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

Optimal. Leaf size=80 \[ -\frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{c \log (x)}{a}+\frac{f x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.120113, antiderivative size = 80, 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} \[ -\frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a b^3}+\frac{x^3 (b e-a f)}{3 b^2}+\frac{c \log (x)}{a}+\frac{f x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0306707, size = 75, normalized size = 0.94 \[ \frac{-2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+a b x^3 \left (-2 a f+2 b e+b f x^3\right )+6 b^3 c \log (x)}{6 a b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 97, normalized size = 1.2 \begin{align*}{\frac{f{x}^{6}}{6\,b}}-{\frac{a{x}^{3}f}{3\,{b}^{2}}}+{\frac{e{x}^{3}}{3\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{3}}}-{\frac{ae\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}}+{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,b}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{c\ln \left ( x \right ) }{a}} \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),x)

[Out]

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

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Maxima [A]  time = 0.958248, size = 104, normalized size = 1.3 \begin{align*} \frac{c \log \left (x^{3}\right )}{3 \, a} + \frac{b f x^{6} + 2 \,{\left (b e - a f\right )} x^{3}}{6 \, b^{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 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58809, size = 171, normalized size = 2.14 \begin{align*} \frac{a b^{2} f x^{6} + 6 \, b^{3} c \log \left (x\right ) + 2 \,{\left (a b^{2} e - a^{2} b f\right )} x^{3} - 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{6 \, a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.91323, size = 68, normalized size = 0.85 \begin{align*} \frac{f x^{6}}{6 b} - \frac{x^{3} \left (a f - b e\right )}{3 b^{2}} + \frac{c \log{\left (x \right )}}{a} + \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 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),x)

[Out]

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

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Giac [A]  time = 1.05752, size = 107, normalized size = 1.34 \begin{align*} \frac{c \log \left ({\left | x \right |}\right )}{a} + \frac{b f x^{6} - 2 \, a f x^{3} + 2 \, b x^{3} e}{6 \, b^{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 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),x, algorithm="giac")

[Out]

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