3.1060 \(\int \frac{b+2 c x}{x^8 (b+c x)^8} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{7 x^7 (b+c x)^7} \]

[Out]

-1/(7*x^7*(b + c*x)^7)

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Rubi [A]  time = 0.002028, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {74} \[ -\frac{1}{7 x^7 (b+c x)^7} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(x^8*(b + c*x)^8),x]

[Out]

-1/(7*x^7*(b + c*x)^7)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [A]  time = 0.0204685, size = 14, normalized size = 1. \[ -\frac{1}{7 x^7 (b+c x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*x^7*(b + c*x)^7)

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Maple [B]  time = 0.02, size = 177, normalized size = 12.6 \begin{align*} 132\,{\frac{{c}^{7}}{{b}^{13} \left ( cx+b \right ) }}+66\,{\frac{{c}^{7}}{{b}^{12} \left ( cx+b \right ) ^{2}}}+30\,{\frac{{c}^{7}}{{b}^{11} \left ( cx+b \right ) ^{3}}}+12\,{\frac{{c}^{7}}{{b}^{10} \left ( cx+b \right ) ^{4}}}+4\,{\frac{{c}^{7}}{{b}^{9} \left ( cx+b \right ) ^{5}}}+{\frac{{c}^{7}}{{b}^{8} \left ( cx+b \right ) ^{6}}}+{\frac{{c}^{7}}{7\,{b}^{7} \left ( cx+b \right ) ^{7}}}-{\frac{1}{7\,{b}^{7}{x}^{7}}}-132\,{\frac{{c}^{6}}{{b}^{13}x}}+66\,{\frac{{c}^{5}}{{b}^{12}{x}^{2}}}-30\,{\frac{{c}^{4}}{{b}^{11}{x}^{3}}}+12\,{\frac{{c}^{3}}{{b}^{10}{x}^{4}}}-4\,{\frac{{c}^{2}}{{b}^{9}{x}^{5}}}+{\frac{c}{{b}^{8}{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/x^8/(c*x+b)^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 [B]  time = 0.971204, size = 109, normalized size = 7.79 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Fricas [B]  time = 0.951381, size = 171, normalized size = 12.21 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/x^8/(c*x+b)^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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Sympy [B]  time = 3.77256, size = 87, normalized size = 6.21 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/x**8/(c*x+b)**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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Giac [A]  time = 1.06779, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/(c*x^2 + b*x)^7