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

Optimal. Leaf size=16 \[ -\frac{1}{7 \left (a+b x+c x^2\right )^7} \]

[Out]

-1/(7*(a + b*x + c*x^2)^7)

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Rubi [A]  time = 0.0045067, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ -\frac{1}{7 \left (a+b x+c x^2\right )^7} \]

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)

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

Mathematica [A]  time = 0.0112869, size = 15, normalized size = 0.94 \[ -\frac{1}{7 (a+x (b+c x))^7} \]

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)

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{7\, \left ( c{x}^{2}+bx+a \right ) ^{7}}} \end{align*}

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

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Maxima [A]  time = 1.03121, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x + a\right )}^{7}} \end{align*}

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

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Fricas [B]  time = 1.23583, size = 737, normalized size = 46.06 \begin{align*} -\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 )}} \end{align*}

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 + 1
5*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)

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Sympy [B]  time = 51.4507, size = 359, normalized size = 22.44 \begin{align*} - \frac{1}{7 a^{7} + 49 a^{6} b x + 49 b c^{6} x^{13} + 7 c^{7} x^{14} + x^{12} \left (49 a c^{6} + 147 b^{2} c^{5}\right ) + x^{11} \left (294 a b c^{5} + 245 b^{3} c^{4}\right ) + x^{10} \left (147 a^{2} c^{5} + 735 a b^{2} c^{4} + 245 b^{4} c^{3}\right ) + x^{9} \left (735 a^{2} b c^{4} + 980 a b^{3} c^{3} + 147 b^{5} c^{2}\right ) + x^{8} \left (245 a^{3} c^{4} + 1470 a^{2} b^{2} c^{3} + 735 a b^{4} c^{2} + 49 b^{6} c\right ) + x^{7} \left (980 a^{3} b c^{3} + 1470 a^{2} b^{3} c^{2} + 294 a b^{5} c + 7 b^{7}\right ) + x^{6} \left (245 a^{4} c^{3} + 1470 a^{3} b^{2} c^{2} + 735 a^{2} b^{4} c + 49 a b^{6}\right ) + x^{5} \left (735 a^{4} b c^{2} + 980 a^{3} b^{3} c + 147 a^{2} b^{5}\right ) + x^{4} \left (147 a^{5} c^{2} + 735 a^{4} b^{2} c + 245 a^{3} b^{4}\right ) + x^{3} \left (294 a^{5} b c + 245 a^{4} b^{3}\right ) + x^{2} \left (49 a^{6} c + 147 a^{5} b^{2}\right )} \end{align*}

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]

-1/(7*a**7 + 49*a**6*b*x + 49*b*c**6*x**13 + 7*c**7*x**14 + x**12*(49*a*c**6 + 147*b**2*c**5) + x**11*(294*a*b
*c**5 + 245*b**3*c**4) + x**10*(147*a**2*c**5 + 735*a*b**2*c**4 + 245*b**4*c**3) + x**9*(735*a**2*b*c**4 + 980
*a*b**3*c**3 + 147*b**5*c**2) + x**8*(245*a**3*c**4 + 1470*a**2*b**2*c**3 + 735*a*b**4*c**2 + 49*b**6*c) + x**
7*(980*a**3*b*c**3 + 1470*a**2*b**3*c**2 + 294*a*b**5*c + 7*b**7) + x**6*(245*a**4*c**3 + 1470*a**3*b**2*c**2
+ 735*a**2*b**4*c + 49*a*b**6) + x**5*(735*a**4*b*c**2 + 980*a**3*b**3*c + 147*a**2*b**5) + x**4*(147*a**5*c**
2 + 735*a**4*b**2*c + 245*a**3*b**4) + x**3*(294*a**5*b*c + 245*a**4*b**3) + x**2*(49*a**6*c + 147*a**5*b**2))

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Giac [A]  time = 1.14072, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x + a\right )}^{7}} \end{align*}

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