3.17.5 $\int \frac{-10x-x^5+(15x+2x^4)\log (x^2)+(-10-x^3){\log }^2(x^2)}{2x^5-4x^4\log (x^2)+2x^3{\log }^2(x^2)}dx$

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

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Rubi [F]  time = 0.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{-10x-x^5+(15x+2x^4)\log \left(x^2\right)+(-10-x^3){\log }^2\left(x^2\right)}{2x^5-4x^4\log \left(x^2\right)+2x^3{\log }^2\left(x^2\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-10*x - x^5 + (15*x + 2*x^4)*Log[x^2] + (-10 - x^3)*Log[x^2]^2)/(2*x^5 - 4*x^4*Log[x^2] + 2*x^3*Log[x^2]^
2),x]

[Out]

5/(2*x^2) - x/2 - 5*Defer[Int][1/(x^2*(x - Log[x^2])^2), x] + (5*Defer[Int][1/(x*(x - Log[x^2])^2), x])/2 + (5
*Defer[Int][1/(x^2*(x - Log[x^2])), x])/2

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-10x-x^5+(15x+2x^4)\log \left(x^2\right)+(-10-x^3){\log }^2\left(x^2\right)}{2x^3{(x-\log \left(x^2\right))}^2}dx\\ & =\frac{1}{2}\int \frac{-10x-x^5+(15x+2x^4)\log \left(x^2\right)+(-10-x^3){\log }^2\left(x^2\right)}{x^3{(x-\log \left(x^2\right))}^2}dx\\ & =\frac{1}{2}\int (\frac{-10-x^3}{x^3}+\frac{5(-2+x)}{x^2{(x-\log \left(x^2\right))}^2}+\frac{5}{x^2(x-\log \left(x^2\right))})dx\\ & =\frac{1}{2}\int \frac{-10-x^3}{x^3}dx+\frac{5}{2}\int \frac{-2+x}{x^2{(x-\log \left(x^2\right))}^2}dx+\frac{5}{2}\int \frac{1}{x^2(x-\log \left(x^2\right))}dx\\ & =\frac{1}{2}\int (-1-\frac{10}{x^3})dx+\frac{5}{2}\int (-\frac{2}{x^2{(x-\log \left(x^2\right))}^2}+\frac{1}{x{(x-\log \left(x^2\right))}^2})dx+\frac{5}{2}\int \frac{1}{x^2(x-\log \left(x^2\right))}dx\\ & =\frac{5}{2x^2}-\frac{x}{2}+\frac{5}{2}\int \frac{1}{x{(x-\log \left(x^2\right))}^2}dx+\frac{5}{2}\int \frac{1}{x^2(x-\log \left(x^2\right))}dx-5\int \frac{1}{x^2{(x-\log \left(x^2\right))}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.15, size = 28, normalized size = 0.93 $$\begin{array}{c}\frac{1}{2}(\frac{5}{x^2}-x+\frac{5}{x(-x+\log \left(x^2\right))})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-10*x - x^5 + (15*x + 2*x^4)*Log[x^2] + (-10 - x^3)*Log[x^2]^2)/(2*x^5 - 4*x^4*Log[x^2] + 2*x^3*Log
[x^2]^2),x]

[Out]

(5/x^2 - x + 5/(x*(-x + Log[x^2])))/2

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fricas [A]  time = 0.55, size = 32, normalized size = 1.07 $$\begin{array}{c}-\frac{x^4-(x^3-5)\log \left(x^2\right)}{2(x^3-x^2\log \left(x^2\right))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="fricas")

[Out]

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

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giac [A]  time = 0.28, size = 24, normalized size = 0.80 $$\begin{array}{c}-\frac{1}{2}x-\frac{5}{2(x^2-x\log \left(x^2\right))}+\frac{5}{2x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 27, normalized size = 0.90

method	result	size
risch	$-\frac{x^3-5}{2x^2}-\frac{5}{2x(-\ln \left(x^2\right)+x)}$	$27$
norman	$\frac{\frac{x^3\ln \left(x^2\right)}{2}-\frac{x^4}{2}-\frac{5\ln \left(x^2\right)}{2}}{x^2(-\ln \left(x^2\right)+x)}$	$36$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-10)*ln(x^2)^2+(2*x^4+15*x)*ln(x^2)-x^5-10*x)/(2*x^3*ln(x^2)^2-4*x^4*ln(x^2)+2*x^5),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 28, normalized size = 0.93 $$\begin{array}{c}-\frac{x^4-2(x^3-5)\log (x)}{2(x^3-2x^2\log (x))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-10)*log(x^2)^2+(2*x^4+15*x)*log(x^2)-x^5-10*x)/(2*x^3*log(x^2)^2-4*x^4*log(x^2)+2*x^5),x, alg
orithm="maxima")

[Out]

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

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mupad [B]  time = 1.15, size = 23, normalized size = 0.77 $$\begin{array}{c}-\frac{x}{2}-\frac{5\ln \left(x^2\right)}{2x^2(x-\ln \left(x^2\right))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x - log(x^2)*(15*x + 2*x^4) + log(x^2)^2*(x^3 + 10) + x^5)/(2*x^5 - 4*x^4*log(x^2) + 2*x^3*log(x^2)^2
),x)

[Out]

- x/2 - (5*log(x^2))/(2*x^2*(x - log(x^2)))

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sympy [A]  time = 0.12, size = 24, normalized size = 0.80 $$\begin{array}{c}-\frac{x}{2}+\frac{5}{-2x^2+2x\log \left(x^2\right)}+\frac{5}{2x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-10)*ln(x**2)**2+(2*x**4+15*x)*ln(x**2)-x**5-10*x)/(2*x**3*ln(x**2)**2-4*x**4*ln(x**2)+2*x**5
),x)

[Out]

-x/2 + 5/(-2*x**2 + 2*x*log(x**2)) + 5/(2*x**2)

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