3.1.88 \(\int \frac {x (b+2 c x^2)}{(a+b x^2+c x^4)^8} \, dx\)

Optimal. Leaf size=18 \[ -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \]

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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1247, 629} \begin {gather*} -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \end {gather*}

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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/14*1/(a + b*x^2 + c*x^4)^7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (b+2 c x^2\right )}{\left (a+b x^2+c x^4\right )^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.05, size = 352, normalized size = 19.56 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 6.78, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{14 \, {\left (c x^{4} + b x^{2} + a\right )}^{7}} \end {gather*}

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

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maple [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} -\frac {1}{14 \left (c \,x^{4}+b \,x^{2}+a \right )^{7}} \end {gather*}

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

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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mupad [B]  time = 12.16, size = 360, normalized size = 20.00 \begin {gather*} -\frac {1}{14\,\left (x^{10}\,\left (105\,a^4\,b\,c^2+140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{18}\,\left (105\,a^2\,b\,c^4+140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{14}\,\left (140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2+42\,a\,b^5\,c+b^7\right )+x^6\,\left (42\,c\,a^5\,b+35\,a^4\,b^3\right )+x^{22}\,\left (35\,b^3\,c^4+42\,a\,b\,c^5\right )+x^8\,\left (21\,a^5\,c^2+105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{20}\,\left (21\,a^2\,c^5+105\,a\,b^2\,c^4+35\,b^4\,c^3\right )+a^7+x^{12}\,\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^{16}\,\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^{28}+x^4\,\left (7\,c\,a^6+21\,a^5\,b^2\right )+x^{24}\,\left (21\,b^2\,c^5+7\,a\,c^6\right )+7\,a^6\,b\,x^2+7\,b\,c^6\,x^{26}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(14*(x^10*(21*a^2*b^5 + 140*a^3*b^3*c + 105*a^4*b*c^2) + x^18*(21*b^5*c^2 + 140*a*b^3*c^3 + 105*a^2*b*c^4)
+ x^14*(b^7 + 140*a^3*b*c^3 + 210*a^2*b^3*c^2 + 42*a*b^5*c) + x^6*(35*a^4*b^3 + 42*a^5*b*c) + x^22*(35*b^3*c^4
+ 42*a*b*c^5) + x^8*(35*a^3*b^4 + 21*a^5*c^2 + 105*a^4*b^2*c) + x^20*(21*a^2*c^5 + 35*b^4*c^3 + 105*a*b^2*c^4
) + a^7 + x^12*(7*a*b^6 + 35*a^4*c^3 + 105*a^2*b^4*c + 210*a^3*b^2*c^2) + x^16*(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^28 + x^4*(7*a^6*c + 21*a^5*b^2) + x^24*(7*a*c^6 + 21*b^2*c^5) + 7*a^6*b*x^2
+ 7*b*c^6*x^26))

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sympy [B]  time = 7.66, size = 360, normalized size = 20.00 \begin {gather*} - \frac {1}{14 a^{7} + 98 a^{6} b x^{2} + 98 b c^{6} x^{26} + 14 c^{7} x^{28} + x^{24} \left (98 a c^{6} + 294 b^{2} c^{5}\right ) + x^{22} \left (588 a b c^{5} + 490 b^{3} c^{4}\right ) + x^{20} \left (294 a^{2} c^{5} + 1470 a b^{2} c^{4} + 490 b^{4} c^{3}\right ) + x^{18} \left (1470 a^{2} b c^{4} + 1960 a b^{3} c^{3} + 294 b^{5} c^{2}\right ) + x^{16} \left (490 a^{3} c^{4} + 2940 a^{2} b^{2} c^{3} + 1470 a b^{4} c^{2} + 98 b^{6} c\right ) + x^{14} \left (1960 a^{3} b c^{3} + 2940 a^{2} b^{3} c^{2} + 588 a b^{5} c + 14 b^{7}\right ) + x^{12} \left (490 a^{4} c^{3} + 2940 a^{3} b^{2} c^{2} + 1470 a^{2} b^{4} c + 98 a b^{6}\right ) + x^{10} \left (1470 a^{4} b c^{2} + 1960 a^{3} b^{3} c + 294 a^{2} b^{5}\right ) + x^{8} \left (294 a^{5} c^{2} + 1470 a^{4} b^{2} c + 490 a^{3} b^{4}\right ) + x^{6} \left (588 a^{5} b c + 490 a^{4} b^{3}\right ) + x^{4} \left (98 a^{6} c + 294 a^{5} b^{2}\right )} \end {gather*}

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]

-1/(14*a**7 + 98*a**6*b*x**2 + 98*b*c**6*x**26 + 14*c**7*x**28 + x**24*(98*a*c**6 + 294*b**2*c**5) + x**22*(58
8*a*b*c**5 + 490*b**3*c**4) + x**20*(294*a**2*c**5 + 1470*a*b**2*c**4 + 490*b**4*c**3) + x**18*(1470*a**2*b*c*
*4 + 1960*a*b**3*c**3 + 294*b**5*c**2) + x**16*(490*a**3*c**4 + 2940*a**2*b**2*c**3 + 1470*a*b**4*c**2 + 98*b*
*6*c) + x**14*(1960*a**3*b*c**3 + 2940*a**2*b**3*c**2 + 588*a*b**5*c + 14*b**7) + x**12*(490*a**4*c**3 + 2940*
a**3*b**2*c**2 + 1470*a**2*b**4*c + 98*a*b**6) + x**10*(1470*a**4*b*c**2 + 1960*a**3*b**3*c + 294*a**2*b**5) +
x**8*(294*a**5*c**2 + 1470*a**4*b**2*c + 490*a**3*b**4) + x**6*(588*a**5*b*c + 490*a**4*b**3) + x**4*(98*a**6
*c + 294*a**5*b**2))

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