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

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

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{21 \left (a+b x^3+c x^6\right )^7} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{21 \left (a+b x^3+c x^6\right )^7} \end {gather*}

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*1/(a + b*x^3 + c*x^6)^7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (b+2 c x^3\right )}{\left (a+b x^3+c x^6\right )^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.01, size = 352, normalized size = 19.56 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 22.37, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{21 \, {\left (c x^{6} + b x^{3} + a\right )}^{7}} \end {gather*}

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]

-1/21/(c*x^6 + b*x^3 + a)^7

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maple [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} -\frac {1}{21 \left (c \,x^{6}+b \,x^{3}+a \right )^{7}} \end {gather*}

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 = 0.95, size = 352, normalized size = 19.56 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 18.21, size = 360, normalized size = 20.00 \begin {gather*} -\frac {1}{21\,\left (x^{15}\,\left (105\,a^4\,b\,c^2+140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{27}\,\left (105\,a^2\,b\,c^4+140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{21}\,\left (140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2+42\,a\,b^5\,c+b^7\right )+x^9\,\left (42\,c\,a^5\,b+35\,a^4\,b^3\right )+x^{33}\,\left (35\,b^3\,c^4+42\,a\,b\,c^5\right )+x^{12}\,\left (21\,a^5\,c^2+105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{30}\,\left (21\,a^2\,c^5+105\,a\,b^2\,c^4+35\,b^4\,c^3\right )+a^7+x^{18}\,\left (35\,a^4\,c^3+210\,a^3\,b^2\,c^2+105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^{24}\,\left (35\,a^3\,c^4+210\,a^2\,b^2\,c^3+105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{42}+x^6\,\left (7\,c\,a^6+21\,a^5\,b^2\right )+x^{36}\,\left (21\,b^2\,c^5+7\,a\,c^6\right )+7\,a^6\,b\,x^3+7\,b\,c^6\,x^{39}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(21*(x^15*(21*a^2*b^5 + 140*a^3*b^3*c + 105*a^4*b*c^2) + x^27*(21*b^5*c^2 + 140*a*b^3*c^3 + 105*a^2*b*c^4)
+ x^21*(b^7 + 140*a^3*b*c^3 + 210*a^2*b^3*c^2 + 42*a*b^5*c) + x^9*(35*a^4*b^3 + 42*a^5*b*c) + x^33*(35*b^3*c^4
+ 42*a*b*c^5) + x^12*(35*a^3*b^4 + 21*a^5*c^2 + 105*a^4*b^2*c) + x^30*(21*a^2*c^5 + 35*b^4*c^3 + 105*a*b^2*c^
4) + a^7 + x^18*(7*a*b^6 + 35*a^4*c^3 + 105*a^2*b^4*c + 210*a^3*b^2*c^2) + x^24*(7*b^6*c + 35*a^3*c^4 + 105*a*
b^4*c^2 + 210*a^2*b^2*c^3) + c^7*x^42 + x^6*(7*a^6*c + 21*a^5*b^2) + x^36*(7*a*c^6 + 21*b^2*c^5) + 7*a^6*b*x^3
+ 7*b*c^6*x^39))

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sympy [B]  time = 11.76, size = 360, normalized size = 20.00 \begin {gather*} - \frac {1}{21 a^{7} + 147 a^{6} b x^{3} + 147 b c^{6} x^{39} + 21 c^{7} x^{42} + x^{36} \left (147 a c^{6} + 441 b^{2} c^{5}\right ) + x^{33} \left (882 a b c^{5} + 735 b^{3} c^{4}\right ) + x^{30} \left (441 a^{2} c^{5} + 2205 a b^{2} c^{4} + 735 b^{4} c^{3}\right ) + x^{27} \left (2205 a^{2} b c^{4} + 2940 a b^{3} c^{3} + 441 b^{5} c^{2}\right ) + x^{24} \left (735 a^{3} c^{4} + 4410 a^{2} b^{2} c^{3} + 2205 a b^{4} c^{2} + 147 b^{6} c\right ) + x^{21} \left (2940 a^{3} b c^{3} + 4410 a^{2} b^{3} c^{2} + 882 a b^{5} c + 21 b^{7}\right ) + x^{18} \left (735 a^{4} c^{3} + 4410 a^{3} b^{2} c^{2} + 2205 a^{2} b^{4} c + 147 a b^{6}\right ) + x^{15} \left (2205 a^{4} b c^{2} + 2940 a^{3} b^{3} c + 441 a^{2} b^{5}\right ) + x^{12} \left (441 a^{5} c^{2} + 2205 a^{4} b^{2} c + 735 a^{3} b^{4}\right ) + x^{9} \left (882 a^{5} b c + 735 a^{4} b^{3}\right ) + x^{6} \left (147 a^{6} c + 441 a^{5} b^{2}\right )} \end {gather*}

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]

-1/(21*a**7 + 147*a**6*b*x**3 + 147*b*c**6*x**39 + 21*c**7*x**42 + x**36*(147*a*c**6 + 441*b**2*c**5) + x**33*
(882*a*b*c**5 + 735*b**3*c**4) + x**30*(441*a**2*c**5 + 2205*a*b**2*c**4 + 735*b**4*c**3) + x**27*(2205*a**2*b
*c**4 + 2940*a*b**3*c**3 + 441*b**5*c**2) + x**24*(735*a**3*c**4 + 4410*a**2*b**2*c**3 + 2205*a*b**4*c**2 + 14
7*b**6*c) + x**21*(2940*a**3*b*c**3 + 4410*a**2*b**3*c**2 + 882*a*b**5*c + 21*b**7) + x**18*(735*a**4*c**3 + 4
410*a**3*b**2*c**2 + 2205*a**2*b**4*c + 147*a*b**6) + x**15*(2205*a**4*b*c**2 + 2940*a**3*b**3*c + 441*a**2*b*
*5) + x**12*(441*a**5*c**2 + 2205*a**4*b**2*c + 735*a**3*b**4) + x**9*(882*a**5*b*c + 735*a**4*b**3) + x**6*(1
47*a**6*c + 441*a**5*b**2))

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