3.41.40 $\int \frac{-6+5x+2x^2+x\log (x^2)}{x}dx$

Optimal. Leaf size=19 $$4+3x+x^2-(3-x)\log \left(x^2\right)$$

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 2, integrand size = 20, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.100, Rules used = {14, 2295} $$\begin{array}{c}{x}^2+x\log \left(x^2\right)+3x-6\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-6 + 5*x + 2*x^2 + x*Log[x^2])/x,x]

[Out]

3*x + x^2 - 6*Log[x] + x*Log[x^2]

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]]

Rule 2295

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{-6+5x+2x^2}{x}+\log \left(x^2\right))dx\\ & =\int \frac{-6+5x+2x^2}{x}dx+\int \log \left(x^2\right)dx\\ & =-2x+x\log \left(x^2\right)+\int (5-\frac{6}{x}+2x)dx\\ & =3x+x^2-6\log (x)+x\log \left(x^2\right)\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.89 $$\begin{array}{c}3x+x^2-6\log (x)+x\log \left(x^2\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 5*x + 2*x^2 + x*Log[x^2])/x,x]

[Out]

3*x + x^2 - 6*Log[x] + x*Log[x^2]

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fricas [A]  time = 0.98, size = 15, normalized size = 0.79 $$\begin{array}{c}{x}^2+(x-3)\log \left(x^2\right)+3x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^2)+2*x^2+5*x-6)/x,x, algorithm="fricas")

[Out]

x^2 + (x - 3)*log(x^2) + 3*x

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giac [A]  time = 0.15, size = 17, normalized size = 0.89 $$\begin{array}{c}{x}^2+x\log \left(x^2\right)+3x-6\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^2)+2*x^2+5*x-6)/x,x, algorithm="giac")

[Out]

x^2 + x*log(x^2) + 3*x - 6*log(x)

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maple [A]  time = 0.02, size = 18, normalized size = 0.95

method	result	size
default	$3x+x^2-6\ln (x)+x\ln \left(x^2\right)$	$18$
norman	$3x+x^2-6\ln (x)+x\ln \left(x^2\right)$	$18$
risch	$3x+x^2-6\ln (x)+x\ln \left(x^2\right)$	$18$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(x^2)+2*x^2+5*x-6)/x,x,method=_RETURNVERBOSE)

[Out]

3*x+x^2-6*ln(x)+x*ln(x^2)

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maxima [A]  time = 0.37, size = 17, normalized size = 0.89 $$\begin{array}{c}{x}^2+x\log \left(x^2\right)+3x-6\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^2)+2*x^2+5*x-6)/x,x, algorithm="maxima")

[Out]

x^2 + x*log(x^2) + 3*x - 6*log(x)

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mupad [B]  time = 3.01, size = 19, normalized size = 1.00 $$\begin{array}{c}3x-3\ln \left(x^2\right)+x\ln \left(x^2\right)+x^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + x*log(x^2) + 2*x^2 - 6)/x,x)

[Out]

3*x - 3*log(x^2) + x*log(x^2) + x^2

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sympy [A]  time = 0.10, size = 17, normalized size = 0.89 $$\begin{array}{c}{x}^2+x\log \left(x^2\right)+3x-6\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x**2)+2*x**2+5*x-6)/x,x)

[Out]

x**2 + x*log(x**2) + 3*x - 6*log(x)

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