3.67.7 \(\int \frac {352-184 x^2+4 x^4+e (-44 x+x^3)+(32-4 e x-16 x^2) \log (e+4 x)}{4 e+16 x} \, dx\)

Optimal. Leaf size=17 \[ \left (11-\frac {x^2}{4}+\log (e+4 x)\right )^2 \]

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Rubi [A]  time = 0.14, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6741, 6686} \begin {gather*} \frac {1}{16} \left (-x^2+4 \log (4 x+e)+44\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(352 - 184*x^2 + 4*x^4 + E*(-44*x + x^3) + (32 - 4*E*x - 16*x^2)*Log[E + 4*x])/(4*E + 16*x),x]

[Out]

(44 - x^2 + 4*Log[E + 4*x])^2/16

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (8-e x-4 x^2\right ) \left (44-x^2+4 \log (e+4 x)\right )}{4 e+16 x} \, dx\\ &=\frac {1}{16} \left (44-x^2+4 \log (e+4 x)\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.12 \begin {gather*} \frac {1}{16} \left (-44+x^2-4 \log (e+4 x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(352 - 184*x^2 + 4*x^4 + E*(-44*x + x^3) + (32 - 4*E*x - 16*x^2)*Log[E + 4*x])/(4*E + 16*x),x]

[Out]

(-44 + x^2 - 4*Log[E + 4*x])^2/16

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fricas [A]  time = 0.59, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {11}{2} \, x^{2} - \frac {1}{2} \, {\left (x^{2} - 44\right )} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1)-16*x^2+32)*log(exp(1)+4*x)+(x^3-44*x)*exp(1)+4*x^4-184*x^2+352)/(4*exp(1)+16*x),x, alg
orithm="fricas")

[Out]

1/16*x^4 - 11/2*x^2 - 1/2*(x^2 - 44)*log(4*x + e) + log(4*x + e)^2

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giac [B]  time = 0.14, size = 41, normalized size = 2.41 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{2} \, x^{2} \log \left (4 \, x + e\right ) - \frac {11}{2} \, x^{2} + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1)-16*x^2+32)*log(exp(1)+4*x)+(x^3-44*x)*exp(1)+4*x^4-184*x^2+352)/(4*exp(1)+16*x),x, alg
orithm="giac")

[Out]

1/16*x^4 - 1/2*x^2*log(4*x + e) - 11/2*x^2 + log(4*x + e)^2 + 22*log(4*x + e)

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maple [B]  time = 0.42, size = 42, normalized size = 2.47

method	result	size
norman	\(\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )-\frac {11 x^{2}}{2}+\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}\)	\(42\)
risch	\(\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )-\frac {11 x^{2}}{2}+\frac {x^{4}}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) x^{2}}{2}\)	\(42\)
derivativedivides	\(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\)	\(144\)
default	\(-\frac {\left ({\mathrm e}+4 x \right ) {\mathrm e}^{3}}{1024}+\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}+4 x \right )^{2}}{2048}-\frac {{\mathrm e} \left ({\mathrm e}+4 x \right )^{3}}{1024}+\frac {\left ({\mathrm e}+4 x \right )^{4}}{4096}+\frac {{\mathrm e} \left (\left ({\mathrm e}+4 x \right ) \ln \left ({\mathrm e}+4 x \right )-{\mathrm e}-4 x \right )}{16}-\frac {\ln \left ({\mathrm e}+4 x \right ) \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {11 \left ({\mathrm e}+4 x \right )^{2}}{32}-\frac {{\mathrm e}^{2} \ln \left ({\mathrm e}+4 x \right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}+4 x \right )}{4}+\ln \left ({\mathrm e}+4 x \right )^{2}+22 \ln \left ({\mathrm e}+4 x \right )\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x*exp(1)-16*x^2+32)*ln(exp(1)+4*x)+(x^3-44*x)*exp(1)+4*x^4-184*x^2+352)/(4*exp(1)+16*x),x,method=_RET
URNVERBOSE)

