\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \,{\left (b^{8} x^{3} + a b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/180*(12*b^6*f*x^18 + 3*(5*b^6*e - 6*a*b^5*f)*x^15 + 5*(4*b^6*d - 5*a*b^5*e + 6*a^2*b^4*f)*x^12 + 10*(3*b^6*c
- 4*a*b^5*d + 5*a^2*b^4*e - 6*a^3*b^3*f)*x^9 + 60*a^3*b^3*c - 60*a^4*b^2*d + 60*a^5*b*e - 60*a^6*f - 30*(3*a*
b^5*c - 4*a^2*b^4*d + 5*a^3*b^3*e - 6*a^4*b^2*f)*x^6 - 60*(2*a^2*b^4*c - 3*a^3*b^3*d + 4*a^4*b^2*e - 5*a^5*b*f
)*x^3 + 60*(3*a^3*b^3*c - 4*a^4*b^2*d + 5*a^5*b*e - 6*a^6*f + (3*a^2*b^4*c - 4*a^3*b^3*d + 5*a^4*b^2*e - 6*a^5
*b*f)*x^3)*log(b*x^3 + a))/(b^8*x^3 + a*b^7)

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Sympy [A]  time = 11.5094, size = 224, normalized size = 1.02 \begin{align*} - \frac{a^{2} \left (6 a^{3} f - 5 a^{2} b e + 4 a b^{2} d - 3 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{7}} - \frac{a^{6} f - a^{5} b e + a^{4} b^{2} d - a^{3} b^{3} c}{3 a b^{7} + 3 b^{8} x^{3}} + \frac{f x^{15}}{15 b^{2}} - \frac{x^{12} \left (2 a f - b e\right )}{12 b^{3}} + \frac{x^{9} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{9 b^{4}} - \frac{x^{6} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{6 b^{5}} + \frac{x^{3} \left (5 a^{4} f - 4 a^{3} b e + 3 a^{2} b^{2} d - 2 a b^{3} c\right )}{3 b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

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

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Giac [A]  time = 1.06618, size = 405, normalized size = 1.84 \begin{align*} \frac{{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d - 6 \, a^{5} f + 5 \, a^{4} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} - \frac{3 \, a^{2} b^{4} c x^{3} - 4 \, a^{3} b^{3} d x^{3} - 6 \, a^{5} b f x^{3} + 5 \, a^{4} b^{2} x^{3} e + 2 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d - 5 \, a^{6} f + 4 \, a^{5} b e}{3 \,{\left (b x^{3} + a\right )} b^{7}} + \frac{12 \, b^{8} f x^{15} - 30 \, a b^{7} f x^{12} + 15 \, b^{8} x^{12} e + 20 \, b^{8} d x^{9} + 60 \, a^{2} b^{6} f x^{9} - 40 \, a b^{7} x^{9} e + 30 \, b^{8} c x^{6} - 60 \, a b^{7} d x^{6} - 120 \, a^{3} b^{5} f x^{6} + 90 \, a^{2} b^{6} x^{6} e - 120 \, a b^{7} c x^{3} + 180 \, a^{2} b^{6} d x^{3} + 300 \, a^{4} b^{4} f x^{3} - 240 \, a^{3} b^{5} x^{3} e}{180 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/3*(3*a^2*b^3*c - 4*a^3*b^2*d - 6*a^5*f + 5*a^4*b*e)*log(abs(b*x^3 + a))/b^7 - 1/3*(3*a^2*b^4*c*x^3 - 4*a^3*b
^3*d*x^3 - 6*a^5*b*f*x^3 + 5*a^4*b^2*x^3*e + 2*a^3*b^3*c - 3*a^4*b^2*d - 5*a^6*f + 4*a^5*b*e)/((b*x^3 + a)*b^7
) + 1/180*(12*b^8*f*x^15 - 30*a*b^7*f*x^12 + 15*b^8*x^12*e + 20*b^8*d*x^9 + 60*a^2*b^6*f*x^9 - 40*a*b^7*x^9*e
+ 30*b^8*c*x^6 - 60*a*b^7*d*x^6 - 120*a^3*b^5*f*x^6 + 90*a^2*b^6*x^6*e - 120*a*b^7*c*x^3 + 180*a^2*b^6*d*x^3 +
300*a^4*b^4*f*x^3 - 240*a^3*b^5*x^3*e)/b^10