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

Optimal. Leaf size=186 \[ \frac{a \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.268936, antiderivative size = 186, 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} \[ \frac{a \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^6 (b e-3 a f)}{6 b^4}+\frac{f x^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0987472, size = 170, normalized size = 0.91 \[ \frac{-\frac{6 a \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )}{a+b x^3}+\frac{3 a^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+6 \log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+6 b x^3 \left (6 a^2 f-3 a b e+b^2 d\right )+3 b^2 x^6 (b e-3 a f)+2 b^3 f x^9}{18 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*(b^2*d - 3*a*b*e + 6*a^2*f)*x^3 + 3*b^2*(b*e - 3*a*f)*x^6 + 2*b^3*f*x^9 + (3*a^2*(-(b^3*c) + a*b^2*d - a^
2*b*e + a^3*f))/(a + b*x^3)^2 - (6*a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f))/(a + b*x^3) + 6*(b^3*c - 3*
a*b^2*d + 6*a^2*b*e - 10*a^3*f)*Log[a + b*x^3])/(18*b^6)

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Maple [A]  time = 0.012, size = 266, normalized size = 1.4 \begin{align*}{\frac{f{x}^{9}}{9\,{b}^{3}}}-{\frac{{x}^{6}af}{2\,{b}^{4}}}+{\frac{e{x}^{6}}{6\,{b}^{3}}}+2\,{\frac{{a}^{2}f{x}^{3}}{{b}^{5}}}-{\frac{ae{x}^{3}}{{b}^{4}}}+{\frac{d{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{5}f}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}e}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}d}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}c}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{10\,\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{6}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ad}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}}-{\frac{5\,{a}^{4}f}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}e}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}d}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ac}{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)^3,x)

[Out]

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

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Maxima [A]  time = 0.95037, size = 258, normalized size = 1.39 \begin{align*} \frac{3 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 7 \, a^{4} b e - 9 \, a^{5} f + 2 \,{\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}}{6 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - 3 \, a b f\right )} x^{6} + 6 \,{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x^{3}}{18 \, b^{5}} + \frac{{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29363, size = 635, normalized size = 3.41 \begin{align*} \frac{2 \, b^{5} f x^{15} +{\left (3 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \,{\left (3 \, b^{5} d - 6 \, a b^{4} e + 10 \, a^{2} b^{3} f\right )} x^{9} + 3 \,{\left (4 \, a b^{4} d - 11 \, a^{2} b^{3} e + 21 \, a^{3} b^{2} f\right )} x^{6} + 9 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 21 \, a^{4} b e - 27 \, a^{5} f + 6 \,{\left (2 \, a b^{4} c - 2 \, a^{2} b^{3} d + a^{3} b^{2} e + a^{4} b f\right )} x^{3} + 6 \,{\left ({\left (b^{5} c - 3 \, a b^{4} d + 6 \, a^{2} b^{3} e - 10 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 6 \, a^{4} b e - 10 \, a^{5} f + 2 \,{\left (a b^{4} c - 3 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 10 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} 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)^3,x, algorithm="fricas")

[Out]

1/18*(2*b^5*f*x^15 + (3*b^5*e - 5*a*b^4*f)*x^12 + 2*(3*b^5*d - 6*a*b^4*e + 10*a^2*b^3*f)*x^9 + 3*(4*a*b^4*d -
11*a^2*b^3*e + 21*a^3*b^2*f)*x^6 + 9*a^2*b^3*c - 15*a^3*b^2*d + 21*a^4*b*e - 27*a^5*f + 6*(2*a*b^4*c - 2*a^2*b
^3*d + a^3*b^2*e + a^4*b*f)*x^3 + 6*((b^5*c - 3*a*b^4*d + 6*a^2*b^3*e - 10*a^3*b^2*f)*x^6 + a^2*b^3*c - 3*a^3*
b^2*d + 6*a^4*b*e - 10*a^5*f + 2*(a*b^4*c - 3*a^2*b^3*d + 6*a^3*b^2*e - 10*a^4*b*f)*x^3)*log(b*x^3 + a))/(b^8*
x^6 + 2*a*b^7*x^3 + a^2*b^6)

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

[Out]

Timed out

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Giac [A]  time = 1.07923, size = 319, normalized size = 1.72 \begin{align*} \frac{{\left (b^{3} c - 3 \, a b^{2} d - 10 \, a^{3} f + 6 \, a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} - \frac{3 \, b^{5} c x^{6} - 9 \, a b^{4} d x^{6} - 30 \, a^{3} b^{2} f x^{6} + 18 \, a^{2} b^{3} x^{6} e + 2 \, a b^{4} c x^{3} - 12 \, a^{2} b^{3} d x^{3} - 50 \, a^{4} b f x^{3} + 28 \, a^{3} b^{2} x^{3} e - 4 \, a^{3} b^{2} d - 21 \, a^{5} f + 11 \, a^{4} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} f x^{9} - 9 \, a b^{5} f x^{6} + 3 \, b^{6} x^{6} e + 6 \, b^{6} d x^{3} + 36 \, a^{2} b^{4} f x^{3} - 18 \, a b^{5} x^{3} e}{18 \, b^{9}} \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)^3,x, algorithm="giac")

[Out]

1/3*(b^3*c - 3*a*b^2*d - 10*a^3*f + 6*a^2*b*e)*log(abs(b*x^3 + a))/b^6 - 1/6*(3*b^5*c*x^6 - 9*a*b^4*d*x^6 - 30
*a^3*b^2*f*x^6 + 18*a^2*b^3*x^6*e + 2*a*b^4*c*x^3 - 12*a^2*b^3*d*x^3 - 50*a^4*b*f*x^3 + 28*a^3*b^2*x^3*e - 4*a
^3*b^2*d - 21*a^5*f + 11*a^4*b*e)/((b*x^3 + a)^2*b^6) + 1/18*(2*b^6*f*x^9 - 9*a*b^5*f*x^6 + 3*b^6*x^6*e + 6*b^
6*d*x^3 + 36*a^2*b^4*f*x^3 - 18*a*b^5*x^3*e)/b^9