[Out]

ln(exp(1)+4*x)^2+22*ln(exp(1)+4*x)-11/2*x^2+1/16*x^4-1/2*ln(exp(1)+4*x)*x^2

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maxima [B]  time = 0.45, size = 225, normalized size = 13.24 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{48} \, x^{3} e + \frac {1}{128} \, x^{2} e^{2} + \frac {1}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e \log \left (4 \, x + e\right ) + \frac {1}{32} \, e^{2} \log \left (4 \, x + e\right )^{2} - \frac {11}{2} \, x^{2} - \frac {1}{256} \, x e^{3} + \frac {1}{3072} \, {\left (64 \, x^{3} - 24 \, x^{2} e + 12 \, x e^{2} - 3 \, e^{3} \log \left (4 \, x + e\right )\right )} e - \frac {1}{32} \, {\left (e \log \left (4 \, x + e\right )^{2} + 2 \, e \log \left (4 \, x + e\right ) - 8 \, x\right )} e + \frac {11}{16} \, {\left (e \log \left (4 \, x + e\right ) - 4 \, x\right )} e + \frac {5}{2} \, x e - \frac {1}{16} \, {\left (8 \, x^{2} - 4 \, x e + e^{2} \log \left (4 \, x + e\right )\right )} \log \left (4 \, x + e\right ) + \frac {1}{1024} \, e^{4} \log \left (4 \, x + e\right ) - \frac {5}{8} \, e^{2} \log \left (4 \, x + e\right ) + \log \left (4 \, x + e\right )^{2} + 22 \, \log \left (4 \, x + e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1)-16*x^2+32)*log(exp(1)+4*x)+(x^3-44*x)*exp(1)+4*x^4-184*x^2+352)/(4*exp(1)+16*x),x, alg
orithm="maxima")

[Out]

1/16*x^4 - 1/48*x^3*e + 1/128*x^2*e^2 + 1/16*(e*log(4*x + e) - 4*x)*e*log(4*x + e) + 1/32*e^2*log(4*x + e)^2 -
11/2*x^2 - 1/256*x*e^3 + 1/3072*(64*x^3 - 24*x^2*e + 12*x*e^2 - 3*e^3*log(4*x + e))*e - 1/32*(e*log(4*x + e)^
2 + 2*e*log(4*x + e) - 8*x)*e + 11/16*(e*log(4*x + e) - 4*x)*e + 5/2*x*e - 1/16*(8*x^2 - 4*x*e + e^2*log(4*x +
e))*log(4*x + e) + 1/1024*e^4*log(4*x + e) - 5/8*e^2*log(4*x + e) + log(4*x + e)^2 + 22*log(4*x + e)

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mupad [B]  time = 0.25, size = 41, normalized size = 2.41 \begin {gather*} \frac {x^4}{16}-\frac {x^2\,\ln \left (4\,x+\mathrm {e}\right )}{2}-\frac {11\,x^2}{2}+{\ln \left (4\,x+\mathrm {e}\right )}^2+22\,\ln \left (4\,x+\mathrm {e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1)*(44*x - x^3) + log(4*x + exp(1))*(4*x*exp(1) + 16*x^2 - 32) + 184*x^2 - 4*x^4 - 352)/(16*x + 4*ex
p(1)),x)

[Out]

22*log(4*x + exp(1)) - (x^2*log(4*x + exp(1)))/2 + log(4*x + exp(1))^2 - (11*x^2)/2 + x^4/16

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sympy [B]  time = 0.18, size = 44, normalized size = 2.59 \begin {gather*} \frac {x^{4}}{16} - \frac {x^{2} \log {\left (4 x + e \right )}}{2} - \frac {11 x^{2}}{2} + \log {\left (4 x + e \right )}^{2} + 22 \log {\left (4 x + e \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1)-16*x**2+32)*ln(exp(1)+4*x)+(x**3-44*x)*exp(1)+4*x**4-184*x**2+352)/(4*exp(1)+16*x),x)

[Out]

x**4/16 - x**2*log(4*x + E)/2 - 11*x**2/2 + log(4*x + E)**2 + 22*log(4*x + E)

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