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

Optimal. Leaf size=163 \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.199532, antiderivative size = 163, 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} \[ \frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^3 b \left (a+b x^3\right )^2}+\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{3 a^5}+\frac{\log (x) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac{3 b c-a d}{3 a^4 x^3}-\frac{c}{6 a^3 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(6*a^3*x^6) + (3*b*c - a*d)/(3*a^4*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(6*a^3*b*(a + b*x^3)^2) + (3*
b^2*c - 2*a*b*d + a^2*e)/(3*a^4*(a + b*x^3)) + ((6*b^2*c - 3*a*b*d + a^2*e)*Log[x])/a^5 - ((6*b^2*c - 3*a*b*d
+ a^2*e)*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^7 \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^3 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a^3 x^3}+\frac{-3 b c+a d}{a^4 x^2}+\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 (a+b x)^3}-\frac{b \left (3 b^2 c-2 a b d+a^2 e\right )}{a^4 (a+b x)^2}-\frac{b \left (6 b^2 c-3 a b d+a^2 e\right )}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{6 a^3 x^6}+\frac{3 b c-a d}{3 a^4 x^3}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^3 b \left (a+b x^3\right )^2}+\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 \left (a+b x^3\right )}+\frac{\left (6 b^2 c-3 a b d+a^2 e\right ) \log (x)}{a^5}-\frac{\left (6 b^2 c-3 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{3 a^5}\\ \end{align*}

Mathematica [A]  time = 0.106458, size = 149, normalized size = 0.91 \[ \frac{\frac{a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b \left (a+b x^3\right )^2}+\frac{2 a \left (a^2 e-2 a b d+3 b^2 c\right )}{a+b x^3}-2 \log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )+6 \log (x) \left (a^2 e-3 a b d+6 b^2 c\right )-\frac{a^2 c}{x^6}-\frac{2 a (a d-3 b c)}{x^3}}{6 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 213, normalized size = 1.3 \begin{align*} -{\frac{f}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{bd}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{b}^{2}c}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) bd}{{a}^{4}}}-2\,{\frac{\ln \left ( b{x}^{3}+a \right ){b}^{2}c}{{a}^{5}}}+{\frac{e}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,bd}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{b}^{2}c}{{a}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{6\,{a}^{3}{x}^{6}}}-{\frac{d}{3\,{a}^{3}{x}^{3}}}+{\frac{bc}{{a}^{4}{x}^{3}}}+{\frac{e\ln \left ( x \right ) }{{a}^{3}}}-3\,{\frac{\ln \left ( x \right ) bd}{{a}^{4}}}+6\,{\frac{\ln \left ( x \right ){b}^{2}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^7/(b*x^3+a)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.980084, size = 246, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (6 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} +{\left (18 \, a b^{3} c - 9 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c + 2 \,{\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} x^{3}}{6 \,{\left (a^{4} b^{3} x^{12} + 2 \, a^{5} b^{2} x^{9} + a^{6} b x^{6}\right )}} - \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \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)^3,x, algorithm="maxima")

[Out]

1/6*(2*(6*b^4*c - 3*a*b^3*d + a^2*b^2*e)*x^9 + (18*a*b^3*c - 9*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^6 - a^3*b*c +
2*(2*a^2*b^2*c - a^3*b*d)*x^3)/(a^4*b^3*x^12 + 2*a^5*b^2*x^9 + a^6*b*x^6) - 1/3*(6*b^2*c - 3*a*b*d + a^2*e)*lo
g(b*x^3 + a)/a^5 + 1/3*(6*b^2*c - 3*a*b*d + a^2*e)*log(x^3)/a^5

________________________________________________________________________________________

Fricas [B]  time = 1.31011, size = 652, normalized size = 4. \begin{align*} \frac{2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (18 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{6} - a^{4} b c + 2 \,{\left (2 \, a^{3} b^{2} c - a^{4} b d\right )} x^{3} - 2 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \,{\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} +{\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{12} + 2 \, a^{6} b^{2} x^{9} + a^{7} b x^{6}\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^7/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (18*a^2*b^3*c - 9*a^3*b^2*d + 3*a^4*b*e - a^5*f)*x^6 - a^4*
b*c + 2*(2*a^3*b^2*c - a^4*b*d)*x^3 - 2*((6*b^5*c - 3*a*b^4*d + a^2*b^3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d +
a^3*b^2*e)*x^9 + (6*a^2*b^3*c - 3*a^3*b^2*d + a^4*b*e)*x^6)*log(b*x^3 + a) + 6*((6*b^5*c - 3*a*b^4*d + a^2*b^
3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (6*a^2*b^3*c - 3*a^3*b^2*d + a^4*b*e)*x^6)*log(x))/(
a^5*b^3*x^12 + 2*a^6*b^2*x^9 + a^7*b*x^6)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.07307, size = 255, normalized size = 1.56 \begin{align*} \frac{{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (6 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} + \frac{12 \, b^{4} c x^{9} - 6 \, a b^{3} d x^{9} + 2 \, a^{2} b^{2} x^{9} e + 18 \, a b^{3} c x^{6} - 9 \, a^{2} b^{2} d x^{6} - a^{4} f x^{6} + 3 \, a^{3} b x^{6} e + 4 \, a^{2} b^{2} c x^{3} - 2 \, a^{3} b d x^{3} - a^{3} b c}{6 \,{\left (b x^{6} + a x^{3}\right )}^{2} a^{4} 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)^3,x, algorithm="giac")

[Out]

(6*b^2*c - 3*a*b*d + a^2*e)*log(abs(x))/a^5 - 1/3*(6*b^3*c - 3*a*b^2*d + a^2*b*e)*log(abs(b*x^3 + a))/(a^5*b)
+ 1/6*(12*b^4*c*x^9 - 6*a*b^3*d*x^9 + 2*a^2*b^2*x^9*e + 18*a*b^3*c*x^6 - 9*a^2*b^2*d*x^6 - a^4*f*x^6 + 3*a^3*b
*x^6*e + 4*a^2*b^2*c*x^3 - 2*a^3*b*d*x^3 - a^3*b*c)/((b*x^6 + a*x^3)^2*a^4*b)