3.4.4 \(\int \frac {\frac {5}{(-5+x)^6}+\frac {3}{(-3+x)^4}+\frac {1}{(-1+x)^2}}{(1+\frac {1}{(-5+x)^5}+\frac {1}{(-3+x)^3}+\frac {1}{-1+x})^2} \, dx\) [304]

Optimal. Leaf size=19 \[ \frac {1}{1+\frac {1}{(-5+x)^5}+\frac {1}{(-3+x)^3}+\frac {1}{-1+x}} \]

[Out]

1/(1+1/(x-1)+1/(-3+x)^3+1/(x-5)^5)

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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6818} \begin {gather*} \frac {1}{\frac {1}{(x-3)^3}+\frac {1}{(x-5)^5}+\frac {1}{x-1}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5/(-5 + x)^6 + 3/(-3 + x)^4 + (-1 + x)^(-2))/(1 + (-5 + x)^(-5) + (-3 + x)^(-3) + (-1 + x)^(-1))^2,x]

[Out]

(1 + (-5 + x)^(-5) + (-3 + x)^(-3) + (-1 + x)^(-1))^(-1)

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{1+\frac {1}{(-5+x)^5}+\frac {1}{(-3+x)^3}+\frac {1}{-1+x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{1+\frac {1}{(-5+x)^5}+\frac {1}{(-3+x)^3}+\frac {1}{-1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5/(-5 + x)^6 + 3/(-3 + x)^4 + (-1 + x)^(-2))/(1 + (-5 + x)^(-5) + (-3 + x)^(-3) + (-1 + x)^(-1))^2,
x]

[Out]

(1 + (-5 + x)^(-5) + (-3 + x)^(-3) + (-1 + x)^(-1))^(-1)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(19)=38\).
time = 0.12, size = 87, normalized size = 4.58

method	result	size
gosper	\(-\frac {x^{8}-34 x^{7}+503 x^{6}-4228 x^{5}+22076 x^{4}-73260 x^{3}+150661 x^{2}-175054 x +87527}{x^{9}-34 x^{8}+502 x^{7}-4201 x^{6}+21774 x^{5}-71474 x^{4}+144740 x^{3}-164339 x^{2}+78071 x +3152}\)	\(86\)
default	\(\frac {-x^{8}+34 x^{7}-503 x^{6}+4228 x^{5}-22076 x^{4}+73260 x^{3}-150661 x^{2}+175054 x -87527}{x^{9}-34 x^{8}+502 x^{7}-4201 x^{6}+21774 x^{5}-71474 x^{4}+144740 x^{3}-164339 x^{2}+78071 x +3152}\)	\(87\)
risch	\(\frac {-x^{8}+34 x^{7}-503 x^{6}+4228 x^{5}-22076 x^{4}+73260 x^{3}-150661 x^{2}+175054 x -87527}{x^{9}-34 x^{8}+502 x^{7}-4201 x^{6}+21774 x^{5}-71474 x^{4}+144740 x^{3}-164339 x^{2}+78071 x +3152}\)	\(87\)
parallelrisch	\(\frac {-x^{8}+34 x^{7}-503 x^{6}+4228 x^{5}-22076 x^{4}+73260 x^{3}-150661 x^{2}+175054 x -87527}{x^{9}-34 x^{8}+502 x^{7}-4201 x^{6}+21774 x^{5}-71474 x^{4}+144740 x^{3}-164339 x^{2}+78071 x +3152}\)	\(87\)
norman	\(\frac {-x^{17}+69 x^{16}-2229 x^{15}+44761 x^{14}-625602 x^{13}+6455858 x^{12}-50913715 x^{11}[phone] x^{10}-1521750430 x^{9}[phone] x^{8}-17915907435 x^{7}[phone] x^{6}-80517901754 x^{5}[phone] x^{4}-118540274125 x^{3}[phone] x^{2}-36925453125 x [phone]}{\left (x -1\right ) \left (-3+x \right )^{3} \left (x -5\right )^{5} \left (x^{9}-34 x^{8}+502 x^{7}-4201 x^{6}+21774 x^{5}-71474 x^{4}+144740 x^{3}-164339 x^{2}+78071 x +3152\right )}\)	\(147\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(x-1)^2+3/(-3+x)^4+5/(x-5)^6)/(1+1/(x-1)+1/(-3+x)^3+1/(x-5)^5)^2,x,method=_RETURNVERBOSE)

[Out]

(-x^8+34*x^7-503*x^6+4228*x^5-22076*x^4+73260*x^3-150661*x^2+175054*x-87527)/(x^9-34*x^8+502*x^7-4201*x^6+2177
4*x^5-71474*x^4+144740*x^3-164339*x^2+78071*x+3152)

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Maxima [A]
time = 0.35, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{\frac {1}{x - 1} + \frac {1}{{\left (x - 3\right )}^{3}} + \frac {1}{{\left (x - 5\right )}^{5}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/(x-1)^2+3/(-3+x)^4+5/(x-5)^6)/(1+1/(x-1)+1/(-3+x)^3+1/(x-5)^5)^2,x, algorithm="maxima")

