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

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)

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Rubi [A]  time = 0.004369, 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, 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 = 0.0133975, 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)

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Maple [A]  time = 0.002, size = 17, 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 = 1.00645, size = 22, normalized size = 1.22 \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 = 1.31757, size = 737, normalized size = 40.94 \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 = 55.2036, size = 359, normalized size = 19.94 \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 = 1.10916, size = 22, normalized size = 1.22 \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