3.257 \(\int \frac{c+d x^3+e x^6+f x^9}{x^7 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=130 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-2 a b d+3 b^2 c\right )}{3 a^4}+\frac{\log (x) \left (a^2 e-2 a b d+3 b^2 c\right )}{a^4}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{6 a^2 x^6} \]

[Out]

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

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Rubi [A]  time = 0.153838, antiderivative size = 130, 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} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^3 b \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-2 a b d+3 b^2 c\right )}{3 a^4}+\frac{\log (x) \left (a^2 e-2 a b d+3 b^2 c\right )}{a^4}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{6 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.114931, size = 118, normalized size = 0.91 \[ -\frac{\frac{2 a \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}+2 \log \left (a+b x^3\right ) \left (a^2 e-2 a b d+3 b^2 c\right )-6 \log (x) \left (a^2 e-2 a b d+3 b^2 c\right )+\frac{a^2 c}{x^6}+\frac{2 a (a d-2 b c)}{x^3}}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^7*(a + b*x^3)^2),x]

[Out]

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

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Maple [A]  time = 0.015, size = 167, normalized size = 1.3 \begin{align*} -{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}}+{\frac{2\,\ln \left ( b{x}^{3}+a \right ) bd}{3\,{a}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{2}c}{{a}^{4}}}-{\frac{f}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{e}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{bd}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}c}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{6\,{a}^{2}{x}^{6}}}-{\frac{d}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bc}{3\,{a}^{3}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-2\,{\frac{\ln \left ( x \right ) bd}{{a}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{4}}} \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^7/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.979299, size = 186, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (3 \, b^{3} c - 2 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - a^{2} b c +{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{3}}{6 \,{\left (a^{3} b^{2} x^{9} + a^{4} b x^{6}\right )}} - \frac{{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} + \frac{{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \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^7/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42823, size = 431, normalized size = 3.32 \begin{align*} \frac{2 \,{\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c +{\left (3 \, a^{2} b^{2} c - 2 \, a^{3} b d\right )} x^{3} - 2 \,{\left ({\left (3 \, b^{4} c - 2 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} +{\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (3 \, b^{4} c - 2 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} +{\left (3 \, a b^{3} c - 2 \, a^{2} b^{2} d + a^{3} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + a^{5} b x^{6}\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^7/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.07027, size = 271, normalized size = 2.08 \begin{align*} \frac{{\left (3 \, b^{2} c - 2 \, a b d + a^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (3 \, b^{3} c - 2 \, a b^{2} d + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac{3 \, b^{4} c x^{3} - 2 \, a b^{3} d x^{3} + a^{2} b^{2} x^{3} e + 4 \, a b^{3} c - 3 \, a^{2} b^{2} d - a^{4} f + 2 \, a^{3} b e}{3 \,{\left (b x^{3} + a\right )} a^{4} b} - \frac{9 \, b^{2} c x^{6} - 6 \, a b d x^{6} + 3 \, a^{2} x^{6} e - 4 \, a b c x^{3} + 2 \, a^{2} d x^{3} + a^{2} c}{6 \, a^{4} x^{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^7/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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