3.228 \(\int \frac{c+d x^3+e x^6+f x^9}{x^4 (a+b x^3)} \, dx\)

Optimal. Leaf size=81 \[ \frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{3 a x^3}+\frac{f x^3}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.116136, antiderivative size = 81, 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} \[ \frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^2 b^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{3 a x^3}+\frac{f x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)),x]

[Out]

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

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

Mathematica [A]  time = 0.0434588, size = 77, normalized size = 0.95 \[ \frac{1}{3} \left (\frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^2 b^2}+\frac{3 \log (x) (a d-b c)}{a^2}-\frac{c}{a x^3}+\frac{f x^3}{b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^4*(a + b*x^3)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 94, normalized size = 1.2 \begin{align*}{\frac{f{x}^{3}}{3\,b}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{2}}}+{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,b}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{2}}}-{\frac{c}{3\,a{x}^{3}}}+{\frac{d\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bc}{{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^4/(b*x^3+a),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.966186, size = 104, normalized size = 1.28 \begin{align*} \frac{f x^{3}}{3 \, b} - \frac{{\left (b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{2}} + \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2} b^{2}} - \frac{c}{3 \, a x^{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^4/(b*x^3+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.50273, size = 180, normalized size = 2.22 \begin{align*} \frac{a^{2} b f x^{6} +{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \log \left (b x^{3} + a\right ) - 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (x\right ) - a b^{2} c}{3 \, a^{2} b^{2} x^{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^4/(b*x^3+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 11.4391, size = 70, normalized size = 0.86 \begin{align*} \frac{f x^{3}}{3 b} - \frac{c}{3 a x^{3}} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2} b^{2}} \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**4/(b*x**3+a),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.05868, size = 128, normalized size = 1.58 \begin{align*} \frac{f x^{3}}{3 \, b} - \frac{{\left (b c - a d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b^{2}} + \frac{b c x^{3} - a d x^{3} - a c}{3 \, a^{2} x^{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^4/(b*x^3+a),x, algorithm="giac")

[Out]

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