3.252 \(\int \frac{x^8 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=180 \[ \frac{x^3 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right ) \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{x^6 \left (3 a^2 f-2 a b e+b^2 d\right )}{6 b^4}+\frac{x^9 (b e-2 a f)}{9 b^3}+\frac{f x^{12}}{12 b^2} \]

[Out]

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

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Rubi [A]  time = 0.26477, antiderivative size = 180, 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} \[ \frac{x^3 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a \log \left (a+b x^3\right ) \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{x^6 \left (3 a^2 f-2 a b e+b^2 d\right )}{6 b^4}+\frac{x^9 (b e-2 a f)}{9 b^3}+\frac{f x^{12}}{12 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^6)/(6*b^4) + ((b*e -
2*a*f)*x^9)/(9*b^3) + (f*x^12)/(12*b^2) - (a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*b^6*(a + b*x^3)) - (a*(
2*b^3*c - 3*a*b^2*d + 4*a^2*b*e - 5*a^3*f)*Log[a + b*x^3])/(3*b^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{x^8 \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^2 \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{b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^2}{b^3}+\frac{f x^3}{b^2}-\frac{a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 (a+b x)^2}+\frac{a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right )}{b^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3}{3 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^6}{6 b^4}+\frac{(b e-2 a f) x^9}{9 b^3}+\frac{f x^{12}}{12 b^2}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 b^6 \left (a+b 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 ) \log \left (a+b x^3\right )}{3 b^6}\\ \end{align*}

Mathematica [A]  time = 0.110585, size = 167, normalized size = 0.93 \[ \frac{12 b x^3 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+\frac{12 a^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+12 a \log \left (a+b x^3\right ) \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )+6 b^2 x^6 \left (3 a^2 f-2 a b e+b^2 d\right )+4 b^3 x^9 (b e-2 a f)+3 b^4 f x^{12}}{36 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(12*b*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3 + 6*b^2*(b^2*d - 2*a*b*e + 3*a^2*f)*x^6 + 4*b^3*(b*e - 2*a
*f)*x^9 + 3*b^4*f*x^12 + (12*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 12*a*(-2*b^3*c + 3*a*b^
2*d - 4*a^2*b*e + 5*a^3*f)*Log[a + b*x^3])/(36*b^6)

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Maple [A]  time = 0.013, size = 240, normalized size = 1.3 \begin{align*}{\frac{f{x}^{12}}{12\,{b}^{2}}}-{\frac{2\,{x}^{9}af}{9\,{b}^{3}}}+{\frac{{x}^{9}e}{9\,{b}^{2}}}+{\frac{{a}^{2}f{x}^{6}}{2\,{b}^{4}}}-{\frac{ae{x}^{6}}{3\,{b}^{3}}}+{\frac{d{x}^{6}}{6\,{b}^{2}}}-{\frac{4\,{a}^{3}f{x}^{3}}{3\,{b}^{5}}}+{\frac{{a}^{2}e{x}^{3}}{{b}^{4}}}-{\frac{2\,ad{x}^{3}}{3\,{b}^{3}}}+{\frac{c{x}^{3}}{3\,{b}^{2}}}+{\frac{5\,{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{6}}}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{5}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{b}^{4}}}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}}+{\frac{{a}^{5}f}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{4}e}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}d}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}c}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.964983, size = 243, normalized size = 1.35 \begin{align*} -\frac{a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f}{3 \,{\left (b^{7} x^{3} + a b^{6}\right )}} + \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - 2 \, a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x^{3}}{36 \, b^{5}} - \frac{{\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/3*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)/(b^7*x^3 + a*b^6) + 1/36*(3*b^3*f*x^12 + 4*(b^3*e - 2*a*b^2*f)*
x^9 + 6*(b^3*d - 2*a*b^2*e + 3*a^2*b*f)*x^6 + 12*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^3)/b^5 - 1/3*(2*a
*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*log(b*x^3 + a)/b^6

