3.3.51 \(\int \frac {-3-x+21 x^2+84 x^3+63 x^4}{5 x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {3+x+21 \left (-x-x^2\right )^2-x \log (x)}{5 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 14} \begin {gather*} \frac {21 x^3}{5}+\frac {42 x^2}{5}+\frac {21 x}{5}+\frac {3}{5 x}-\frac {\log (x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - x + 21*x^2 + 84*x^3 + 63*x^4)/(5*x^2),x]

[Out]

3/(5*x) + (21*x)/5 + (42*x^2)/5 + (21*x^3)/5 - 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}{5} \int \frac {-3-x+21 x^2+84 x^3+63 x^4}{x^2} \, dx\\ &=\frac {1}{5} \int \left (21-\frac {3}{x^2}-\frac {1}{x}+84 x+63 x^2\right ) \, dx\\ &=\frac {3}{5 x}+\frac {21 x}{5}+\frac {42 x^2}{5}+\frac {21 x^3}{5}-\frac {\log (x)}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 0.96 \begin {gather*} \frac {1}{5} \left (\frac {3}{x}+21 x+42 x^2+21 x^3-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - x + 21*x^2 + 84*x^3 + 63*x^4)/(5*x^2),x]

[Out]

(3/x + 21*x + 42*x^2 + 21*x^3 - Log[x])/5

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fricas [A]  time = 1.28, size = 27, normalized size = 0.96 \begin {gather*} \frac {21 \, x^{4} + 42 \, x^{3} + 21 \, x^{2} - x \log \relax (x) + 3}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(63*x^4+84*x^3+21*x^2-x-3)/x^2,x, algorithm="fricas")

[Out]

1/5*(21*x^4 + 42*x^3 + 21*x^2 - x*log(x) + 3)/x

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giac [A]  time = 0.45, size = 24, normalized size = 0.86 \begin {gather*} \frac {21}{5} \, x^{3} + \frac {42}{5} \, x^{2} + \frac {21}{5} \, x + \frac {3}{5 \, x} - \frac {1}{5} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(63*x^4+84*x^3+21*x^2-x-3)/x^2,x, algorithm="giac")

[Out]

21/5*x^3 + 42/5*x^2 + 21/5*x + 3/5/x - 1/5*log(abs(x))

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maple [A]  time = 0.03, size = 24, normalized size = 0.86

method	result	size
default	\(\frac {21 x^{3}}{5}+\frac {42 x^{2}}{5}+\frac {21 x}{5}-\frac {\ln \relax (x )}{5}+\frac {3}{5 x}\)	\(24\)
risch	\(\frac {21 x^{3}}{5}+\frac {42 x^{2}}{5}+\frac {21 x}{5}-\frac {\ln \relax (x )}{5}+\frac {3}{5 x}\)	\(24\)
norman	\(\frac {\frac {3}{5}+\frac {21}{5} x^{2}+\frac {42}{5} x^{3}+\frac {21}{5} x^{4}}{x}-\frac {\ln \relax (x )}{5}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(63*x^4+84*x^3+21*x^2-x-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

21/5*x^3+42/5*x^2+21/5*x-1/5*ln(x)+3/5/x

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maxima [A]  time = 0.45, size = 23, normalized size = 0.82 \begin {gather*} \frac {21}{5} \, x^{3} + \frac {42}{5} \, x^{2} + \frac {21}{5} \, x + \frac {3}{5 \, x} - \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(63*x^4+84*x^3+21*x^2-x-3)/x^2,x, algorithm="maxima")

[Out]

21/5*x^3 + 42/5*x^2 + 21/5*x + 3/5/x - 1/5*log(x)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.82 \begin {gather*} \frac {21\,x}{5}-\frac {\ln \relax (x)}{5}+\frac {3}{5\,x}+\frac {42\,x^2}{5}+\frac {21\,x^3}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((21*x^2)/5 - x/5 + (84*x^3)/5 + (63*x^4)/5 - 3/5)/x^2,x)

[Out]

(21*x)/5 - log(x)/5 + 3/(5*x) + (42*x^2)/5 + (21*x^3)/5

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sympy [A]  time = 0.08, size = 27, normalized size = 0.96 \begin {gather*} \frac {21 x^{3}}{5} + \frac {42 x^{2}}{5} + \frac {21 x}{5} - \frac {\log {\relax (x )}}{5} + \frac {3}{5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(63*x**4+84*x**3+21*x**2-x-3)/x**2,x)

[Out]

21*x**3/5 + 42*x**2/5 + 21*x/5 - log(x)/5 + 3/(5*x)

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