3.43.33 \(\int (e^{4 x+4 x^2+x^3} (1296+2880 x+2016 x^2+432 x^3)+e^{8 x+8 x^2+2 x^3} (144+392 x+384 x^2+160 x^3+24 x^4)) \, dx\) [4233]

Optimal. Leaf size=18 \[ \left (36+2 e^{x (2+x)^2} (2+x)\right )^2 \]

[Out]

(36+2*exp(x*(2+x)^2)*(2+x))^2

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Rubi [F]
time = 0.70, 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 (e^{4 x+4 x^2+x^3} \left (1296+2880 x+2016 x^2+432 x^3\right )+e^{8 x+8 x^2+2 x^3} \left (144+392 x+384 x^2+160 x^3+24 x^4\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(4*x + 4*x^2 + x^3)*(1296 + 2880*x + 2016*x^2 + 432*x^3) + E^(8*x + 8*x^2 + 2*x^3)*(144 + 392*x + 384*x^
2 + 160*x^3 + 24*x^4),x]

[Out]

1296*Defer[Int][E^(x*(2 + x)^2), x] + 144*Defer[Int][E^(2*x*(2 + x)^2), x] + 2880*Defer[Int][E^(x*(2 + x)^2)*x
, x] + 392*Defer[Int][E^(2*x*(2 + x)^2)*x, x] + 2016*Defer[Int][E^(x*(2 + x)^2)*x^2, x] + 384*Defer[Int][E^(2*
x*(2 + x)^2)*x^2, x] + 432*Defer[Int][E^(x*(2 + x)^2)*x^3, x] + 160*Defer[Int][E^(2*x*(2 + x)^2)*x^3, x] + 24*
Defer[Int][E^(2*x*(2 + x)^2)*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{4 x+4 x^2+x^3} \left (1296+2880 x+2016 x^2+432 x^3\right ) \, dx+\int e^{8 x+8 x^2+2 x^3} \left (144+392 x+384 x^2+160 x^3+24 x^4\right ) \, dx\\ &=\int \left (1296 e^{4 x+4 x^2+x^3}+2880 e^{4 x+4 x^2+x^3} x+2016 e^{4 x+4 x^2+x^3} x^2+432 e^{4 x+4 x^2+x^3} x^3\right ) \, dx+\int \left (144 e^{8 x+8 x^2+2 x^3}+392 e^{8 x+8 x^2+2 x^3} x+384 e^{8 x+8 x^2+2 x^3} x^2+160 e^{8 x+8 x^2+2 x^3} x^3+24 e^{8 x+8 x^2+2 x^3} x^4\right ) \, dx\\ &=24 \int e^{8 x+8 x^2+2 x^3} x^4 \, dx+144 \int e^{8 x+8 x^2+2 x^3} \, dx+160 \int e^{8 x+8 x^2+2 x^3} x^3 \, dx+384 \int e^{8 x+8 x^2+2 x^3} x^2 \, dx+392 \int e^{8 x+8 x^2+2 x^3} x \, dx+432 \int e^{4 x+4 x^2+x^3} x^3 \, dx+1296 \int e^{4 x+4 x^2+x^3} \, dx+2016 \int e^{4 x+4 x^2+x^3} x^2 \, dx+2880 \int e^{4 x+4 x^2+x^3} x \, dx\\ &=24 \int e^{2 x (2+x)^2} x^4 \, dx+144 \int e^{2 x (2+x)^2} \, dx+160 \int e^{2 x (2+x)^2} x^3 \, dx+384 \int e^{2 x (2+x)^2} x^2 \, dx+392 \int e^{2 x (2+x)^2} x \, dx+432 \int e^{x (2+x)^2} x^3 \, dx+1296 \int e^{x (2+x)^2} \, dx+2016 \int e^{x (2+x)^2} x^2 \, dx+2880 \int e^{x (2+x)^2} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.07, size = 19, normalized size = 1.06 \begin {gather*} 4 \left (18+e^{x (2+x)^2} (2+x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x + 4*x^2 + x^3)*(1296 + 2880*x + 2016*x^2 + 432*x^3) + E^(8*x + 8*x^2 + 2*x^3)*(144 + 392*x +
384*x^2 + 160*x^3 + 24*x^4),x]

[Out]

