3.277 \(\int \frac{x^{11} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=226 \[ \frac{x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}-\frac{a^2 \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a \log \left (a+b x^3\right ) \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}+\frac{x^9 (b e-3 a f)}{9 b^4}+\frac{f x^{12}}{12 b^3} \]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/(3*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^6)/(6*b^5) + ((b*e -
3*a*f)*x^9)/(9*b^4) + (f*x^12)/(12*b^3) + (a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^7*(a + b*x^3)^2) - (
a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f))/(3*b^7*(a + b*x^3)) - (a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e -
15*a^3*f)*Log[a + b*x^3])/(3*b^7)

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Rubi [A]  time = 0.330679, antiderivative size = 226, 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} \[ \frac{x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}-\frac{a^2 \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac{a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a \log \left (a+b x^3\right ) \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{x^6 \left (6 a^2 f-3 a b e+b^2 d\right )}{6 b^5}+\frac{x^9 (b e-3 a f)}{9 b^4}+\frac{f x^{12}}{12 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/(3*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^6)/(6*b^5) + ((b*e -
3*a*f)*x^9)/(9*b^4) + (f*x^12)/(12*b^3) + (a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^7*(a + b*x^3)^2) - (
a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f))/(3*b^7*(a + b*x^3)) - (a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e -
15*a^3*f)*Log[a + b*x^3])/(3*b^7)

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{x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x}{b^5}+\frac{(b e-3 a f) x^2}{b^4}+\frac{f x^3}{b^3}+\frac{a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6 (a+b x)^3}-\frac{a^2 \left (-3 b^3 c+4 a b^2 d-5 a^2 b e+6 a^3 f\right )}{b^6 (a+b x)^2}+\frac{a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right )}{b^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^3}{3 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^6}{6 b^5}+\frac{(b e-3 a f) x^9}{9 b^4}+\frac{f x^{12}}{12 b^3}+\frac{a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^7 \left (a+b x^3\right )^2}-\frac{a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right )}{3 b^7 \left (a+b x^3\right )}-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7}\\ \end{align*}

Mathematica [A]  time = 0.120015, size = 208, normalized size = 0.92 \[ \frac{12 b x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+\frac{12 a^2 \left (-5 a^2 b e+6 a^3 f+4 a b^2 d-3 b^3 c\right )}{a+b x^3}+\frac{6 a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+12 a \log \left (a+b x^3\right ) \left (-10 a^2 b e+15 a^3 f+6 a b^2 d-3 b^3 c\right )+6 b^2 x^6 \left (6 a^2 f-3 a b e+b^2 d\right )+4 b^3 x^9 (b e-3 a f)+3 b^4 f x^{12}}{36 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(12*b*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3 + 6*b^2*(b^2*d - 3*a*b*e + 6*a^2*f)*x^6 + 4*b^3*(b*e - 3*
a*f)*x^9 + 3*b^4*f*x^12 + (6*a^3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(a + b*x^3)^2 + (12*a^2*(-3*b^3*c + 4*a*
b^2*d - 5*a^2*b*e + 6*a^3*f))/(a + b*x^3) + 12*a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*Log[a + b*x^3]
)/(36*b^7)

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Maple [A]  time = 0.013, size = 313, normalized size = 1.4 \begin{align*}{\frac{f{x}^{12}}{12\,{b}^{3}}}-{\frac{{x}^{9}af}{3\,{b}^{4}}}+{\frac{{x}^{9}e}{9\,{b}^{3}}}+{\frac{{a}^{2}f{x}^{6}}{{b}^{5}}}-{\frac{ae{x}^{6}}{2\,{b}^{4}}}+{\frac{d{x}^{6}}{6\,{b}^{3}}}-{\frac{10\,{a}^{3}f{x}^{3}}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}e{x}^{3}}{{b}^{5}}}-{\frac{ad{x}^{3}}{{b}^{4}}}+{\frac{c{x}^{3}}{3\,{b}^{3}}}-{\frac{{a}^{6}f}{6\,{b}^{7} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{5}e}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}d}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{3}c}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+5\,{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{{b}^{7}}}-{\frac{10\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{{b}^{5}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) c}{{b}^{4}}}+2\,{\frac{{a}^{5}f}{{b}^{7} \left ( b{x}^{3}+a \right ) }}-{\frac{5\,{a}^{4}e}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}d}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}c}{{b}^{4} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