[Out]

1/(1/(x - 1) + 1/(x - 3)^3 + 1/(x - 5)^5 + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (19) = 38\).
time = 0.57, size = 85, normalized size = 4.47 \begin {gather*} -\frac {x^{8} - 34 \, x^{7} + 503 \, x^{6} - 4228 \, x^{5} + 22076 \, x^{4} - 73260 \, x^{3} + 150661 \, x^{2} - 175054 \, x + 87527}{x^{9} - 34 \, x^{8} + 502 \, x^{7} - 4201 \, x^{6} + 21774 \, x^{5} - 71474 \, x^{4} + 144740 \, x^{3} - 164339 \, x^{2} + 78071 \, x + 3152} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/(x-1)^2+3/(-3+x)^4+5/(x-5)^6)/(1+1/(x-1)+1/(-3+x)^3+1/(x-5)^5)^2,x, algorithm="fricas")

[Out]

-(x^8 - 34*x^7 + 503*x^6 - 4228*x^5 + 22076*x^4 - 73260*x^3 + 150661*x^2 - 175054*x + 87527)/(x^9 - 34*x^8 + 5
02*x^7 - 4201*x^6 + 21774*x^5 - 71474*x^4 + 144740*x^3 - 164339*x^2 + 78071*x + 3152)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (20) = 40\).
time = 0.50, size = 82, normalized size = 4.32 \begin {gather*} \frac {- x^{8} + 34 x^{7} - 503 x^{6} + 4228 x^{5} - 22076 x^{4} + 73260 x^{3} - 150661 x^{2} + 175054 x - 87527}{x^{9} - 34 x^{8} + 502 x^{7} - 4201 x^{6} + 21774 x^{5} - 71474 x^{4} + 144740 x^{3} - 164339 x^{2} + 78071 x + 3152} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/(x-1)**2+3/(-3+x)**4+5/(x-5)**6)/(1+1/(x-1)+1/(-3+x)**3+1/(x-5)**5)**2,x)

[Out]

(-x**8 + 34*x**7 - 503*x**6 + 4228*x**5 - 22076*x**4 + 73260*x**3 - 150661*x**2 + 175054*x - 87527)/(x**9 - 34
*x**8 + 502*x**7 - 4201*x**6 + 21774*x**5 - 71474*x**4 + 144740*x**3 - 164339*x**2 + 78071*x + 3152)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (19) = 38\).
time = 0.48, size = 85, normalized size = 4.47 \begin {gather*} -\frac {x^{8} - 34 \, x^{7} + 503 \, x^{6} - 4228 \, x^{5} + 22076 \, x^{4} - 73260 \, x^{3} + 150661 \, x^{2} - 175054 \, x + 87527}{x^{9} - 34 \, x^{8} + 502 \, x^{7} - 4201 \, x^{6} + 21774 \, x^{5} - 71474 \, x^{4} + 144740 \, x^{3} - 164339 \, x^{2} + 78071 \, x + 3152} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/(x-1)^2+3/(-3+x)^4+5/(x-5)^6)/(1+1/(x-1)+1/(-3+x)^3+1/(x-5)^5)^2,x, algorithm="giac")

[Out]

-(x^8 - 34*x^7 + 503*x^6 - 4228*x^5 + 22076*x^4 - 73260*x^3 + 150661*x^2 - 175054*x + 87527)/(x^9 - 34*x^8 + 5
02*x^7 - 4201*x^6 + 21774*x^5 - 71474*x^4 + 144740*x^3 - 164339*x^2 + 78071*x + 3152)

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Mupad [B]
time = 0.31, size = 85, normalized size = 4.47 \begin {gather*} -\frac {x^8-34\,x^7+503\,x^6-4228\,x^5+22076\,x^4-73260\,x^3+150661\,x^2-175054\,x+87527}{x^9-34\,x^8+502\,x^7-4201\,x^6+21774\,x^5-71474\,x^4+144740\,x^3-164339\,x^2+78071\,x+3152} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(x - 1)^2 + 3/(x - 3)^4 + 5/(x - 5)^6)/(1/(x - 1) + 1/(x - 3)^3 + 1/(x - 5)^5 + 1)^2,x)

[Out]

-(150661*x^2 - 175054*x - 73260*x^3 + 22076*x^4 - 4228*x^5 + 503*x^6 - 34*x^7 + x^8 + 87527)/(78071*x - 164339
*x^2 + 144740*x^3 - 71474*x^4 + 21774*x^5 - 4201*x^6 + 502*x^7 - 34*x^8 + x^9 + 3152)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((1/(x-1)^2+3/(x-3)^4+5/(x-5)^6)/(1+1/(x-1)+1/(x-3)^3+1/(x-5)^5)^2,x)

[Out]

not solved

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