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

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

[Out]

-1/(14*(a + b*x^2 + c*x^4)^7)

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Rubi [A]  time = 0.0120266, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{1}{14 \left (a+b x^2+c x^4\right )^7} \]

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)

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Rubi in Sympy [A]  time = 4.9566, size = 17, normalized size = 0.94 \[ - \frac{1}{14 \left (a + b x^{2} + c x^{4}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(2*c*x**2+b)/(c*x**4+b*x**2+a)**8,x)

[Out]

-1/(14*(a + b*x**2 + c*x**4)**7)

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Mathematica [A]  time = 0.0204354, size = 18, 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 = 0.002, size = 17, normalized size = 0.9 \[ -{\frac{1}{14\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{7}}} \]

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 [A]  time = 0.858208, size = 475, normalized size = 26.39 \[ -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{16} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{14} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{12} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} + a^{7} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{6} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*x/(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 [A]  time = 0.371777, size = 475, normalized size = 26.39 \[ -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{16} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{14} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{12} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} + a^{7} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{6} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)*x/(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. \[ \text{Timed out} \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError