3.279 \(\int \frac{x^5 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=146 \[ -\frac{3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.200995, antiderivative size = 146, 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} \[ -\frac{3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c}{3 b^5 \left (a+b x^3\right )}+\frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^5 \left (a+b x^3\right )^2}+\frac{\log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

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

Mathematica [A]  time = 0.0737187, size = 145, normalized size = 0.99 \[ \frac{a^2 b^2 \left (3 d-4 e x^3-11 f x^6\right )+2 \left (a+b x^3\right )^2 \log \left (a+b x^3\right ) \left (6 a^2 f-3 a b e+b^2 d\right )+a^3 b \left (2 f x^3-5 e\right )+7 a^4 f-a b^3 \left (c-4 x^3 \left (d+e x^3-f x^6\right )\right )+b^4 x^3 \left (-2 c+2 e x^6+f x^9\right )}{6 b^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(7*a^4*f + a^3*b*(-5*e + 2*f*x^3) + a^2*b^2*(3*d - 4*e*x^3 - 11*f*x^6) + b^4*x^3*(-2*c + 2*e*x^6 + f*x^9) - a*
b^3*(c - 4*x^3*(d + e*x^3 - f*x^6)) + 2*(b^2*d - 3*a*b*e + 6*a^2*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^5*(a +
b*x^3)^2)

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Maple [A]  time = 0.013, size = 213, normalized size = 1.5 \begin{align*}{\frac{f{x}^{6}}{6\,{b}^{3}}}-{\frac{a{x}^{3}f}{{b}^{4}}}+{\frac{{x}^{3}e}{3\,{b}^{3}}}-{\frac{{a}^{4}f}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}e}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}d}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ac}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+2\,{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}f}{{b}^{5}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ae}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{3}}}+{\frac{4\,{a}^{3}f}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}e}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ad}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.952873, size = 198, normalized size = 1.36 \begin{align*} -\frac{a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac{b f x^{6} + 2 \,{\left (b e - 3 \, a f\right )} x^{3}}{6 \, b^{4}} + \frac{{\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.25994, size = 470, normalized size = 3.22 \begin{align*} \frac{b^{4} f x^{12} + 2 \,{\left (b^{4} e - 2 \, a b^{3} f\right )} x^{9} +{\left (4 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{6} - a b^{3} c + 3 \, a^{2} b^{2} d - 5 \, a^{3} b e + 7 \, a^{4} f - 2 \,{\left (b^{4} c - 2 \, a b^{3} d + 2 \, a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 2 \,{\left ({\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{6} + a^{2} b^{2} d - 3 \, a^{3} b e + 6 \, a^{4} f + 2 \,{\left (a b^{3} d - 3 \, a^{2} b^{2} e + 6 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

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

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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**5*(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 = 1.08806, size = 197, normalized size = 1.35 \begin{align*} \frac{{\left (b^{2} d + 6 \, a^{2} f - 3 \, a b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} + \frac{b^{3} f x^{6} - 6 \, a b^{2} f x^{3} + 2 \, b^{3} x^{3} e}{6 \, b^{6}} - \frac{a b^{3} c - 3 \, a^{2} b^{2} d - 7 \, a^{4} f + 5 \, a^{3} b e + 2 \,{\left (b^{4} c - 2 \, a b^{3} d - 4 \, a^{3} b f + 3 \, a^{2} b^{2} e\right )} x^{3}}{6 \,{\left (b x^{3} + a\right )}^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

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