3.119 \(\int \frac{x^2 \left (b+2 c x^3\right )}{\left (-a+b x^3+c x^6\right )^8} \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{21 \left (a-b x^3-c x^6\right )^7} \]

[Out]

1/(21*(a - b*x^3 - c*x^6)^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.0136783, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{1}{21 \left (a-b x^3-c x^6\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

1/(21*(a - b*x^3 - c*x^6)^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.48165, size = 15, normalized size = 0.75 \[ \frac{1}{21 \left (a - b x^{3} - c x^{6}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(2*c*x**3+b)/(c*x**6+b*x**3-a)**8,x)

[Out]

1/(21*(a - b*x**3 - c*x**6)**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0275861, size = 20, normalized size = 1. \[ -\frac{1}{21 \left (-a+b x^3+c x^6\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

-1/(21*(-a + b*x^3 + c*x^6)^7)

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 19, normalized size = 1. \[ -{\frac{1}{21\, \left ( c{x}^{6}+b{x}^{3}-a \right ) ^{7}}} \]

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-a)^8,x)

[Out]

-1/21/(c*x^6+b*x^3-a)^7

_______________________________________________________________________________________

Maxima [A]  time = 0.874977, size = 481, normalized size = 24.05 \[ -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \,{\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \,{\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3 - a)^8,x, algorithm="maxima")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 - a*c^6)*x^36 + 7*(5*b^3*c^4 - 6*a
*b*c^5)*x^33 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3*b^5*c^2 - 20
*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 5*a
^3*c^4)*x^24 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3)*x^21 - 7*(a*
b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x^18 + 7*(3*a^2*b^5 - 20*a^3*b^
3*c + 15*a^4*b*c^2)*x^15 - 7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6
*b*x^3 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^9 - a^7 - 7*(3*a^5*b^2 - a^6*c)*x^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.334645, size = 481, normalized size = 24.05 \[ -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{36} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{33} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{30} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{27} + 7 \,{\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{24} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{21} - 7 \,{\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{18} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{15} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{12} + 7 \, a^{6} b x^{3} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{9} - a^{7} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3 - a)^8,x, algorithm="fricas")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 7*(3*b^2*c^5 - a*c^6)*x^36 + 7*(5*b^3*c^4 - 6*a
*b*c^5)*x^33 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^30 + 7*(3*b^5*c^2 - 20
*a*b^3*c^3 + 15*a^2*b*c^4)*x^27 + 7*(b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 5*a
^3*c^4)*x^24 + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3)*x^21 - 7*(a*
b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x^18 + 7*(3*a^2*b^5 - 20*a^3*b^
3*c + 15*a^4*b*c^2)*x^15 - 7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*x^12 + 7*a^6
*b*x^3 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^9 - a^7 - 7*(3*a^5*b^2 - a^6*c)*x^6)

_______________________________________________________________________________________

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*(2*c*x**3+b)/(c*x**6+b*x**3-a)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.36293, size = 24, normalized size = 1.2 \[ -\frac{1}{21 \,{\left (c x^{6} + b x^{3} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3 - a)^8,x, algorithm="giac")

[Out]

-1/21/(c*x^6 + b*x^3 - a)^7