1/12*f*x^12/b^3-1/3/b^4*x^9*a*f+1/9/b^3*x^9*e+1/b^5*x^6*a^2*f-1/2/b^4*x^6*a*e+1/6/b^3*x^6*d-10/3/b^6*a^3*f*x^3
+2/b^5*a^2*e*x^3-1/b^4*a*d*x^3+1/3/b^3*c*x^3-1/6*a^6/b^7/(b*x^3+a)^2*f+1/6*a^5/b^6/(b*x^3+a)^2*e-1/6*a^4/b^5/(
b*x^3+a)^2*d+1/6*a^3/b^4/(b*x^3+a)^2*c+5*a^4/b^7*ln(b*x^3+a)*f-10/3*a^3/b^6*ln(b*x^3+a)*e+2*a^2/b^5*ln(b*x^3+a
)*d-a/b^4*ln(b*x^3+a)*c+2*a^5/b^7/(b*x^3+a)*f-5/3*a^4/b^6/(b*x^3+a)*e+4/3*a^3/b^5/(b*x^3+a)*d-a^2/b^4/(b*x^3+a
)*c

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Maxima [A]  time = 0.963488, size = 315, normalized size = 1.39 \begin{align*} -\frac{5 \, a^{3} b^{3} c - 7 \, a^{4} b^{2} d + 9 \, a^{5} b e - 11 \, a^{6} f + 2 \,{\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}}{6 \,{\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} + \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - 3 \, a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x^{3}}{36 \, b^{6}} - \frac{{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/6*(5*a^3*b^3*c - 7*a^4*b^2*d + 9*a^5*b*e - 11*a^6*f + 2*(3*a^2*b^4*c - 4*a^3*b^3*d + 5*a^4*b^2*e - 6*a^5*b*
f)*x^3)/(b^9*x^6 + 2*a*b^8*x^3 + a^2*b^7) + 1/36*(3*b^3*f*x^12 + 4*(b^3*e - 3*a*b^2*f)*x^9 + 6*(b^3*d - 3*a*b^
2*e + 6*a^2*b*f)*x^6 + 12*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/b^6 - 1/3*(3*a*b^3*c - 6*a^2*b^2*d +
10*a^3*b*e - 15*a^4*f)*log(b*x^3 + a)/b^7

