3.90.72 \(\int \frac {1-144 x-96 x^2+(-720 x-480 x^2) \log (5)+(-900 x-600 x^2) \log ^2(5)}{x} \, dx\) [8972]

Optimal. Leaf size=17 \[ -12 x (3+x) (2+5 \log (5))^2+\log (x) \]

[Out]

ln(x)-12*(3+x)*(2+5*ln(5))^2*x

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.59, number of steps used = 2, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {14} \begin {gather*} -12 x^2 (2+5 \log (5))^2-36 x (2+5 \log (5))^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 144*x - 96*x^2 + (-720*x - 480*x^2)*Log[5] + (-900*x - 600*x^2)*Log[5]^2)/x,x]

[Out]

-36*x*(2 + 5*Log[5])^2 - 12*x^2*(2 + 5*Log[5])^2 + Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}-36 (2+5 \log (5))^2-24 x (2+5 \log (5))^2\right ) \, dx\\ &=-36 x (2+5 \log (5))^2-12 x^2 (2+5 \log (5))^2+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -12 x (3+x) (2+5 \log (5))^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 144*x - 96*x^2 + (-720*x - 480*x^2)*Log[5] + (-900*x - 600*x^2)*Log[5]^2)/x,x]

[Out]

-12*x*(3 + x)*(2 + 5*Log[5])^2 + Log[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(17)=34\).
time = 0.62, size = 40, normalized size = 2.35

method	result	size
norman	\(\left (-900 \ln \left (5\right )^{2}-720 \ln \left (5\right )-144\right ) x +\left (-300 \ln \left (5\right )^{2}-240 \ln \left (5\right )-48\right ) x^{2}+\ln \left (x \right )\)	\(34\)
default	\(-300 x^{2} \ln \left (5\right )^{2}-900 x \ln \left (5\right )^{2}-240 x^{2} \ln \left (5\right )-720 x \ln \left (5\right )-48 x^{2}-144 x +\ln \left (x \right )\)	\(40\)
risch	\(-300 x^{2} \ln \left (5\right )^{2}-900 x \ln \left (5\right )^{2}-240 x^{2} \ln \left (5\right )-720 x \ln \left (5\right )-48 x^{2}-144 x +\ln \left (x \right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-600*x^2-900*x)*ln(5)^2+(-480*x^2-720*x)*ln(5)-96*x^2-144*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

-300*x^2*ln(5)^2-900*x*ln(5)^2-240*x^2*ln(5)-720*x*ln(5)-48*x^2-144*x+ln(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.25, size = 35, normalized size = 2.06 \begin {gather*} -12 \, {\left (25 \, \log \left (5\right )^{2} + 20 \, \log \left (5\right ) + 4\right )} x^{2} - 36 \, {\left (25 \, \log \left (5\right )^{2} + 20 \, \log \left (5\right ) + 4\right )} x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-600*x^2-900*x)*log(5)^2+(-480*x^2-720*x)*log(5)-96*x^2-144*x+1)/x,x, algorithm="maxima")

[Out]

-12*(25*log(5)^2 + 20*log(5) + 4)*x^2 - 36*(25*log(5)^2 + 20*log(5) + 4)*x + log(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.35, size = 35, normalized size = 2.06 \begin {gather*} -300 \, {\left (x^{2} + 3 \, x\right )} \log \left (5\right )^{2} - 48 \, x^{2} - 240 \, {\left (x^{2} + 3 \, x\right )} \log \left (5\right ) - 144 \, x + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-600*x^2-900*x)*log(5)^2+(-480*x^2-720*x)*log(5)-96*x^2-144*x+1)/x,x, algorithm="fricas")

[Out]

-300*(x^2 + 3*x)*log(5)^2 - 48*x^2 - 240*(x^2 + 3*x)*log(5) - 144*x + log(x)

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Sympy [A]
time = 0.04, size = 34, normalized size = 2.00 \begin {gather*} - x^{2} \cdot \left (48 + 240 \log {\left (5 \right )} + 300 \log {\left (5 \right )}^{2}\right ) - x \left (144 + 720 \log {\left (5 \right )} + 900 \log {\left (5 \right )}^{2}\right ) + \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-600*x**2-900*x)*ln(5)**2+(-480*x**2-720*x)*ln(5)-96*x**2-144*x+1)/x,x)

[Out]

-x**2*(48 + 240*log(5) + 300*log(5)**2) - x*(144 + 720*log(5) + 900*log(5)**2) + log(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (17) = 34\).
time = 0.41, size = 40, normalized size = 2.35 \begin {gather*} -300 \, x^{2} \log \left (5\right )^{2} - 240 \, x^{2} \log \left (5\right ) - 900 \, x \log \left (5\right )^{2} - 48 \, x^{2} - 720 \, x \log \left (5\right ) - 144 \, x + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-600*x^2-900*x)*log(5)^2+(-480*x^2-720*x)*log(5)-96*x^2-144*x+1)/x,x, algorithm="giac")

[Out]

-300*x^2*log(5)^2 - 240*x^2*log(5) - 900*x*log(5)^2 - 48*x^2 - 720*x*log(5) - 144*x + log(abs(x))

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Mupad [B]
time = 0.07, size = 27, normalized size = 1.59 \begin {gather*} \ln \left (x\right )-36\,x\,{\left (5\,\ln \left (5\right )+2\right )}^2-12\,x^2\,{\left (5\,\ln \left (5\right )+2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(144*x + log(5)*(720*x + 480*x^2) + log(5)^2*(900*x + 600*x^2) + 96*x^2 - 1)/x,x)

[Out]

log(x) - 36*x*(5*log(5) + 2)^2 - 12*x^2*(5*log(5) + 2)^2

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