3.38.64 \(\int \frac {900+90 x+902 x^2-60 x^3-45 x^4+90 x^5-18 x^6}{45 x^5} \, dx\)

Optimal. Leaf size=30 \[ \frac {4}{3}-\frac {1}{5} \left (5+\frac {5+\frac {x}{3}}{x^2}-x\right )^2-\log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 14} \begin {gather*} -\frac {5}{x^4}-\frac {2}{3 x^3}-\frac {x^2}{5}-\frac {451}{45 x^2}+2 x+\frac {4}{3 x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(900 + 90*x + 902*x^2 - 60*x^3 - 45*x^4 + 90*x^5 - 18*x^6)/(45*x^5),x]

[Out]

-5/x^4 - 2/(3*x^3) - 451/(45*x^2) + 4/(3*x) + 2*x - x^2/5 - Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{45} \int \frac {900+90 x+902 x^2-60 x^3-45 x^4+90 x^5-18 x^6}{x^5} \, dx\\ &=\frac {1}{45} \int \left (90+\frac {900}{x^5}+\frac {90}{x^4}+\frac {902}{x^3}-\frac {60}{x^2}-\frac {45}{x}-18 x\right ) \, dx\\ &=-\frac {5}{x^4}-\frac {2}{3 x^3}-\frac {451}{45 x^2}+\frac {4}{3 x}+2 x-\frac {x^2}{5}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 41, normalized size = 1.37 \begin {gather*} -\frac {5}{x^4}-\frac {2}{3 x^3}-\frac {451}{45 x^2}+\frac {4}{3 x}+2 x-\frac {x^2}{5}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(900 + 90*x + 902*x^2 - 60*x^3 - 45*x^4 + 90*x^5 - 18*x^6)/(45*x^5),x]

[Out]

-5/x^4 - 2/(3*x^3) - 451/(45*x^2) + 4/(3*x) + 2*x - x^2/5 - Log[x]

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fricas [A]  time = 0.73, size = 37, normalized size = 1.23 \begin {gather*} -\frac {9 \, x^{6} - 90 \, x^{5} + 45 \, x^{4} \log \relax (x) - 60 \, x^{3} + 451 \, x^{2} + 30 \, x + 225}{45 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-18*x^6+90*x^5-45*x^4-60*x^3+902*x^2+90*x+900)/x^5,x, algorithm="fricas")

[Out]

-1/45*(9*x^6 - 90*x^5 + 45*x^4*log(x) - 60*x^3 + 451*x^2 + 30*x + 225)/x^4

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giac [A]  time = 0.15, size = 34, normalized size = 1.13 \begin {gather*} -\frac {1}{5} \, x^{2} + 2 \, x + \frac {60 \, x^{3} - 451 \, x^{2} - 30 \, x - 225}{45 \, x^{4}} - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-18*x^6+90*x^5-45*x^4-60*x^3+902*x^2+90*x+900)/x^5,x, algorithm="giac")

[Out]

-1/5*x^2 + 2*x + 1/45*(60*x^3 - 451*x^2 - 30*x - 225)/x^4 - log(abs(x))

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maple [A]  time = 0.02, size = 34, normalized size = 1.13

method	result	size
default	\(-\frac {x^{2}}{5}+2 x -\ln \relax (x )-\frac {5}{x^{4}}+\frac {4}{3 x}-\frac {451}{45 x^{2}}-\frac {2}{3 x^{3}}\)	\(34\)
risch	\(-\frac {x^{2}}{5}+2 x +\frac {60 x^{3}-451 x^{2}-30 x -225}{45 x^{4}}-\ln \relax (x )\)	\(34\)
norman	\(\frac {-5-\frac {2}{3} x -\frac {451}{45} x^{2}+\frac {4}{3} x^{3}+2 x^{5}-\frac {1}{5} x^{6}}{x^{4}}-\ln \relax (x )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/45*(-18*x^6+90*x^5-45*x^4-60*x^3+902*x^2+90*x+900)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/5*x^2+2*x-ln(x)-5/x^4+4/3/x-451/45/x^2-2/3/x^3

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maxima [A]  time = 0.36, size = 33, normalized size = 1.10 \begin {gather*} -\frac {1}{5} \, x^{2} + 2 \, x + \frac {60 \, x^{3} - 451 \, x^{2} - 30 \, x - 225}{45 \, x^{4}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-18*x^6+90*x^5-45*x^4-60*x^3+902*x^2+90*x+900)/x^5,x, algorithm="maxima")

[Out]

-1/5*x^2 + 2*x + 1/45*(60*x^3 - 451*x^2 - 30*x - 225)/x^4 - log(x)

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mupad [B]  time = 2.05, size = 33, normalized size = 1.10 \begin {gather*} 2\,x-\ln \relax (x)-\frac {x^2}{5}-\frac {-\frac {4\,x^3}{3}+\frac {451\,x^2}{45}+\frac {2\,x}{3}+5}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + (902*x^2)/45 - (4*x^3)/3 - x^4 + 2*x^5 - (2*x^6)/5 + 20)/x^5,x)

[Out]

2*x - log(x) - x^2/5 - ((2*x)/3 + (451*x^2)/45 - (4*x^3)/3 + 5)/x^4

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sympy [A]  time = 0.10, size = 31, normalized size = 1.03 \begin {gather*} - \frac {x^{2}}{5} + 2 x - \log {\relax (x )} - \frac {- 60 x^{3} + 451 x^{2} + 30 x + 225}{45 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/45*(-18*x**6+90*x**5-45*x**4-60*x**3+902*x**2+90*x+900)/x**5,x)

[Out]

-x**2/5 + 2*x - log(x) - (-60*x**3 + 451*x**2 + 30*x + 225)/(45*x**4)

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