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

Optimal. Leaf size=95 \[ -\frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}+\frac{b c-a d}{3 a^2 x^3}-\frac{c}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.128976, antiderivative size = 95, 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} \[ -\frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b}+\frac{\log (x) \left (a^2 e-a b d+b^2 c\right )}{a^3}+\frac{b c-a d}{3 a^2 x^3}-\frac{c}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0686564, size = 88, normalized size = 0.93 \[ \frac{\log \left (a+b x^3\right ) \left (\frac{2 a^3 f}{b}-2 a^2 e+2 a b d-2 b^2 c\right )+6 \log (x) \left (a^2 e-a b d+b^2 c\right )-\frac{a \left (a c+2 a d x^3-2 b c x^3\right )}{x^6}}{6 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 116, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( b{x}^{3}+a \right ) f}{3\,b}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,a}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{3}}}-{\frac{c}{6\,a{x}^{6}}}-{\frac{d}{3\,a{x}^{3}}}+{\frac{bc}{3\,{x}^{3}{a}^{2}}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{3}}} \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),x)

[Out]

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

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Maxima [A]  time = 0.948345, size = 126, normalized size = 1.33 \begin{align*} \frac{{\left (b^{2} c - a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{3}} - \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^{3} b} + \frac{2 \,{\left (b c - a d\right )} x^{3} - a c}{6 \, a^{2} 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52655, size = 213, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} \log \left (b x^{3} + a\right ) - 6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e\right )} x^{6} \log \left (x\right ) + a^{2} b c - 2 \,{\left (a b^{2} c - a^{2} b d\right )} x^{3}}{6 \, a^{3} b 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 51.4964, size = 85, normalized size = 0.89 \begin{align*} - \frac{a c + x^{3} \left (2 a d - 2 b c\right )}{6 a^{2} x^{6}} + \frac{\left (a^{2} e - a b d + b^{2} c\right ) \log{\left (x \right )}}{a^{3}} + \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^{3} b} \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),x)

[Out]

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

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Giac [A]  time = 1.05506, size = 170, normalized size = 1.79 \begin{align*} \frac{{\left (b^{2} c - a b d + a^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \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^{3} b} - \frac{3 \, b^{2} c x^{6} - 3 \, a b d x^{6} + 3 \, a^{2} x^{6} e - 2 \, a b c x^{3} + 2 \, a^{2} d x^{3} + a^{2} c}{6 \, a^{3} 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),x, algorithm="giac")

[Out]

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