3.231 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{13} (a+b x^3)} \, dx\)

Optimal. Leaf size=164 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{b \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5}+\frac{b \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5}-\frac{a^2 e-a b d+b^2 c}{6 a^3 x^6}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{12 a x^{12}} \]

[Out]

-c/(12*a*x^12) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(6*a^3*x^6) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(3*a^4*x^3) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x])/a^5 - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*
f)*Log[a + b*x^3])/(3*a^5)

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Rubi [A]  time = 0.181439, antiderivative size = 164, 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} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{3 a^4 x^3}-\frac{b \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5}+\frac{b \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^5}-\frac{a^2 e-a b d+b^2 c}{6 a^3 x^6}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]

[Out]

-c/(12*a*x^12) + (b*c - a*d)/(9*a^2*x^9) - (b^2*c - a*b*d + a^2*e)/(6*a^3*x^6) + (b^3*c - a*b^2*d + a^2*b*e -
a^3*f)/(3*a^4*x^3) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[x])/a^5 - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*
f)*Log[a + b*x^3])/(3*a^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{c+d x^3+e x^6+f x^9}{x^{13} \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^5 (a+b x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a x^5}+\frac{-b c+a d}{a^2 x^4}+\frac{b^2 c-a b d+a^2 e}{a^3 x^3}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^2}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{12 a x^{12}}+\frac{b c-a d}{9 a^2 x^9}-\frac{b^2 c-a b d+a^2 e}{6 a^3 x^6}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^4 x^3}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^5}-\frac{b \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^5}\\ \end{align*}

Mathematica [A]  time = 0.0624507, size = 164, normalized size = 1. \[ \frac{36 b x^{12} \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )-12 b x^{12} \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )-6 a^2 b^2 x^6 \left (c+2 d x^3\right )+2 a^3 b x^3 \left (2 c+3 d x^3+6 e x^6\right )-a^4 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+12 a b^3 c x^9}{36 a^5 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)),x]

[Out]

(12*a*b^3*c*x^9 - 6*a^2*b^2*x^6*(c + 2*d*x^3) + 2*a^3*b*x^3*(2*c + 3*d*x^3 + 6*e*x^6) - a^4*(3*c + 4*d*x^3 + 6
*e*x^6 + 12*f*x^9) + 36*b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^12*Log[x] - 12*b*(b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*x^12*Log[a + b*x^3])/(36*a^5*x^12)

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Maple [A]  time = 0.009, size = 210, normalized size = 1.3 \begin{align*}{\frac{b\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) e}{3\,{a}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{4}}}-{\frac{{b}^{4}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{5}}}-{\frac{c}{12\,a{x}^{12}}}-{\frac{d}{9\,a{x}^{9}}}+{\frac{bc}{9\,{a}^{2}{x}^{9}}}-{\frac{e}{6\,a{x}^{6}}}+{\frac{bd}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}c}{6\,{a}^{3}{x}^{6}}}-{\frac{f}{3\,a{x}^{3}}}+{\frac{be}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}d}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}c}{3\,{a}^{4}{x}^{3}}}-{\frac{b\ln \left ( x \right ) f}{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) e}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) d}{{a}^{4}}}+{\frac{{b}^{4}\ln \left ( x \right ) c}{{a}^{5}}} \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^13/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.955154, size = 224, normalized size = 1.37 \begin{align*} -\frac{{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac{{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{5}} + \frac{12 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 6 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 3 \, a^{3} c + 4 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}}{36 \, a^{4} x^{12}} \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^13/(b*x^3+a),x, algorithm="maxima")

[Out]

-1/3*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*log(b*x^3 + a)/a^5 + 1/3*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*
log(x^3)/a^5 + 1/36*(12*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^9 - 6*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 3*a^3*c
+ 4*(a^2*b*c - a^3*d)*x^3)/(a^4*x^12)

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Fricas [A]  time = 1.5128, size = 355, normalized size = 2.16 \begin{align*} -\frac{12 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (b x^{3} + a\right ) - 36 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} \log \left (x\right ) - 12 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 6 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 3 \, a^{4} c - 4 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}}{36 \, a^{5} x^{12}} \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^13/(b*x^3+a),x, algorithm="fricas")

[Out]

-1/36*(12*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^12*log(b*x^3 + a) - 36*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*
b*f)*x^12*log(x) - 12*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^9 + 6*(a^2*b^2*c - a^3*b*d + a^4*e)*x^6 + 3*a^
4*c - 4*(a^3*b*c - a^4*d)*x^3)/(a^5*x^12)

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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**13/(b*x**3+a),x)

[Out]

Timed out

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Giac [A]  time = 1.06825, size = 317, normalized size = 1.93 \begin{align*} \frac{{\left (b^{4} c - a b^{3} d - a^{3} b f + a^{2} b^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} - \frac{25 \, b^{4} c x^{12} - 25 \, a b^{3} d x^{12} - 25 \, a^{3} b f x^{12} + 25 \, a^{2} b^{2} x^{12} e - 12 \, a b^{3} c x^{9} + 12 \, a^{2} b^{2} d x^{9} + 12 \, a^{4} f x^{9} - 12 \, a^{3} b x^{9} e + 6 \, a^{2} b^{2} c x^{6} - 6 \, a^{3} b d x^{6} + 6 \, a^{4} x^{6} e - 4 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{5} x^{12}} \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^13/(b*x^3+a),x, algorithm="giac")

[Out]

(b^4*c - a*b^3*d - a^3*b*f + a^2*b^2*e)*log(abs(x))/a^5 - 1/3*(b^5*c - a*b^4*d - a^3*b^2*f + a^2*b^3*e)*log(ab
s(b*x^3 + a))/(a^5*b) - 1/36*(25*b^4*c*x^12 - 25*a*b^3*d*x^12 - 25*a^3*b*f*x^12 + 25*a^2*b^2*x^12*e - 12*a*b^3
*c*x^9 + 12*a^2*b^2*d*x^9 + 12*a^4*f*x^9 - 12*a^3*b*x^9*e + 6*a^2*b^2*c*x^6 - 6*a^3*b*d*x^6 + 6*a^4*x^6*e - 4*
a^3*b*c*x^3 + 4*a^4*d*x^3 + 3*a^4*c)/(a^5*x^12)