4*(18 + E^(x*(2 + x)^2)*(2 + x))^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(17)=34\).
time = 0.12, size = 88, normalized size = 4.89

method	result	size
risch	\(\left (4 x^{2}+16 x +16\right ) {\mathrm e}^{2 x \left (2+x \right )^{2}}+\left (288+144 x \right ) {\mathrm e}^{x \left (2+x \right )^{2}}\)	\(36\)
default	\(144 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x} x +288 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x}+16 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x}+16 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x} x +4 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x} x^{2}\)	\(88\)
norman	\(144 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x} x +288 \,{\mathrm e}^{x^{3}+4 x^{2}+4 x}+16 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x}+16 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x} x +4 \,{\mathrm e}^{2 x^{3}+8 x^{2}+8 x} x^{2}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^4+160*x^3+384*x^2+392*x+144)*exp(x^3+4*x^2+4*x)^2+(432*x^3+2016*x^2+2880*x+1296)*exp(x^3+4*x^2+4*x),
x,method=_RETURNVERBOSE)

[Out]

144*exp(x^3+4*x^2+4*x)*x+288*exp(x^3+4*x^2+4*x)+16*exp(x^3+4*x^2+4*x)^2+16*exp(x^3+4*x^2+4*x)^2*x+4*exp(x^3+4*
x^2+4*x)^2*x^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18) = 36\).
time = 0.32, size = 44, normalized size = 2.44 \begin {gather*} 4 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x^{3} + 8 \, x^{2} + 8 \, x\right )} + 144 \, {\left (x + 2\right )} e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^4+160*x^3+384*x^2+392*x+144)*exp(x^3+4*x^2+4*x)^2+(432*x^3+2016*x^2+2880*x+1296)*exp(x^3+4*x^2
+4*x),x, algorithm="maxima")

[Out]

4*(x^2 + 4*x + 4)*e^(2*x^3 + 8*x^2 + 8*x) + 144*(x + 2)*e^(x^3 + 4*x^2 + 4*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18) = 36\).
time = 0.35, size = 44, normalized size = 2.44 \begin {gather*} 4 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x^{3} + 8 \, x^{2} + 8 \, x\right )} + 144 \, {\left (x + 2\right )} e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^4+160*x^3+384*x^2+392*x+144)*exp(x^3+4*x^2+4*x)^2+(432*x^3+2016*x^2+2880*x+1296)*exp(x^3+4*x^2
+4*x),x, algorithm="fricas")

[Out]

4*(x^2 + 4*x + 4)*e^(2*x^3 + 8*x^2 + 8*x) + 144*(x + 2)*e^(x^3 + 4*x^2 + 4*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
time = 0.06, size = 42, normalized size = 2.33 \begin {gather*} \left (144 x + 288\right ) e^{x^{3} + 4 x^{2} + 4 x} + \left (4 x^{2} + 16 x + 16\right ) e^{2 x^{3} + 8 x^{2} + 8 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x**4+160*x**3+384*x**2+392*x+144)*exp(x**3+4*x**2+4*x)**2+(432*x**3+2016*x**2+2880*x+1296)*exp(x
**3+4*x**2+4*x),x)

[Out]

(144*x + 288)*exp(x**3 + 4*x**2 + 4*x) + (4*x**2 + 16*x + 16)*exp(2*x**3 + 8*x**2 + 8*x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (18) = 36\).
time = 0.42, size = 87, normalized size = 4.83 \begin {gather*} 4 \, x^{2} e^{\left (2 \, x^{3} + 8 \, x^{2} + 8 \, x\right )} + 16 \, x e^{\left (2 \, x^{3} + 8 \, x^{2} + 8 \, x\right )} + 144 \, x e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} + 16 \, e^{\left (2 \, x^{3} + 8 \, x^{2} + 8 \, x\right )} + 288 \, e^{\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^4+160*x^3+384*x^2+392*x+144)*exp(x^3+4*x^2+4*x)^2+(432*x^3+2016*x^2+2880*x+1296)*exp(x^3+4*x^2
+4*x),x, algorithm="giac")

[Out]

4*x^2*e^(2*x^3 + 8*x^2 + 8*x) + 16*x*e^(2*x^3 + 8*x^2 + 8*x) + 144*x*e^(x^3 + 4*x^2 + 4*x) + 16*e^(2*x^3 + 8*x
^2 + 8*x) + 288*e^(x^3 + 4*x^2 + 4*x)

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Mupad [B]
time = 3.04, size = 50, normalized size = 2.78 \begin {gather*} 4\,{\mathrm {e}}^{x^3+4\,x^2+4\,x}\,\left (x+2\right )\,\left (2\,{\mathrm {e}}^{x^3+4\,x^2+4\,x}+x\,{\mathrm {e}}^{x^3+4\,x^2+4\,x}+36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x + 8*x^2 + 2*x^3)*(392*x + 384*x^2 + 160*x^3 + 24*x^4 + 144) + exp(4*x + 4*x^2 + x^3)*(2880*x + 201
6*x^2 + 432*x^3 + 1296),x)

[Out]

4*exp(4*x + 4*x^2 + x^3)*(x + 2)*(2*exp(4*x + 4*x^2 + x^3) + x*exp(4*x + 4*x^2 + x^3) + 36)

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