3.27.68 \(\int (x^2)^{16 x^2+8 x^3+x^4} (-32 x-16 x^2-2 x^3+(-32 x-24 x^2-4 x^3) \log (x^2)) \, dx\)

Optimal. Leaf size=18 \[ -\left (x^2\right )^{x^2 (4+x)^2}+\log (3) \]

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Rubi [F]  time = 0.68, 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 \left (x^2\right )^{16 x^2+8 x^3+x^4} \left (-32 x-16 x^2-2 x^3+\left (-32 x-24 x^2-4 x^3\right ) \log \left (x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2)^(16*x^2 + 8*x^3 + x^4)*(-32*x - 16*x^2 - 2*x^3 + (-32*x - 24*x^2 - 4*x^3)*Log[x^2]),x]

[Out]

-32*Defer[Int][x*(x^2)^(x^2*(4 + x)^2), x] - 32*Log[x^2]*Defer[Int][x*(x^2)^(x^2*(4 + x)^2), x] - 2*Defer[Int]
[x^3*(x^2)^(x^2*(4 + x)^2), x] - 4*Log[x^2]*Defer[Int][x^3*(x^2)^(x^2*(4 + x)^2), x] - 16*Defer[Int][(x^2)^(1
+ 16*x^2 + 8*x^3 + x^4), x] - 24*Log[x^2]*Defer[Int][(x^2)^(1 + 16*x^2 + 8*x^3 + x^4), x] + 64*Defer[Int][Defe
r[Int][x*(x^2)^(x^2*(4 + x)^2), x]/x, x] + 8*Defer[Int][Defer[Int][x^3*(x^2)^(x^2*(4 + x)^2), x]/x, x] + 48*De
fer[Int][Defer[Int][(x^2)^(1 + 16*x^2 + 8*x^3 + x^4), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 x \left (x^2\right )^{x^2 (4+x)^2} (4+x) \left (-4-x-2 (2+x) \log \left (x^2\right )\right ) \, dx\\ &=2 \int x \left (x^2\right )^{x^2 (4+x)^2} (4+x) \left (-4-x-2 (2+x) \log \left (x^2\right )\right ) \, dx\\ &=2 \int \left (-x \left (x^2\right )^{x^2 (4+x)^2} (4+x)^2-2 x \left (x^2\right )^{x^2 (4+x)^2} (2+x) (4+x) \log \left (x^2\right )\right ) \, dx\\ &=-\left (2 \int x \left (x^2\right )^{x^2 (4+x)^2} (4+x)^2 \, dx\right )-4 \int x \left (x^2\right )^{x^2 (4+x)^2} (2+x) (4+x) \log \left (x^2\right ) \, dx\\ &=-\left (2 \int \left (16 x \left (x^2\right )^{x^2 (4+x)^2}+x^3 \left (x^2\right )^{x^2 (4+x)^2}+8 \left (x^2\right )^{1+x^2 (4+x)^2}\right ) \, dx\right )+4 \int \frac {2 \left (8 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx+6 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx\right )}{x} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ &=-\left (2 \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx\right )+8 \int \frac {8 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx+6 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx}{x} \, dx-16 \int \left (x^2\right )^{1+x^2 (4+x)^2} \, dx-32 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ &=-\left (2 \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx\right )+8 \int \left (\frac {8 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x}+\frac {6 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx}{x}\right ) \, dx-16 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-32 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ &=-\left (2 \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx\right )+8 \int \frac {8 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x} \, dx-16 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-32 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+48 \int \frac {\int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx}{x} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ &=-\left (2 \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx\right )+8 \int \left (\frac {8 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x}+\frac {\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x}\right ) \, dx-16 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-32 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+48 \int \frac {\int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx}{x} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ &=-\left (2 \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx\right )+8 \int \frac {\int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x} \, dx-16 \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-32 \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx+48 \int \frac {\int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx}{x} \, dx+64 \int \frac {\int x \left (x^2\right )^{x^2 (4+x)^2} \, dx}{x} \, dx-\left (4 \log \left (x^2\right )\right ) \int x^3 \left (x^2\right )^{x^2 (4+x)^2} \, dx-\left (24 \log \left (x^2\right )\right ) \int \left (x^2\right )^{1+16 x^2+8 x^3+x^4} \, dx-\left (32 \log \left (x^2\right )\right ) \int x \left (x^2\right )^{x^2 (4+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 15, normalized size = 0.83 \begin {gather*} -\left (x^2\right )^{x^2 (4+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2)^(16*x^2 + 8*x^3 + x^4)*(-32*x - 16*x^2 - 2*x^3 + (-32*x - 24*x^2 - 4*x^3)*Log[x^2]),x]

