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

Optimal. Leaf size=16 \[ -\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1584, 446, 74} \begin {gather*} -\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(21*x^21*(b + c*x^3)^7)

Rule 74

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

Rule 446

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^2 \left (b+2 c x^3\right )}{\left (b x^3+c x^6\right )^8} \, dx &=\int \frac {b+2 c x^3}{x^{22} \left (b+c x^3\right )^8} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^3\right )\\ &=-\frac {1}{21 x^{21} \left (b+c x^3\right )^7}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{21 x^{21} \left (b+c x^3\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*1/(x^21*(b + c*x^3)^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 (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))/(b*x^3 + c*x^6)^8,x]

[Out]

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

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fricas [B]  time = 0.71, size = 81, normalized size = 5.06 \begin {gather*} -\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\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)^8,x, algorithm="fricas")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*x^30 + 21*b^5*c^2*x^27 + 7*b^6
*c*x^24 + b^7*x^21)

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giac [A]  time = 0.61, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{21 \, {\left (c x^{6} + b x^{3}\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)^8,x, algorithm="giac")

[Out]

-1/21/(c*x^6 + b*x^3)^7

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maple [B]  time = 0.01, size = 197, normalized size = 12.31 \begin {gather*} -\frac {\left (-\frac {b^{6}}{7 \left (c \,x^{3}+b \right )^{7} c}-\frac {b^{5}}{\left (c \,x^{3}+b \right )^{6} c}-\frac {4 b^{4}}{\left (c \,x^{3}+b \right )^{5} c}-\frac {12 b^{3}}{\left (c \,x^{3}+b \right )^{4} c}-\frac {30 b^{2}}{\left (c \,x^{3}+b \right )^{3} c}-\frac {66 b}{\left (c \,x^{3}+b \right )^{2} c}-\frac {132}{\left (c \,x^{3}+b \right ) c}\right ) c^{8}}{3 b^{13}}-\frac {44 c^{6}}{b^{13} x^{3}}+\frac {22 c^{5}}{b^{12} x^{6}}-\frac {10 c^{4}}{b^{11} x^{9}}+\frac {4 c^{3}}{b^{10} x^{12}}-\frac {4 c^{2}}{3 b^{9} x^{15}}+\frac {c}{3 b^{8} x^{18}}-\frac {1}{21 b^{7} x^{21}} \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)^8,x)

[Out]

-1/21/b^7/x^21-44/b^13*c^6/x^3+22/b^12*c^5/x^6-10/b^11*c^4/x^9+4/b^10*c^3/x^12-4/3/b^9*c^2/x^15+1/3/b^8*c/x^18
-1/3*c^8/b^13*(-12*b^3/c/(c*x^3+b)^4-30*b^2/c/(c*x^3+b)^3-132/c/(c*x^3+b)-b^5/c/(c*x^3+b)^6-4*b^4/c/(c*x^3+b)^
5-66*b/c/(c*x^3+b)^2-1/7*b^6/c/(c*x^3+b)^7)

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maxima [B]  time = 0.51, size = 81, normalized size = 5.06 \begin {gather*} -\frac {1}{21 \, {\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\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)^8,x, algorithm="maxima")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*x^30 + 21*b^5*c^2*x^27 + 7*b^6
*c*x^24 + b^7*x^21)

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mupad [B]  time = 5.07, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{21\,x^{21}\,{\left (c\,x^3+b\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(21*x^21*(b + c*x^3)^7)

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sympy [B]  time = 1.90, size = 87, normalized size = 5.44 \begin {gather*} - \frac {1}{21 b^{7} x^{21} + 147 b^{6} c x^{24} + 441 b^{5} c^{2} x^{27} + 735 b^{4} c^{3} x^{30} + 735 b^{3} c^{4} x^{33} + 441 b^{2} c^{5} x^{36} + 147 b c^{6} x^{39} + 21 c^{7} x^{42}} \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)**8,x)

[Out]

-1/(21*b**7*x**21 + 147*b**6*c*x**24 + 441*b**5*c**2*x**27 + 735*b**4*c**3*x**30 + 735*b**3*c**4*x**33 + 441*b
**2*c**5*x**36 + 147*b*c**6*x**39 + 21*c**7*x**42)

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