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

Optimal. Leaf size=100 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (-a^2 b e+2 a^3 f+b^3 c\right )}{3 a^2 b^3}+\frac{c \log (x)}{a^2}+\frac{f x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.125302, antiderivative size = 100, 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} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a b^3 \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (-a^2 b e+2 a^3 f+b^3 c\right )}{3 a^2 b^3}+\frac{c \log (x)}{a^2}+\frac{f x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.122146, size = 95, normalized size = 0.95 \[ \frac{\frac{\frac{a \left (a^2 b \left (e+f x^3\right )+a^3 (-f)+a b^2 \left (f x^6-d\right )+b^3 c\right )}{a+b x^3}+\log \left (a+b x^3\right ) \left (a^2 b e-2 a^3 f-b^3 c\right )}{b^3}+3 c \log (x)}{3 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 125, normalized size = 1.3 \begin{align*}{\frac{f{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{2}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}}-{\frac{{a}^{2}f}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ae}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{d}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{c}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{c\ln \left ( x \right ) }{{a}^{2}}} \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/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.95645, size = 135, normalized size = 1.35 \begin{align*} \frac{f x^{3}}{3 \, b^{2}} + \frac{b^{3} c - a b^{2} d + a^{2} b e - a^{3} f}{3 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac{c \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{{\left (b^{3} c - a^{2} b e + 2 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{3}} \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/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43537, size = 293, normalized size = 2.93 \begin{align*} \frac{a^{2} b^{2} f x^{6} + a^{3} b f x^{3} + a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f -{\left (a b^{3} c - a^{3} b e + 2 \, a^{4} f +{\left (b^{4} c - a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \,{\left (b^{4} c x^{3} + a b^{3} c\right )} \log \left (x\right )}{3 \,{\left (a^{2} b^{4} x^{3} + a^{3} b^{3}\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/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 33.8253, size = 95, normalized size = 0.95 \begin{align*} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \frac{f x^{3}}{3 b^{2}} + \frac{c \log{\left (x \right )}}{a^{2}} - \frac{\left (2 a^{3} f - a^{2} b e + b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2} b^{3}} \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/(b*x**3+a)**2,x)

[Out]

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

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Giac [A]  time = 1.07303, size = 169, normalized size = 1.69 \begin{align*} \frac{f x^{3}}{3 \, b^{2}} + \frac{c \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (b^{3} c + 2 \, a^{3} f - a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{3}} + \frac{b^{4} c x^{3} + 2 \, a^{3} b f x^{3} - a^{2} b^{2} x^{3} e + 2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f}{3 \,{\left (b x^{3} + a\right )} a^{2} b^{3}} \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/(b*x^3+a)^2,x, algorithm="giac")

[Out]

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