[Out]

-(x^2)^(x^2*(4 + x)^2)

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fricas [A]  time = 0.62, size = 20, normalized size = 1.11 \begin {gather*} -{\left (x^{2}\right )}^{x^{4} + 8 \, x^{3} + 16 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-24*x^2-32*x)*log(x^2)-2*x^3-16*x^2-32*x)*exp((x^4+8*x^3+16*x^2)*log(x^2)),x, algorithm="fri
cas")

[Out]

-(x^2)^(x^4 + 8*x^3 + 16*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -2 \, {\left (x^{3} + 8 \, x^{2} + 2 \, {\left (x^{3} + 6 \, x^{2} + 8 \, x\right )} \log \left (x^{2}\right ) + 16 \, x\right )} {\left (x^{2}\right )}^{x^{4} + 8 \, x^{3} + 16 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-24*x^2-32*x)*log(x^2)-2*x^3-16*x^2-32*x)*exp((x^4+8*x^3+16*x^2)*log(x^2)),x, algorithm="gia
c")

[Out]

integrate(-2*(x^3 + 8*x^2 + 2*(x^3 + 6*x^2 + 8*x)*log(x^2) + 16*x)*(x^2)^(x^4 + 8*x^3 + 16*x^2), x)

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maple [A]  time = 0.06, size = 16, normalized size = 0.89

method	result	size
risch	\(-\left (x^{2}\right )^{\left (4+x \right )^{2} x^{2}}\)	\(16\)
default	\(-{\mathrm e}^{\left (x^{4}+8 x^{3}+16 x^{2}\right ) \ln \left (x^{2}\right )}\)	\(23\)
norman	\(-{\mathrm e}^{\left (x^{4}+8 x^{3}+16 x^{2}\right ) \ln \left (x^{2}\right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3-24*x^2-32*x)*ln(x^2)-2*x^3-16*x^2-32*x)*exp((x^4+8*x^3+16*x^2)*ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

-(x^2)^((4+x)^2*x^2)

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maxima [A]  time = 0.45, size = 25, normalized size = 1.39 \begin {gather*} -e^{\left (2 \, x^{4} \log \relax (x) + 16 \, x^{3} \log \relax (x) + 32 \, x^{2} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-24*x^2-32*x)*log(x^2)-2*x^3-16*x^2-32*x)*exp((x^4+8*x^3+16*x^2)*log(x^2)),x, algorithm="max
ima")

[Out]

-e^(2*x^4*log(x) + 16*x^3*log(x) + 32*x^2*log(x))

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mupad [B]  time = 1.58, size = 20, normalized size = 1.11 \begin {gather*} -{\left (x^2\right )}^{x^4+8\,x^3+16\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(log(x^2)*(16*x^2 + 8*x^3 + x^4))*(32*x + log(x^2)*(32*x + 24*x^2 + 4*x^3) + 16*x^2 + 2*x^3),x)

[Out]

-(x^2)^(16*x^2 + 8*x^3 + x^4)

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sympy [A]  time = 0.34, size = 20, normalized size = 1.11 \begin {gather*} - e^{\left (x^{4} + 8 x^{3} + 16 x^{2}\right ) \log {\left (x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3-24*x**2-32*x)*ln(x**2)-2*x**3-16*x**2-32*x)*exp((x**4+8*x**3+16*x**2)*ln(x**2)),x)

[Out]

-exp((x**4 + 8*x**3 + 16*x**2)*log(x**2))

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