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

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{7 \left (a+b x+c x^2\right )^7} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{7 (a+x (b+c x))^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/(a + x*(b + c*x))^7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)/(a + b*x + c*x^2)^8,x]

[Out]

IntegrateAlgebraic[(b + 2*c*x)/(a + b*x + c*x^2)^8, x]

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fricas [B]  time = 0.66, size = 350, normalized size = 21.88 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.40, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{7 \, {\left (c x^{2} + b x + a\right )}^{7}} \end {gather*}

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}} \end {gather*}

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 = 0.44, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{7 \, {\left (c x^{2} + b x + a\right )}^{7}} \end {gather*}

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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mupad [B]  time = 3.62, size = 358, normalized size = 22.38 \begin {gather*} -\frac {1}{7\,\left (x^5\,\left (105\,a^4\,b\,c^2+140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^9\,\left (105\,a^2\,b\,c^4+140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^7\,\left (140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2+42\,a\,b^5\,c+b^7\right )+x^3\,\left (42\,c\,a^5\,b+35\,a^4\,b^3\right )+x^{11}\,\left (35\,b^3\,c^4+42\,a\,b\,c^5\right )+x^4\,\left (21\,a^5\,c^2+105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{10}\,\left (21\,a^2\,c^5+105\,a\,b^2\,c^4+35\,b^4\,c^3\right )+a^7+x^6\,\left (35\,a^4\,c^3+210\,a^3\,b^2\,c^2+105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^8\,\left (35\,a^3\,c^4+210\,a^2\,b^2\,c^3+105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{14}+x^2\,\left (7\,c\,a^6+21\,a^5\,b^2\right )+x^{12}\,\left (21\,b^2\,c^5+7\,a\,c^6\right )+7\,b\,c^6\,x^{13}+7\,a^6\,b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(7*(x^5*(21*a^2*b^5 + 140*a^3*b^3*c + 105*a^4*b*c^2) + x^9*(21*b^5*c^2 + 140*a*b^3*c^3 + 105*a^2*b*c^4) + x
^7*(b^7 + 140*a^3*b*c^3 + 210*a^2*b^3*c^2 + 42*a*b^5*c) + x^3*(35*a^4*b^3 + 42*a^5*b*c) + x^11*(35*b^3*c^4 + 4
2*a*b*c^5) + x^4*(35*a^3*b^4 + 21*a^5*c^2 + 105*a^4*b^2*c) + x^10*(21*a^2*c^5 + 35*b^4*c^3 + 105*a*b^2*c^4) +
a^7 + x^6*(7*a*b^6 + 35*a^4*c^3 + 105*a^2*b^4*c + 210*a^3*b^2*c^2) + x^8*(7*b^6*c + 35*a^3*c^4 + 105*a*b^4*c^2
+ 210*a^2*b^2*c^3) + c^7*x^14 + x^2*(7*a^6*c + 21*a^5*b^2) + x^12*(7*a*c^6 + 21*b^2*c^5) + 7*b*c^6*x^13 + 7*a
^6*b*x))

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sympy [B]  time = 4.79, size = 359, normalized size = 22.44 \begin {gather*} - \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 {gather*}

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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