### 3.126 $$\int \frac{x (b+2 c x^2)}{(b x^2+c x^4)^8} \, dx$$

Optimal. Leaf size=16 $-\frac{1}{14 x^{14} \left (b+c x^2\right )^7}$

[Out]

-1/(14*x^14*(b + c*x^2)^7)

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Rubi [A]  time = 0.0208838, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {1584, 446, 74} $-\frac{1}{14 x^{14} \left (b+c x^2\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4)^8,x]

[Out]

-1/(14*x^14*(b + c*x^2)^7)

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]

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 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]

Rubi steps

\begin{align*} \int \frac{x \left (b+2 c x^2\right )}{\left (b x^2+c x^4\right )^8} \, dx &=\int \frac{b+2 c x^2}{x^{15} \left (b+c x^2\right )^8} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^2\right )\\ &=-\frac{1}{14 x^{14} \left (b+c x^2\right )^7}\\ \end{align*}

Mathematica [A]  time = 0.0277516, size = 16, normalized size = 1. $-\frac{1}{14 x^{14} \left (b+c x^2\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x*(b + 2*c*x^2))/(b*x^2 + c*x^4)^8,x]

[Out]

-1/(14*x^14*(b + c*x^2)^7)

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Maple [B]  time = 0.019, size = 197, normalized size = 12.3 \begin{align*} -{\frac{1}{14\,{b}^{7}{x}^{14}}}-66\,{\frac{{c}^{6}}{{b}^{13}{x}^{2}}}+33\,{\frac{{c}^{5}}{{b}^{12}{x}^{4}}}-15\,{\frac{{c}^{4}}{{b}^{11}{x}^{6}}}+6\,{\frac{{c}^{3}}{{b}^{10}{x}^{8}}}-2\,{\frac{{c}^{2}}{{b}^{9}{x}^{10}}}+{\frac{c}{2\,{b}^{8}{x}^{12}}}-{\frac{{c}^{8}}{2\,{b}^{13}} \left ( -132\,{\frac{1}{c \left ( c{x}^{2}+b \right ) }}-{\frac{{b}^{5}}{c \left ( c{x}^{2}+b \right ) ^{6}}}-4\,{\frac{{b}^{4}}{c \left ( c{x}^{2}+b \right ) ^{5}}}-12\,{\frac{{b}^{3}}{c \left ( c{x}^{2}+b \right ) ^{4}}}-30\,{\frac{{b}^{2}}{c \left ( c{x}^{2}+b \right ) ^{3}}}-{\frac{{b}^{6}}{7\,c \left ( c{x}^{2}+b \right ) ^{7}}}-66\,{\frac{b}{c \left ( c{x}^{2}+b \right ) ^{2}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*c*x^2+b)/(c*x^4+b*x^2)^8,x)

[Out]

-1/14/b^7/x^14-66/b^13*c^6/x^2+33/b^12*c^5/x^4-15/b^11*c^4/x^6+6/b^10*c^3/x^8-2/b^9*c^2/x^10+1/2/b^8*c/x^12-1/
2*c^8/b^13*(-132/c/(c*x^2+b)-1/c*b^5/(c*x^2+b)^6-4/c*b^4/(c*x^2+b)^5-12/c*b^3/(c*x^2+b)^4-30*b^2/c/(c*x^2+b)^3
-1/7/c*b^6/(c*x^2+b)^7-66/c*b/(c*x^2+b)^2)

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Maxima [B]  time = 1.03812, size = 109, normalized size = 6.81 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2)^8,x, algorithm="maxima")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*x^20 + 21*b^5*c^2*x^18 + 7*b^6
*c*x^16 + b^7*x^14)

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Fricas [B]  time = 1.05018, size = 177, normalized size = 11.06 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2)^8,x, algorithm="fricas")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*x^20 + 21*b^5*c^2*x^18 + 7*b^6
*c*x^16 + b^7*x^14)

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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*c*x**2+b)/(c*x**4+b*x**2)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.12524, size = 20, normalized size = 1.25 \begin{align*} -\frac{1}{14 \,{\left (c x^{4} + b x^{2}\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*c*x^2+b)/(c*x^4+b*x^2)^8,x, algorithm="giac")

[Out]

-1/14/(c*x^4 + b*x^2)^7