3.71.84 \(\int \frac {-180000+30000 x-x^3+180000 \log (5)+(-45000-7500 x) \log ^2(5)}{x^3} \, dx\)

Optimal. Leaf size=23 \[ -x+625 \left (-4+\frac {(6+x) (2-\log (5))}{x}\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {14} \begin {gather*} \frac {22500 (2-\log (5))^2}{x^2}-x-\frac {7500 \left (4-\log ^2(5)\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-180000 + 30000*x - x^3 + 180000*Log[5] + (-45000 - 7500*x)*Log[5]^2)/x^3,x]

[Out]

-x + (22500*(2 - Log[5])^2)/x^2 - (7500*(4 - Log[5]^2))/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} &=\int \left (-1-\frac {45000 (-2+\log (5))^2}{x^3}-\frac {7500 \left (-4+\log ^2(5)\right )}{x^2}\right ) \, dx\\ &=-x+\frac {22500 (2-\log (5))^2}{x^2}-\frac {7500 \left (4-\log ^2(5)\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.13 \begin {gather*} -x+\frac {22500 (-2+\log (5))^2}{x^2}+\frac {7500 \left (-4+\log ^2(5)\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-180000 + 30000*x - x^3 + 180000*Log[5] + (-45000 - 7500*x)*Log[5]^2)/x^3,x]

[Out]

-x + (22500*(-2 + Log[5])^2)/x^2 + (7500*(-4 + Log[5]^2))/x

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fricas [A]  time = 0.65, size = 26, normalized size = 1.13 \begin {gather*} -\frac {x^{3} - 7500 \, {\left (x + 3\right )} \log \relax (5)^{2} + 30000 \, x + 90000 \, \log \relax (5) - 90000}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x-45000)*log(5)^2+180000*log(5)-x^3+30000*x-180000)/x^3,x, algorithm="fricas")

[Out]

-(x^3 - 7500*(x + 3)*log(5)^2 + 30000*x + 90000*log(5) - 90000)/x^2

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giac [A]  time = 0.19, size = 30, normalized size = 1.30 \begin {gather*} -x + \frac {7500 \, {\left (x \log \relax (5)^{2} + 3 \, \log \relax (5)^{2} - 4 \, x - 12 \, \log \relax (5) + 12\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x-45000)*log(5)^2+180000*log(5)-x^3+30000*x-180000)/x^3,x, algorithm="giac")

[Out]

-x + 7500*(x*log(5)^2 + 3*log(5)^2 - 4*x - 12*log(5) + 12)/x^2

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maple [A]  time = 0.04, size = 31, normalized size = 1.35

method	result	size
risch	\(-x +\frac {\left (7500 \ln \relax (5)^{2}-30000\right ) x +22500 \ln \relax (5)^{2}-90000 \ln \relax (5)+90000}{x^{2}}\)	\(31\)
gosper	\(\frac {7500 x \ln \relax (5)^{2}-x^{3}+22500 \ln \relax (5)^{2}-90000 \ln \relax (5)-30000 x +90000}{x^{2}}\)	\(32\)
default	\(-x -\frac {-45000 \ln \relax (5)^{2}+180000 \ln \relax (5)-180000}{2 x^{2}}-\frac {-7500 \ln \relax (5)^{2}+30000}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7500*x-45000)*ln(5)^2+180000*ln(5)-x^3+30000*x-180000)/x^3,x,method=_RETURNVERBOSE)

[Out]

-x+((7500*ln(5)^2-30000)*x+22500*ln(5)^2-90000*ln(5)+90000)/x^2

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maxima [A]  time = 0.36, size = 29, normalized size = 1.26 \begin {gather*} -x + \frac {7500 \, {\left ({\left (\log \relax (5)^{2} - 4\right )} x + 3 \, \log \relax (5)^{2} - 12 \, \log \relax (5) + 12\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x-45000)*log(5)^2+180000*log(5)-x^3+30000*x-180000)/x^3,x, algorithm="maxima")

[Out]

-x + 7500*((log(5)^2 - 4)*x + 3*log(5)^2 - 12*log(5) + 12)/x^2

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mupad [B]  time = 4.15, size = 27, normalized size = 1.17 \begin {gather*} \frac {22500\,{\left (\ln \relax (5)-2\right )}^2}{x^2}-x+\frac {7500\,{\ln \relax (5)}^2-30000}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)^2*(7500*x + 45000) - 180000*log(5) - 30000*x + x^3 + 180000)/x^3,x)

[Out]

(22500*(log(5) - 2)^2)/x^2 - x + (7500*log(5)^2 - 30000)/x

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sympy [A]  time = 0.28, size = 29, normalized size = 1.26 \begin {gather*} - x - \frac {x \left (30000 - 7500 \log {\relax (5 )}^{2}\right ) - 90000 - 22500 \log {\relax (5 )}^{2} + 90000 \log {\relax (5 )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7500*x-45000)*ln(5)**2+180000*ln(5)-x**3+30000*x-180000)/x**3,x)

[Out]

-x - (x*(30000 - 7500*log(5)**2) - 90000 - 22500*log(5)**2 + 90000*log(5))/x**2

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