### 3.118 $$\int \frac{x (b+2 c x^2)}{(-a+b x^2+c x^4)^8} \, dx$$

Optimal. Leaf size=20 $\frac{1}{14 \left (a-b x^2-c x^4\right )^7}$

[Out]

1/(14*(a - b*x^2 - c*x^4)^7)

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Rubi [A]  time = 0.0195398, antiderivative size = 20, 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 = {1247, 629} $\frac{1}{14 \left (a-b x^2-c x^4\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(14*(a - b*x^2 - c*x^4)^7)

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

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

Mathematica [A]  time = 0.0169894, size = 20, normalized size = 1. $-\frac{1}{14 \left (-a+b x^2+c x^4\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(14*(-a + b*x^2 + c*x^4)^7)

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Maple [A]  time = 0.001, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{14\, \left ( c{x}^{4}+b{x}^{2}-a \right ) ^{7}}} \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-a)^8,x)

[Out]

-1/14/(c*x^4+b*x^2-a)^7

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Maxima [B]  time = 1.48109, size = 481, normalized size = 24.05 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 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^{16} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 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^{12} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\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-a)^8,x, algorithm="maxima")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 7*(3*b^2*c^5 - a*c^6)*x^24 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^22 + 7*(5*b^4*c^3 -
15*a*b^2*c^4 + 3*a^2*c^5)*x^20 + 7*(3*b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^18 + 7*(b^6*c - 15*a*b^4*c^2 +
30*a^2*b^2*c^3 - 5*a^3*c^4)*x^16 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3)*x^14 - 7*(a*b^6 - 15*a
^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x^12 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^10 + 7*a^6*b*x^2 -
7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^8 - a^7 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^6 - 7*(3*a^5*b^2 - a^6*c)*x^
4)

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Fricas [B]  time = 1.32395, size = 748, normalized size = 37.4 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 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^{16} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 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^{12} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\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-a)^8,x, algorithm="fricas")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 7*(3*b^2*c^5 - a*c^6)*x^24 + 7*(5*b^3*c^4 - 6*a*b*c^5)*x^22 + 7*(5*b^4*c^3 -
15*a*b^2*c^4 + 3*a^2*c^5)*x^20 + 7*(3*b^5*c^2 - 20*a*b^3*c^3 + 15*a^2*b*c^4)*x^18 + 7*(b^6*c - 15*a*b^4*c^2 +
30*a^2*b^2*c^3 - 5*a^3*c^4)*x^16 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3)*x^14 - 7*(a*b^6 - 15*a
^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x^12 + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^10 + 7*a^6*b*x^2 -
7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^8 - a^7 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^6 - 7*(3*a^5*b^2 - a^6*c)*x^
4)

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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-a)**8,x)

[Out]

Timed out

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Giac [A]  time = 55.4883, size = 24, normalized size = 1.2 \begin{align*} -\frac{1}{14 \,{\left (c x^{4} + b x^{2} - a\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-a)^8,x, algorithm="giac")

[Out]

-1/14/(c*x^4 + b*x^2 - a)^7