3.43.50 \(\int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+(2500 x^4-32 x^8+8 x^9) \log (x)+4 x^8 \log ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {5 x}{-4+\frac {625}{2 x^4}+x+\log (x)} \]

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Rubi [F]  time = 1.02, 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 {gather*} \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(31250*x^4 - 100*x^8 + 20*x^8*Log[x])/(390625 - 10000*x^4 + 2500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + (2500*x^
4 - 32*x^8 + 8*x^9)*Log[x] + 4*x^8*Log[x]^2),x]

[Out]

25000*Defer[Int][x^4/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])^2, x] - 20*Defer[Int][x^8/(625 - 8*x^4 + 2*x^5 + 2*x
^4*Log[x])^2, x] - 20*Defer[Int][x^9/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])^2, x] + 10*Defer[Int][x^4/(625 - 8*x
^4 + 2*x^5 + 2*x^4*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x^4 \left (3125-10 x^4+2 x^4 \log (x)\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx\\ &=10 \int \frac {x^4 \left (3125-10 x^4+2 x^4 \log (x)\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx\\ &=10 \int \left (-\frac {2 x^4 \left (-1250+x^4+x^5\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)}\right ) \, dx\\ &=10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \frac {x^4 \left (-1250+x^4+x^5\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx\\ &=10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \left (-\frac {1250 x^4}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^8}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^9}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}\right ) \, dx\\ &=10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \frac {x^8}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx-20 \int \frac {x^9}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx+25000 \int \frac {x^4}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.12, size = 26, normalized size = 1.53 \begin {gather*} \frac {10 x^5}{625-8 x^4+2 x^5+2 x^4 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(31250*x^4 - 100*x^8 + 20*x^8*Log[x])/(390625 - 10000*x^4 + 2500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + (2
500*x^4 - 32*x^8 + 8*x^9)*Log[x] + 4*x^8*Log[x]^2),x]

[Out]

(10*x^5)/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])

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fricas [A]  time = 0.46, size = 26, normalized size = 1.53 \begin {gather*} \frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \relax (x) - 8 \, x^{4} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="fricas")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

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giac [A]  time = 0.18, size = 26, normalized size = 1.53 \begin {gather*} \frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \relax (x) - 8 \, x^{4} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="giac")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

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maple [A]  time = 0.02, size = 27, normalized size = 1.59

method	result	size
risch	\(\frac {10 x^{5}}{2 x^{4} \ln \relax (x )+2 x^{5}-8 x^{4}+625}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x^8*ln(x)-100*x^8+31250*x^4)/(4*x^8*ln(x)^2+(8*x^9-32*x^8+2500*x^4)*ln(x)+4*x^10-32*x^9+64*x^8+2500*x^
5-10000*x^4+390625),x,method=_RETURNVERBOSE)

[Out]

10*x^5/(2*x^4*ln(x)+2*x^5-8*x^4+625)

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maxima [A]  time = 0.38, size = 26, normalized size = 1.53 \begin {gather*} \frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \relax (x) - 8 \, x^{4} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="maxima")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

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mupad [B]  time = 3.15, size = 26, normalized size = 1.53 \begin {gather*} \frac {10\,x^5}{2\,x^4\,\ln \relax (x)-8\,x^4+2\,x^5+625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x^8*log(x) + 31250*x^4 - 100*x^8)/(log(x)*(2500*x^4 - 32*x^8 + 8*x^9) + 4*x^8*log(x)^2 - 10000*x^4 + 2
500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + 390625),x)

[Out]

(10*x^5)/(2*x^4*log(x) - 8*x^4 + 2*x^5 + 625)

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sympy [A]  time = 0.16, size = 24, normalized size = 1.41 \begin {gather*} \frac {10 x^{5}}{2 x^{5} + 2 x^{4} \log {\relax (x )} - 8 x^{4} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x**8*ln(x)-100*x**8+31250*x**4)/(4*x**8*ln(x)**2+(8*x**9-32*x**8+2500*x**4)*ln(x)+4*x**10-32*x**
9+64*x**8+2500*x**5-10000*x**4+390625),x)

[Out]

10*x**5/(2*x**5 + 2*x**4*log(x) - 8*x**4 + 625)

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