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

Optimal. Leaf size=15 \[ -\frac {1}{7 \left (b x+c x^2\right )^7} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {629} \[ -\frac {1}{7 \left (b x+c x^2\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(b*x + c*x^2)^8,x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 14, normalized size = 0.93 \[ -\frac {1}{7 x^7 (b+c x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/(x^7*(b + c*x)^7)

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fricas [B]  time = 0.74, size = 81, normalized size = 5.40 \[ -\frac {1}{7 \, {\left (c^{7} x^{14} + 7 \, b c^{6} x^{13} + 21 \, b^{2} c^{5} x^{12} + 35 \, b^{3} c^{4} x^{11} + 35 \, b^{4} c^{3} x^{10} + 21 \, b^{5} c^{2} x^{9} + 7 \, b^{6} c x^{8} + b^{7} x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x)^8,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 13, normalized size = 0.87 \[ -\frac {1}{7 \, {\left (c x^{2} + b x\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x)^8,x, algorithm="giac")

[Out]

-1/7/(c*x^2 + b*x)^7

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maple [B]  time = 0.02, size = 177, normalized size = 11.80 \[ \frac {c^{7}}{7 \left (c x +b \right )^{7} b^{7}}+\frac {c^{7}}{\left (c x +b \right )^{6} b^{8}}+\frac {4 c^{7}}{\left (c x +b \right )^{5} b^{9}}+\frac {12 c^{7}}{\left (c x +b \right )^{4} b^{10}}+\frac {30 c^{7}}{\left (c x +b \right )^{3} b^{11}}+\frac {66 c^{7}}{\left (c x +b \right )^{2} b^{12}}+\frac {132 c^{7}}{\left (c x +b \right ) b^{13}}-\frac {132 c^{6}}{b^{13} x}+\frac {66 c^{5}}{b^{12} x^{2}}-\frac {30 c^{4}}{b^{11} x^{3}}+\frac {12 c^{3}}{b^{10} x^{4}}-\frac {4 c^{2}}{b^{9} x^{5}}+\frac {c}{b^{8} x^{6}}-\frac {1}{7 b^{7} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

132/b^13*c^7/(c*x+b)+66/b^12*c^7/(c*x+b)^2+30/b^11*c^7/(c*x+b)^3+12/b^10*c^7/(c*x+b)^4+4/b^9*c^7/(c*x+b)^5+1/b
^8*c^7/(c*x+b)^6+1/7/b^7*c^7/(c*x+b)^7-1/7/b^7/x^7-132/b^13*c^6/x+66/b^12*c^5/x^2-30/b^11*c^4/x^3+12/b^10*c^3/
x^4-4/b^9*c^2/x^5+1/b^8*c/x^6

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maxima [A]  time = 0.53, size = 13, normalized size = 0.87 \[ -\frac {1}{7 \, {\left (c x^{2} + b x\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x)^8,x, algorithm="maxima")

[Out]

-1/7/(c*x^2 + b*x)^7

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mupad [B]  time = 4.38, size = 12, normalized size = 0.80 \[ -\frac {1}{7\,x^7\,{\left (b+c\,x\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(b*x + c*x^2)^8,x)

[Out]

-1/(7*x^7*(b + c*x)^7)

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sympy [B]  time = 0.93, size = 87, normalized size = 5.80 \[ - \frac {1}{7 b^{7} x^{7} + 49 b^{6} c x^{8} + 147 b^{5} c^{2} x^{9} + 245 b^{4} c^{3} x^{10} + 245 b^{3} c^{4} x^{11} + 147 b^{2} c^{5} x^{12} + 49 b c^{6} x^{13} + 7 c^{7} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x**2+b*x)**8,x)

[Out]

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

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