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Fricas [A]  time = 1.34569, size = 765, normalized size = 3.38 \begin{align*} \frac{3 \, b^{6} f x^{18} + 2 \,{\left (2 \, b^{6} e - 3 \, a b^{5} f\right )} x^{15} +{\left (6 \, b^{6} d - 10 \, a b^{5} e + 15 \, a^{2} b^{4} f\right )} x^{12} + 4 \,{\left (3 \, b^{6} c - 6 \, a b^{5} d + 10 \, a^{2} b^{4} e - 15 \, a^{3} b^{3} f\right )} x^{9} - 30 \, a^{3} b^{3} c + 42 \, a^{4} b^{2} d - 54 \, a^{5} b e + 66 \, a^{6} f + 6 \,{\left (4 \, a b^{5} c - 11 \, a^{2} b^{4} d + 21 \, a^{3} b^{3} e - 34 \, a^{4} b^{2} f\right )} x^{6} - 12 \,{\left (2 \, a^{2} b^{4} c - a^{3} b^{3} d - a^{4} b^{2} e + 4 \, a^{5} b f\right )} x^{3} - 12 \,{\left (3 \, a^{3} b^{3} c - 6 \, a^{4} b^{2} d + 10 \, a^{5} b e - 15 \, a^{6} f +{\left (3 \, a b^{5} c - 6 \, a^{2} b^{4} d + 10 \, a^{3} b^{3} e - 15 \, a^{4} b^{2} f\right )} x^{6} + 2 \,{\left (3 \, a^{2} b^{4} c - 6 \, a^{3} b^{3} d + 10 \, a^{4} b^{2} e - 15 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{36 \,{\left (b^{9} x^{6} + 2 \, a b^{8} x^{3} + a^{2} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/36*(3*b^6*f*x^18 + 2*(2*b^6*e - 3*a*b^5*f)*x^15 + (6*b^6*d - 10*a*b^5*e + 15*a^2*b^4*f)*x^12 + 4*(3*b^6*c -
6*a*b^5*d + 10*a^2*b^4*e - 15*a^3*b^3*f)*x^9 - 30*a^3*b^3*c + 42*a^4*b^2*d - 54*a^5*b*e + 66*a^6*f + 6*(4*a*b^
5*c - 11*a^2*b^4*d + 21*a^3*b^3*e - 34*a^4*b^2*f)*x^6 - 12*(2*a^2*b^4*c - a^3*b^3*d - a^4*b^2*e + 4*a^5*b*f)*x
^3 - 12*(3*a^3*b^3*c - 6*a^4*b^2*d + 10*a^5*b*e - 15*a^6*f + (3*a*b^5*c - 6*a^2*b^4*d + 10*a^3*b^3*e - 15*a^4*
b^2*f)*x^6 + 2*(3*a^2*b^4*c - 6*a^3*b^3*d + 10*a^4*b^2*e - 15*a^5*b*f)*x^3)*log(b*x^3 + a))/(b^9*x^6 + 2*a*b^8
*x^3 + a^2*b^7)

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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(x**11*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.09688, size = 402, normalized size = 1.78 \begin{align*} -\frac{{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d - 15 \, a^{4} f + 10 \, a^{3} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} + \frac{9 \, a b^{5} c x^{6} - 18 \, a^{2} b^{4} d x^{6} - 45 \, a^{4} b^{2} f x^{6} + 30 \, a^{3} b^{3} x^{6} e + 12 \, a^{2} b^{4} c x^{3} - 28 \, a^{3} b^{3} d x^{3} - 78 \, a^{5} b f x^{3} + 50 \, a^{4} b^{2} x^{3} e + 4 \, a^{3} b^{3} c - 11 \, a^{4} b^{2} d - 34 \, a^{6} f + 21 \, a^{5} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{7}} + \frac{3 \, b^{9} f x^{12} - 12 \, a b^{8} f x^{9} + 4 \, b^{9} x^{9} e + 6 \, b^{9} d x^{6} + 36 \, a^{2} b^{7} f x^{6} - 18 \, a b^{8} x^{6} e + 12 \, b^{9} c x^{3} - 36 \, a b^{8} d x^{3} - 120 \, a^{3} b^{6} f x^{3} + 72 \, a^{2} b^{7} x^{3} e}{36 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/3*(3*a*b^3*c - 6*a^2*b^2*d - 15*a^4*f + 10*a^3*b*e)*log(abs(b*x^3 + a))/b^7 + 1/6*(9*a*b^5*c*x^6 - 18*a^2*b
^4*d*x^6 - 45*a^4*b^2*f*x^6 + 30*a^3*b^3*x^6*e + 12*a^2*b^4*c*x^3 - 28*a^3*b^3*d*x^3 - 78*a^5*b*f*x^3 + 50*a^4
*b^2*x^3*e + 4*a^3*b^3*c - 11*a^4*b^2*d - 34*a^6*f + 21*a^5*b*e)/((b*x^3 + a)^2*b^7) + 1/36*(3*b^9*f*x^12 - 12
*a*b^8*f*x^9 + 4*b^9*x^9*e + 6*b^9*d*x^6 + 36*a^2*b^7*f*x^6 - 18*a*b^8*x^6*e + 12*b^9*c*x^3 - 36*a*b^8*d*x^3 -
120*a^3*b^6*f*x^3 + 72*a^2*b^7*x^3*e)/b^12