3.15.34 \(\int (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+(1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)) \log (5)) \, dx\)

Optimal. Leaf size=22 \[ 4 \left (36 x^2 (-x+x \log (4))^2+\log (5)\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 9, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6, 12, 30} \begin {gather*} 5184 x^8 (1-\log (4))^4+288 x^4 (1-\log (4))^2 \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[41472*x^7 - 165888*x^7*Log[4] + 248832*x^7*Log[4]^2 - 165888*x^7*Log[4]^3 + 41472*x^7*Log[4]^4 + (1152*x^3
- 2304*x^3*Log[4] + 1152*x^3*Log[4]^2)*Log[5],x]

[Out]

5184*x^8*(1 - Log[4])^4 + 288*x^4*(1 - Log[4])^2*Log[5]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (x^7 (41472-165888 \log (4))+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx\\ &=\int \left (-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+x^7 \left (41472-165888 \log (4)+248832 \log ^2(4)\right )+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx\\ &=\int \left (x^7 \left (41472-165888 \log (4)+248832 \log ^2(4)\right )+x^7 \left (-165888 \log ^3(4)+41472 \log ^4(4)\right )+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx\\ &=\int \left (x^7 \left (41472-165888 \log (4)+248832 \log ^2(4)-165888 \log ^3(4)+41472 \log ^4(4)\right )+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx\\ &=5184 x^8 (1-\log (4))^4+\log (5) \int \left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \, dx\\ &=5184 x^8 (1-\log (4))^4+\log (5) \int \left (x^3 (1152-2304 \log (4))+1152 x^3 \log ^2(4)\right ) \, dx\\ &=5184 x^8 (1-\log (4))^4+\log (5) \int x^3 \left (1152-2304 \log (4)+1152 \log ^2(4)\right ) \, dx\\ &=5184 x^8 (1-\log (4))^4+\left (1152 (1-\log (4))^2 \log (5)\right ) \int x^3 \, dx\\ &=5184 x^8 (1-\log (4))^4+288 x^4 (1-\log (4))^2 \log (5)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.41 \begin {gather*} 1152 (-1+\log (4))^2 \left (\frac {9}{2} x^8 (-1+\log (4))^2+\frac {1}{4} x^4 \log (5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[41472*x^7 - 165888*x^7*Log[4] + 248832*x^7*Log[4]^2 - 165888*x^7*Log[4]^3 + 41472*x^7*Log[4]^4 + (11
52*x^3 - 2304*x^3*Log[4] + 1152*x^3*Log[4]^2)*Log[5],x]

[Out]

1152*(-1 + Log[4])^2*((9*x^8*(-1 + Log[4])^2)/2 + (x^4*Log[5])/4)

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fricas [B]  time = 1.37, size = 64, normalized size = 2.91 \begin {gather*} 82944 \, x^{8} \log \relax (2)^{4} - 165888 \, x^{8} \log \relax (2)^{3} + 124416 \, x^{8} \log \relax (2)^{2} - 41472 \, x^{8} \log \relax (2) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (2) + x^{4}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*log(2)^4-1327104*x^7*log(2)^3+995328*
x^7*log(2)^2-331776*x^7*log(2)+41472*x^7,x, algorithm="fricas")

[Out]

82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8*log(2) + 5184*x^8 + 288*(4*x^4*log(
2)^2 - 4*x^4*log(2) + x^4)*log(5)

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giac [B]  time = 0.23, size = 64, normalized size = 2.91 \begin {gather*} 82944 \, x^{8} \log \relax (2)^{4} - 165888 \, x^{8} \log \relax (2)^{3} + 124416 \, x^{8} \log \relax (2)^{2} - 41472 \, x^{8} \log \relax (2) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (2) + x^{4}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*log(2)^4-1327104*x^7*log(2)^3+995328*
x^7*log(2)^2-331776*x^7*log(2)+41472*x^7,x, algorithm="giac")

[Out]

82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8*log(2) + 5184*x^8 + 288*(4*x^4*log(
2)^2 - 4*x^4*log(2) + x^4)*log(5)