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Fricas [A]  time = 1.2588, size = 555, normalized size = 3.08 \begin{align*} \frac{3 \, b^{5} f x^{15} +{\left (4 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \,{\left (3 \, b^{5} d - 4 \, a b^{4} e + 5 \, a^{2} b^{3} f\right )} x^{9} + 6 \,{\left (2 \, b^{5} c - 3 \, a b^{4} d + 4 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{6} - 12 \, a^{2} b^{3} c + 12 \, a^{3} b^{2} d - 12 \, a^{4} b e + 12 \, a^{5} f + 12 \,{\left (a b^{4} c - 2 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 4 \, a^{4} b f\right )} x^{3} - 12 \,{\left (2 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 4 \, a^{4} b e - 5 \, a^{5} f +{\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{36 \,{\left (b^{7} x^{3} + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(3*b^5*f*x^15 + (4*b^5*e - 5*a*b^4*f)*x^12 + 2*(3*b^5*d - 4*a*b^4*e + 5*a^2*b^3*f)*x^9 + 6*(2*b^5*c - 3*a
*b^4*d + 4*a^2*b^3*e - 5*a^3*b^2*f)*x^6 - 12*a^2*b^3*c + 12*a^3*b^2*d - 12*a^4*b*e + 12*a^5*f + 12*(a*b^4*c -
2*a^2*b^3*d + 3*a^3*b^2*e - 4*a^4*b*f)*x^3 - 12*(2*a^2*b^3*c - 3*a^3*b^2*d + 4*a^4*b*e - 5*a^5*f + (2*a*b^4*c
- 3*a^2*b^3*d + 4*a^3*b^2*e - 5*a^4*b*f)*x^3)*log(b*x^3 + a))/(b^7*x^3 + a*b^6)

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Sympy [A]  time = 11.0927, size = 180, normalized size = 1. \begin{align*} \frac{a \left (5 a^{3} f - 4 a^{2} b e + 3 a b^{2} d - 2 b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{6}} + \frac{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c}{3 a b^{6} + 3 b^{7} x^{3}} + \frac{f x^{12}}{12 b^{2}} - \frac{x^{9} \left (2 a f - b e\right )}{9 b^{3}} + \frac{x^{6} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{6 b^{4}} - \frac{x^{3} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{3 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

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

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Giac [A]  time = 1.05787, size = 335, normalized size = 1.86 \begin{align*} -\frac{{\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d - 5 \, a^{4} f + 4 \, a^{3} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} + \frac{2 \, a b^{4} c x^{3} - 3 \, a^{2} b^{3} d x^{3} - 5 \, a^{4} b f x^{3} + 4 \, a^{3} b^{2} x^{3} e + a^{2} b^{3} c - 2 \, a^{3} b^{2} d - 4 \, a^{5} f + 3 \, a^{4} b e}{3 \,{\left (b x^{3} + a\right )} b^{6}} + \frac{3 \, b^{6} f x^{12} - 8 \, a b^{5} f x^{9} + 4 \, b^{6} x^{9} e + 6 \, b^{6} d x^{6} + 18 \, a^{2} b^{4} f x^{6} - 12 \, a b^{5} x^{6} e + 12 \, b^{6} c x^{3} - 24 \, a b^{5} d x^{3} - 48 \, a^{3} b^{3} f x^{3} + 36 \, a^{2} b^{4} x^{3} e}{36 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/3*(2*a*b^3*c - 3*a^2*b^2*d - 5*a^4*f + 4*a^3*b*e)*log(abs(b*x^3 + a))/b^6 + 1/3*(2*a*b^4*c*x^3 - 3*a^2*b^3*
d*x^3 - 5*a^4*b*f*x^3 + 4*a^3*b^2*x^3*e + a^2*b^3*c - 2*a^3*b^2*d - 4*a^5*f + 3*a^4*b*e)/((b*x^3 + a)*b^6) + 1
/36*(3*b^6*f*x^12 - 8*a*b^5*f*x^9 + 4*b^6*x^9*e + 6*b^6*d*x^6 + 18*a^2*b^4*f*x^6 - 12*a*b^5*x^6*e + 12*b^6*c*x
^3 - 24*a*b^5*d*x^3 - 48*a^3*b^3*f*x^3 + 36*a^2*b^4*x^3*e)/b^8