### 3.111 $$\int \frac{x^2 (b+2 c x^3)}{(a+b x^3+c x^6)^8} \, dx$$

Optimal. Leaf size=18 $-\frac{1}{21 \left (a+b x^3+c x^6\right )^7}$

[Out]

-1/(21*(a + b*x^3 + c*x^6)^7)

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Rubi [A]  time = 0.023087, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {1468, 629} $-\frac{1}{21 \left (a+b x^3+c x^6\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6)^8,x]

[Out]

-1/(21*(a + b*x^3 + c*x^6)^7)

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^2 \left (b+2 c x^3\right )}{\left (a+b x^3+c x^6\right )^8} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx,x,x^3\right )\\ &=-\frac{1}{21 \left (a+b x^3+c x^6\right )^7}\\ \end{align*}

Mathematica [A]  time = 0.0130597, size = 18, normalized size = 1. $-\frac{1}{21 \left (a+b x^3+c x^6\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6)^8,x]

[Out]

-1/(21*(a + b*x^3 + c*x^6)^7)

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Maple [A]  time = 0.001, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{21\, \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a)^8,x)

[Out]

-1/21/(c*x^6+b*x^3+a)^7

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Maxima [B]  time = 1.70515, size = 475, normalized size = 26.39 \begin{align*} -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{36} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{33} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{24} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{21} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{18} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{9} + a^{7} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a)^8,x, algorithm="maxima")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 + a*c^6)*x^36 + 7*(5*b^3*c^4 + 6*a*b*c^5)*x^33 + 7*(5*b^4*c^3 +
15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3*b^5*c^2 + 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c + 15*a*b^4*c^2 +
30*a^2*b^2*c^3 + 5*a^3*c^4)*x^24 + (b^7 + 42*a*b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*c^3)*x^21 + 7*(a*b^6 + 15*a
^2*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*x^18 + 7*(3*a^2*b^5 + 20*a^3*b^3*c + 15*a^4*b*c^2)*x^15 + 7*(5*a^3*b^4
+ 15*a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6*b*x^3 + 7*(5*a^4*b^3 + 6*a^5*b*c)*x^9 + a^7 + 7*(3*a^5*b^2 + a^6*c)*x
^6)

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Fricas [B]  time = 1.26847, size = 749, normalized size = 41.61 \begin{align*} -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{36} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{33} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{24} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{21} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{18} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{9} + a^{7} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a)^8,x, algorithm="fricas")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 + a*c^6)*x^36 + 7*(5*b^3*c^4 + 6*a*b*c^5)*x^33 + 7*(5*b^4*c^3 +
15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3*b^5*c^2 + 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c + 15*a*b^4*c^2 +
30*a^2*b^2*c^3 + 5*a^3*c^4)*x^24 + (b^7 + 42*a*b^5*c + 210*a^2*b^3*c^2 + 140*a^3*b*c^3)*x^21 + 7*(a*b^6 + 15*a
^2*b^4*c + 30*a^3*b^2*c^2 + 5*a^4*c^3)*x^18 + 7*(3*a^2*b^5 + 20*a^3*b^3*c + 15*a^4*b*c^2)*x^15 + 7*(5*a^3*b^4
+ 15*a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6*b*x^3 + 7*(5*a^4*b^3 + 6*a^5*b*c)*x^9 + a^7 + 7*(3*a^5*b^2 + a^6*c)*x
^6)

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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**2*(2*c*x**3+b)/(c*x**6+b*x**3+a)**8,x)

[Out]

Timed out

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Giac [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^2*(2*c*x^3+b)/(c*x^6+b*x^3+a)^8,x, algorithm="giac")

[Out]

Timed out