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

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

[Out]

1/(7*(a - b*x - c*x^2)^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.010642, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{7 \left (a-b x-c x^2\right )^7} \]

Antiderivative was successfully verified.

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

[Out]

1/(7*(a - b*x - c*x^2)^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.02, size = 15, normalized size = 0.83 \[ - \frac{1}{7 \left (- a + b x + c x^{2}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(7*(-a + b*x + c*x**2)**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.021195, size = 16, normalized size = 0.89 \[ \frac{1}{7 (a-x (b+c x))^7} \]

Antiderivative was successfully verified.

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

[Out]

1/(7*(a - x*(b + c*x))^7)

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 17, normalized size = 0.9 \[ -{\frac{1}{7\, \left ( c{x}^{2}+bx-a \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.757407, size = 22, normalized size = 1.22 \[ -\frac{1}{7 \,{\left (c x^{2} + b x - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7/(c*x^2 + b*x - a)^7

_______________________________________________________________________________________

Fricas [A]  time = 0.321819, size = 478, normalized size = 26.56 \[ -\frac{1}{7 \,{\left (c^{7} x^{14} + 7 \, b c^{6} x^{13} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{12} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{11} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{10} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{9} + 7 \,{\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{8} + 7 \, a^{6} b x +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{7} - a^{7} - 7 \,{\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{6} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{5} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{4} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{3} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7/(c^7*x^14 + 7*b*c^6*x^13 + 7*(3*b^2*c^5 - a*c^6)*x^12 + 7*(5*b^3*c^4 - 6*a*
b*c^5)*x^11 + 7*(5*b^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*x^10 + 7*(3*b^5*c^2 - 20*
a*b^3*c^3 + 15*a^2*b*c^4)*x^9 + 7*(b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 5*a^3
*c^4)*x^8 + 7*a^6*b*x + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3)*x^7
- a^7 - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*x^6 + 7*(3*a^2*b^
5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*x^5 - 7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*
x^4 + 7*(5*a^4*b^3 - 6*a^5*b*c)*x^3 - 7*(3*a^5*b^2 - a^6*c)*x^2)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27453, size = 22, normalized size = 1.22 \[ -\frac{1}{7 \,{\left (c x^{2} + b x - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7/(c*x^2 + b*x - a)^7