3.25.95 \(\int \frac {-20+5 x^2-174 x^3}{25 x^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{5} \left (\frac {2}{x^2}-\frac {174 x}{5}+\log (x)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14} \begin {gather*} \frac {2}{5 x^2}-\frac {174 x}{25}+\frac {\log (x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20 + 5*x^2 - 174*x^3)/(25*x^3),x]

[Out]

2/(5*x^2) - (174*x)/25 + Log[x]/5

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}{25} \int \frac {-20+5 x^2-174 x^3}{x^3} \, dx\\ &=\frac {1}{25} \int \left (-174-\frac {20}{x^3}+\frac {5}{x}\right ) \, dx\\ &=\frac {2}{5 x^2}-\frac {174 x}{25}+\frac {\log (x)}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{25} \left (\frac {10}{x^2}-174 x+5 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 5*x^2 - 174*x^3)/(25*x^3),x]

[Out]

(10/x^2 - 174*x + 5*Log[x])/25

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fricas [A]  time = 0.55, size = 19, normalized size = 1.12 \begin {gather*} -\frac {174 \, x^{3} - 5 \, x^{2} \log \relax (x) - 10}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-174*x^3+5*x^2-20)/x^3,x, algorithm="fricas")

[Out]

-1/25*(174*x^3 - 5*x^2*log(x) - 10)/x^2

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giac [A]  time = 0.17, size = 14, normalized size = 0.82 \begin {gather*} -\frac {174}{25} \, x + \frac {2}{5 \, x^{2}} + \frac {1}{5} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-174*x^3+5*x^2-20)/x^3,x, algorithm="giac")

[Out]

-174/25*x + 2/5/x^2 + 1/5*log(abs(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {174 x}{25}+\frac {2}{5 x^{2}}+\frac {\ln \relax (x )}{5}\)	\(14\)
risch	\(-\frac {174 x}{25}+\frac {2}{5 x^{2}}+\frac {\ln \relax (x )}{5}\)	\(14\)
norman	\(\frac {\frac {2}{5}-\frac {174 x^{3}}{25}}{x^{2}}+\frac {\ln \relax (x )}{5}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(-174*x^3+5*x^2-20)/x^3,x,method=_RETURNVERBOSE)

[Out]

-174/25*x+2/5/x^2+1/5*ln(x)

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maxima [A]  time = 0.39, size = 13, normalized size = 0.76 \begin {gather*} -\frac {174}{25} \, x + \frac {2}{5 \, x^{2}} + \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-174*x^3+5*x^2-20)/x^3,x, algorithm="maxima")

[Out]

-174/25*x + 2/5/x^2 + 1/5*log(x)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \begin {gather*} \frac {\ln \relax (x)}{5}-\frac {174\,x}{25}+\frac {2}{5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((174*x^3)/25 - x^2/5 + 4/5)/x^3,x)

[Out]

log(x)/5 - (174*x)/25 + 2/(5*x^2)

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sympy [A]  time = 0.08, size = 15, normalized size = 0.88 \begin {gather*} - \frac {174 x}{25} + \frac {\log {\relax (x )}}{5} + \frac {2}{5 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-174*x**3+5*x**2-20)/x**3,x)

[Out]

-174*x/25 + log(x)/5 + 2/(5*x**2)

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