3.11.60 \(\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx\) [1060]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {75} \begin {gather*} -\frac {1}{7 x^7 (b+c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(x^8*(b + c*x)^8),x]

[Out]

-1/7*1/(x^7*(b + c*x)^7)

Rule 75

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*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{7 x^7 (b+c x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/(x^7*(b + c*x)^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(12)=24\).
time = 0.32, size = 177, normalized size = 12.64

method	result	size
gosper	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
norman	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
risch	\(-\frac {1}{7 x^{7} \left (c x +b \right )^{7}}\)	\(13\)
default	\(\frac {132 c^{7}}{b^{13} \left (c x +b \right )}+\frac {66 c^{7}}{b^{12} \left (c x +b \right )^{2}}+\frac {30 c^{7}}{b^{11} \left (c x +b \right )^{3}}+\frac {12 c^{7}}{b^{10} \left (c x +b \right )^{4}}+\frac {4 c^{7}}{b^{9} \left (c x +b \right )^{5}}+\frac {c^{7}}{b^{8} \left (c x +b \right )^{6}}+\frac {c^{7}}{7 b^{7} \left (c x +b \right )^{7}}-\frac {1}{7 b^{7} x^{7}}-\frac {132 c^{6}}{b^{13} x}+\frac {66 c^{5}}{b^{12} x^{2}}-\frac {30 c^{4}}{b^{11} x^{3}}+\frac {12 c^{3}}{b^{10} x^{4}}-\frac {4 c^{2}}{b^{9} x^{5}}+\frac {c}{b^{8} x^{6}}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+c^7
/b^8/(c*x+b)^6+1/7*c^7/b^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] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (12) = 24\).
time = 0.32, size = 81, normalized size = 5.79 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (12) = 24\).
time = 1.75, size = 81, normalized size = 5.79 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (14) = 28\).
time = 0.43, size = 87, normalized size = 6.21 \begin {gather*} - \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 {gather*}

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 = 0.62, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{7 \, {\left (c x^{2} + b x\right )}^{7}} \end {gather*}

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

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Mupad [B]
time = 7.09, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{7\,x^7\,{\left (b+c\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(x^8*(b + c*x)^8),x)

[Out]

-1/(7*x^7*(b + c*x)^7)

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