3.41.46 $\int \frac{-8+22x+13x^2-13x\log (\frac{3e^x{x}^2}{2})+2x{\log }^2(\frac{3e^x{x}^2}{2})}{x{\log }^2(\frac{3e^x{x}^2}{2})}dx$

Optimal. Leaf size=23 $$2x+\frac{4-13x}{\log \left(\frac{3e^x{x}^2}{2}\right)}$$

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Rubi [F]  time = 0.38, 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 = {} $$\begin{array}{c}\int \frac{-8+22x+13x^2-13x\log \left(\frac{3e^x{x}^2}{2}\right)+2x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}{x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x^2)/2]^2)/(x*Log[(3*E^x*x^2)/2]^2),x]

[Out]

2*x + 22*Defer[Int][Log[(3*E^x*x^2)/2]^(-2), x] - 8*Defer[Int][1/(x*Log[(3*E^x*x^2)/2]^2), x] + 13*Defer[Int][
x/Log[(3*E^x*x^2)/2]^2, x] - 13*Defer[Int][Log[(3*E^x*x^2)/2]^(-1), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (2+\frac{-8+22x+13x^2}{x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}-\frac{13}{\log \left(\frac{3e^x{x}^2}{2}\right)})dx\\ & =2x-13\int \frac{1}{\log \left(\frac{3e^x{x}^2}{2}\right)}dx+\int \frac{-8+22x+13x^2}{x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}dx\\ & =2x-13\int \frac{1}{\log \left(\frac{3e^x{x}^2}{2}\right)}dx+\int (\frac{22}{{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}-\frac{8}{x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}+\frac{13x}{{\log }^2\left(\frac{3e^x{x}^2}{2}\right)})dx\\ & =2x-8\int \frac{1}{x{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}dx+13\int \frac{x}{{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}dx-13\int \frac{1}{\log \left(\frac{3e^x{x}^2}{2}\right)}dx+22\int \frac{1}{{\log }^2\left(\frac{3e^x{x}^2}{2}\right)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.67, size = 23, normalized size = 1.00 $$\begin{array}{c}2x+\frac{4-13x}{\log \left(\frac{3e^x{x}^2}{2}\right)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x^2)/2]^2)/(x*Log[(3*E^x*x^2)/2]^2),x
]

[Out]

2*x + (4 - 13*x)/Log[(3*E^x*x^2)/2]

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fricas [A]  time = 1.29, size = 27, normalized size = 1.17 $$\begin{array}{c}\frac{2x\log \left(\frac{3}{2}{x}^2{e}^x\right)-13x+4}{\log \left(\frac{3}{2}{x}^2{e}^x\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="fricas")

[Out]

(2*x*log(3/2*x^2*e^x) - 13*x + 4)/log(3/2*x^2*e^x)

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giac [A]  time = 0.22, size = 21, normalized size = 0.91 $$\begin{array}{c}2x-\frac{13x-4}{x+\log \left(\frac{3}{2}{x}^2\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="giac")

[Out]

2*x - (13*x - 4)/(x + log(3/2*x^2))

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maple [A]  time = 0.25, size = 28, normalized size = 1.22

method	result	size
norman	$\frac{4+2x\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)-13x}{\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)}$	$28$
default	$\frac{(-4\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)+8\ln (x)+4x+26)\ln (x)+4x\ln (x)+2x^2+4-2{(\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)-2\ln (x)-x)}^2+13\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)-26\ln (x)-13x}{\ln \left(\frac{3{\mathrm{e}}^x{x}^2}{2}\right)}$	$83$
risch	$2x-\frac{2i(13x-4)}{\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2+\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+\pi \mathrm{csgn}\left(i{x}^2\right)\mathrm{csgn}\left(i{\mathrm{e}}^x\right)\mathrm{csgn}\left(i{x}^2{\mathrm{e}}^x\right)-\pi \mathrm{csgn}\left(i{x}^2\right)\mathrm{csgn}{\left(i{x}^2{\mathrm{e}}^x\right)}^2-\pi \mathrm{csgn}\left(i{\mathrm{e}}^x\right)\mathrm{csgn}{\left(i{x}^2{\mathrm{e}}^x\right)}^2+\pi \mathrm{csgn}{\left(i{x}^2{\mathrm{e}}^x\right)}^3-2i\ln (2)+2i\ln (3)+4i\ln (x)+2i\ln \left({\mathrm{e}}^x\right)}$	$159$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(3/2*exp(x)*x^2)^2-13*x*ln(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/ln(3/2*exp(x)*x^2)^2,x,method=_RETURNVE
RBOSE)

[Out]

(4+2*x*ln(3/2*exp(x)*x^2)-13*x)/ln(3/2*exp(x)*x^2)

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maxima [A]  time = 0.49, size = 25, normalized size = 1.09 $$\begin{array}{c}2x-\frac{13x-4}{x+\log (3)-\log (2)+2\log (x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="maxima")

[Out]

2*x - (13*x - 4)/(x + log(3) - log(2) + 2*log(x))

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mupad [B]  time = 3.08, size = 21, normalized size = 0.91 $$\begin{array}{c}2x-\frac{13x-4}{\ln \left(\frac{3x^2{\mathrm{e}}^x}{2}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22*x - 13*x*log((3*x^2*exp(x))/2) + 2*x*log((3*x^2*exp(x))/2)^2 + 13*x^2 - 8)/(x*log((3*x^2*exp(x))/2)^2)
,x)

[Out]

2*x - (13*x - 4)/log((3*x^2*exp(x))/2)

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sympy [A]  time = 0.13, size = 19, normalized size = 0.83 $$\begin{array}{c}2x+\frac{4-13x}{\log \left(\frac{3x^2{e}^x}{2}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(3/2*exp(x)*x**2)**2-13*x*ln(3/2*exp(x)*x**2)+13*x**2+22*x-8)/x/ln(3/2*exp(x)*x**2)**2,x)

[Out]

2*x + (4 - 13*x)/log(3*x**2*exp(x)/2)

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