3.74.70 \(\int \frac {1}{4} (x^2)^{\frac {1}{4} (125 x+100 x \log (x))} (250+200 \log (x)+(225+100 \log (x)) \log (x^2)) \, dx\)

Optimal. Leaf size=16 \[ \left (x^2\right )^{25 \left (\frac {5 x}{4}+x \log (x)\right )} \]

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Rubi [F]  time = 0.58, 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 {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((x^2)^((125*x + 100*x*Log[x])/4)*(250 + 200*Log[x] + (225 + 100*Log[x])*Log[x^2]))/4,x]

[Out]

(125*Defer[Int][(x^2)^((25*x*(5 + 4*Log[x]))/4), x])/2 + 50*Defer[Int][(x^2)^((25*x*(5 + 4*Log[x]))/4)*Log[x],
x] + (225*Defer[Int][(x^2)^((25*x*(5 + 4*Log[x]))/4)*Log[x^2], x])/4 + 25*Defer[Int][(x^2)^((25*x*(5 + 4*Log[
x]))/4)*Log[x]*Log[x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \left (250 \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))}+200 \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x)+25 \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} (9+4 \log (x)) \log \left (x^2\right )\right ) \, dx\\ &=\frac {25}{4} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} (9+4 \log (x)) \log \left (x^2\right ) \, dx+50 \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x) \, dx+\frac {125}{2} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \, dx\\ &=\frac {25}{4} \int \left (9 \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log \left (x^2\right )+4 \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x) \log \left (x^2\right )\right ) \, dx+50 \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x) \, dx+\frac {125}{2} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \, dx\\ &=25 \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x) \log \left (x^2\right ) \, dx+50 \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log (x) \, dx+\frac {225}{4} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \log \left (x^2\right ) \, dx+\frac {125}{2} \int \left (x^2\right )^{\frac {25}{4} x (5+4 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 19, normalized size = 1.19 \begin {gather*} x^{25 x \log \left (x^2\right )} \left (x^2\right )^{125 x/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((x^2)^((125*x + 100*x*Log[x])/4)*(250 + 200*Log[x] + (225 + 100*Log[x])*Log[x^2]))/4,x]

[Out]

x^(25*x*Log[x^2])*(x^2)^((125*x)/4)

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fricas [A]  time = 0.65, size = 14, normalized size = 0.88 \begin {gather*} e^{\left (50 \, x \log \relax (x)^{2} + \frac {125}{2} \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*log(x)+125*x)*log(x^2)),x, algorithm="
fricas")

[Out]

e^(50*x*log(x)^2 + 125/2*x*log(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {25}{4} \, {\left ({\left (4 \, \log \relax (x) + 9\right )} \log \left (x^{2}\right ) + 8 \, \log \relax (x) + 10\right )} {\left (x^{2}\right )}^{25 \, x \log \relax (x) + \frac {125}{4} \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*log(x)+125*x)*log(x^2)),x, algorithm="
giac")

[Out]

integrate(25/4*((4*log(x) + 9)*log(x^2) + 8*log(x) + 10)*(x^2)^(25*x*log(x) + 125/4*x), x)

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maple [A]  time = 0.07, size = 17, normalized size = 1.06

method	result	size
default	\({\mathrm e}^{\frac {\left (100 x \ln \relax (x )+125 x \right ) \ln \left (x^{2}\right )}{4}}\)	\(17\)
norman	\({\mathrm e}^{\frac {\left (100 x \ln \relax (x )+125 x \right ) \ln \left (x^{2}\right )}{4}}\)	\(17\)
risch	\({\mathrm e}^{\frac {25 x \left (4 \ln \relax (x )+5\right ) \left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 \ln \relax (x )\right )}{8}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((100*ln(x)+225)*ln(x^2)+200*ln(x)+250)*exp(1/4*(100*x*ln(x)+125*x)*ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

exp(1/4*(100*x*ln(x)+125*x)*ln(x^2))

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maxima [A]  time = 0.46, size = 14, normalized size = 0.88 \begin {gather*} e^{\left (50 \, x \log \relax (x)^{2} + \frac {125}{2} \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*log(x)+125*x)*log(x^2)),x, algorithm="
maxima")

[Out]

e^(50*x*log(x)^2 + 125/2*x*log(x))

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mupad [B]  time = 5.33, size = 17, normalized size = 1.06 \begin {gather*} x^{25\,x\,\ln \left (x^2\right )}\,{\left (x^2\right )}^{\frac {125\,x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x^2)*(125*x + 100*x*log(x)))/4)*(200*log(x) + log(x^2)*(100*log(x) + 225) + 250))/4,x)

[Out]

x^(25*x*log(x^2))*(x^2)^((125*x)/4)

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sympy [A]  time = 0.31, size = 17, normalized size = 1.06 \begin {gather*} e^{2 \left (25 x \log {\relax (x )} + \frac {125 x}{4}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((100*ln(x)+225)*ln(x**2)+200*ln(x)+250)*exp(1/4*(100*x*ln(x)+125*x)*ln(x**2)),x)

[Out]

exp(2*(25*x*log(x) + 125*x/4)*log(x))

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