3.57.15 \(\int \frac {5+9 x+8 x^2-2 x^3+(5-x)^4 (-10 x^2-9 x^3+7 x^4)+(5 x-x^2) \log (x)}{-5 x+x^2} \, dx\)

Optimal. Leaf size=23 \[ (1+x) \left (-x+(5-x)^4 x^2-\log (x)\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 41, normalized size of antiderivative = 1.78, number of steps used = 4, number of rules used = 3, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1593, 6688, 2295} \begin {gather*} x^7-19 x^6+130 x^5-350 x^4+125 x^3+624 x^2-x-x \log (x)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 9*x + 8*x^2 - 2*x^3 + (5 - x)^4*(-10*x^2 - 9*x^3 + 7*x^4) + (5*x - x^2)*Log[x])/(-5*x + x^2),x]

[Out]

-x + 624*x^2 + 125*x^3 - 350*x^4 + 130*x^5 - 19*x^6 + x^7 - Log[x] - x*Log[x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5+9 x+8 x^2-2 x^3+(5-x)^4 \left (-10 x^2-9 x^3+7 x^4\right )+\left (5 x-x^2\right ) \log (x)}{(-5+x) x} \, dx\\ &=\int \left (-2-\frac {1}{x}+1248 x+375 x^2-1400 x^3+650 x^4-114 x^5+7 x^6-\log (x)\right ) \, dx\\ &=-2 x+624 x^2+125 x^3-350 x^4+130 x^5-19 x^6+x^7-\log (x)-\int \log (x) \, dx\\ &=-x+624 x^2+125 x^3-350 x^4+130 x^5-19 x^6+x^7-\log (x)-x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.78 \begin {gather*} -x+624 x^2+125 x^3-350 x^4+130 x^5-19 x^6+x^7-\log (x)-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 9*x + 8*x^2 - 2*x^3 + (5 - x)^4*(-10*x^2 - 9*x^3 + 7*x^4) + (5*x - x^2)*Log[x])/(-5*x + x^2),x]

[Out]

-x + 624*x^2 + 125*x^3 - 350*x^4 + 130*x^5 - 19*x^6 + x^7 - Log[x] - x*Log[x]

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fricas [A]  time = 0.51, size = 39, normalized size = 1.70 \begin {gather*} x^{7} - 19 \, x^{6} + 130 \, x^{5} - 350 \, x^{4} + 125 \, x^{3} + 624 \, x^{2} - {\left (x + 1\right )} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4-9*x^3-10*x^2)*(5-x)^4+(-x^2+5*x)*log(x)-2*x^3+8*x^2+9*x+5)/(x^2-5*x),x, algorithm="fricas")

[Out]

x^7 - 19*x^6 + 130*x^5 - 350*x^4 + 125*x^3 + 624*x^2 - (x + 1)*log(x) - x

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giac [A]  time = 0.17, size = 41, normalized size = 1.78 \begin {gather*} x^{7} - 19 \, x^{6} + 130 \, x^{5} - 350 \, x^{4} + 125 \, x^{3} + 624 \, x^{2} - x \log \relax (x) - x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4-9*x^3-10*x^2)*(5-x)^4+(-x^2+5*x)*log(x)-2*x^3+8*x^2+9*x+5)/(x^2-5*x),x, algorithm="giac")

[Out]

x^7 - 19*x^6 + 130*x^5 - 350*x^4 + 125*x^3 + 624*x^2 - x*log(x) - x - log(x)

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maple [A]  time = 0.33, size = 42, normalized size = 1.83

method	result	size
default	\(x^{7}-19 x^{6}+130 x^{5}-350 x^{4}+125 x^{3}-x -x \ln \relax (x )+624 x^{2}-\ln \relax (x )\)	\(42\)
norman	\(x^{7}-19 x^{6}+130 x^{5}-350 x^{4}+125 x^{3}-x -x \ln \relax (x )+624 x^{2}-\ln \relax (x )\)	\(42\)
risch	\(x^{7}-19 x^{6}+130 x^{5}-350 x^{4}+125 x^{3}-x -x \ln \relax (x )+624 x^{2}-\ln \relax (x )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*x^4-9*x^3-10*x^2)*(5-x)^4+(-x^2+5*x)*ln(x)-2*x^3+8*x^2+9*x+5)/(x^2-5*x),x,method=_RETURNVERBOSE)

[Out]

x^7-19*x^6+130*x^5-350*x^4+125*x^3-x-x*ln(x)+624*x^2-ln(x)

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maxima [A]  time = 0.46, size = 41, normalized size = 1.78 \begin {gather*} x^{7} - 19 \, x^{6} + 130 \, x^{5} - 350 \, x^{4} + 125 \, x^{3} + 624 \, x^{2} - x \log \relax (x) - x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^4-9*x^3-10*x^2)*(5-x)^4+(-x^2+5*x)*log(x)-2*x^3+8*x^2+9*x+5)/(x^2-5*x),x, algorithm="maxima")

[Out]

x^7 - 19*x^6 + 130*x^5 - 350*x^4 + 125*x^3 + 624*x^2 - x*log(x) - x - log(x)

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mupad [B]  time = 3.49, size = 34, normalized size = 1.48 \begin {gather*} -\left (x+1\right )\,\left (x+\ln \relax (x)-625\,x^2+500\,x^3-150\,x^4+20\,x^5-x^6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x - (x - 5)^4*(10*x^2 + 9*x^3 - 7*x^4) + log(x)*(5*x - x^2) + 8*x^2 - 2*x^3 + 5)/(5*x - x^2),x)

[Out]

-(x + 1)*(x + log(x) - 625*x^2 + 500*x^3 - 150*x^4 + 20*x^5 - x^6)

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sympy [B]  time = 0.14, size = 37, normalized size = 1.61 \begin {gather*} x^{7} - 19 x^{6} + 130 x^{5} - 350 x^{4} + 125 x^{3} + 624 x^{2} - x \log {\relax (x )} - x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x**4-9*x**3-10*x**2)*(5-x)**4+(-x**2+5*x)*ln(x)-2*x**3+8*x**2+9*x+5)/(x**2-5*x),x)

[Out]

x**7 - 19*x**6 + 130*x**5 - 350*x**4 + 125*x**3 + 624*x**2 - x*log(x) - x - log(x)

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