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maple [A]  time = 0.03, size = 42, normalized size = 1.91

method	result	size
gosper	\(288 \left (4 \ln \relax (2)^{2}-4 \ln \relax (2)+1\right ) \left (72 x^{4} \ln \relax (2)^{2}-72 x^{4} \ln \relax (2)+18 x^{4}+\ln \relax (5)\right ) x^{4}\)	\(42\)
norman	\(\left (1152 \ln \relax (2)^{2} \ln \relax (5)-1152 \ln \relax (2) \ln \relax (5)+288 \ln \relax (5)\right ) x^{4}+\left (82944 \ln \relax (2)^{4}-165888 \ln \relax (2)^{3}+124416 \ln \relax (2)^{2}-41472 \ln \relax (2)+5184\right ) x^{8}\)	\(53\)
default	\(\ln \relax (5) \left (1152 x^{4} \ln \relax (2)^{2}-1152 x^{4} \ln \relax (2)+288 x^{4}\right )+82944 x^{8} \ln \relax (2)^{4}-165888 x^{8} \ln \relax (2)^{3}+124416 x^{8} \ln \relax (2)^{2}-41472 x^{8} \ln \relax (2)+5184 x^{8}\)	\(66\)
risch	\(82944 x^{8} \ln \relax (2)^{4}-165888 x^{8} \ln \relax (2)^{3}+124416 x^{8} \ln \relax (2)^{2}-41472 x^{8} \ln \relax (2)+5184 x^{8}+1152 x^{4} \ln \relax (5) \ln \relax (2)^{2}-1152 x^{4} \ln \relax (5) \ln \relax (2)+288 x^{4} \ln \relax (5)\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4608*x^3*ln(2)^2-4608*x^3*ln(2)+1152*x^3)*ln(5)+663552*x^7*ln(2)^4-1327104*x^7*ln(2)^3+995328*x^7*ln(2)^2
-331776*x^7*ln(2)+41472*x^7,x,method=_RETURNVERBOSE)

[Out]

288*(4*ln(2)^2-4*ln(2)+1)*(72*x^4*ln(2)^2-72*x^4*ln(2)+18*x^4+ln(5))*x^4

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maxima [B]  time = 0.60, size = 64, normalized size = 2.91 \begin {gather*} 82944 \, x^{8} \log \relax (2)^{4} - 165888 \, x^{8} \log \relax (2)^{3} + 124416 \, x^{8} \log \relax (2)^{2} - 41472 \, x^{8} \log \relax (2) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (2) + x^{4}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*log(2)^4-1327104*x^7*log(2)^3+995328*
x^7*log(2)^2-331776*x^7*log(2)+41472*x^7,x, algorithm="maxima")

[Out]

82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8*log(2) + 5184*x^8 + 288*(4*x^4*log(
2)^2 - 4*x^4*log(2) + x^4)*log(5)

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mupad [B]  time = 1.08, size = 37, normalized size = 1.68 \begin {gather*} 288\,x^4\,{\left (2\,\ln \relax (2)-1\right )}^2\,\left (\ln \relax (5)+72\,x^4\,{\ln \relax (2)}^2-72\,x^4\,\ln \relax (2)+18\,x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(995328*x^7*log(2)^2 - 1327104*x^7*log(2)^3 + 663552*x^7*log(2)^4 + log(5)*(4608*x^3*log(2)^2 - 4608*x^3*lo
g(2) + 1152*x^3) - 331776*x^7*log(2) + 41472*x^7,x)

[Out]

288*x^4*(2*log(2) - 1)^2*(log(5) + 72*x^4*log(2)^2 - 72*x^4*log(2) + 18*x^4)

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sympy [B]  time = 0.07, size = 56, normalized size = 2.55 \begin {gather*} x^{8} \left (- 165888 \log {\relax (2 )}^{3} - 41472 \log {\relax (2 )} + 5184 + 82944 \log {\relax (2 )}^{4} + 124416 \log {\relax (2 )}^{2}\right ) + x^{4} \left (- 1152 \log {\relax (2 )} \log {\relax (5 )} + 288 \log {\relax (5 )} + 1152 \log {\relax (2 )}^{2} \log {\relax (5 )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4608*x**3*ln(2)**2-4608*x**3*ln(2)+1152*x**3)*ln(5)+663552*x**7*ln(2)**4-1327104*x**7*ln(2)**3+9953
28*x**7*ln(2)**2-331776*x**7*ln(2)+41472*x**7,x)

[Out]

x**8*(-165888*log(2)**3 - 41472*log(2) + 5184 + 82944*log(2)**4 + 124416*log(2)**2) + x**4*(-1152*log(2)*log(5
) + 288*log(5) + 1152*log(2)**2*log(5))

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