3.41 \(\int (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4)^2 \, dx\)

Optimal. Leaf size=107 \[ 64 a^2 e^4 x-\frac{16}{5} e^2 x^5 \left (d^4-8 a e^3\right )-8 a d^3 e^2 x^2+32 a d e^4 x^4+\frac{64}{7} d^2 e^4 x^7-\frac{8}{3} d^3 e^3 x^6+\frac{d^6 x^3}{3}+16 d e^5 x^8+\frac{64 e^6 x^9}{9} \]

[Out]

64*a^2*e^4*x - 8*a*d^3*e^2*x^2 + (d^6*x^3)/3 + 32*a*d*e^4*x^4 - (16*e^2*(d^4 - 8*a*e^3)*x^5)/5 - (8*d^3*e^3*x^
6)/3 + (64*d^2*e^4*x^7)/7 + 16*d*e^5*x^8 + (64*e^6*x^9)/9

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Rubi [A]  time = 0.0509105, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {2061} \[ 64 a^2 e^4 x-\frac{16}{5} e^2 x^5 \left (d^4-8 a e^3\right )-8 a d^3 e^2 x^2+32 a d e^4 x^4+\frac{64}{7} d^2 e^4 x^7-\frac{8}{3} d^3 e^3 x^6+\frac{d^6 x^3}{3}+16 d e^5 x^8+\frac{64 e^6 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]

[Out]

64*a^2*e^4*x - 8*a*d^3*e^2*x^2 + (d^6*x^3)/3 + 32*a*d*e^4*x^4 - (16*e^2*(d^4 - 8*a*e^3)*x^5)/5 - (8*d^3*e^3*x^
6)/3 + (64*d^2*e^4*x^7)/7 + 16*d*e^5*x^8 + (64*e^6*x^9)/9

Rule 2061

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )^2 \, dx &=\int \left (64 a^2 e^4-16 a d^3 e^2 x+d^6 x^2+128 a d e^4 x^3-16 e^2 \left (d^4-8 a e^3\right ) x^4-16 d^3 e^3 x^5+64 d^2 e^4 x^6+128 d e^5 x^7+64 e^6 x^8\right ) \, dx\\ &=64 a^2 e^4 x-8 a d^3 e^2 x^2+\frac{d^6 x^3}{3}+32 a d e^4 x^4-\frac{16}{5} e^2 \left (d^4-8 a e^3\right ) x^5-\frac{8}{3} d^3 e^3 x^6+\frac{64}{7} d^2 e^4 x^7+16 d e^5 x^8+\frac{64 e^6 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0125221, size = 109, normalized size = 1.02 \[ 64 a^2 e^4 x+\frac{16}{5} e^2 x^5 \left (8 a e^3-d^4\right )-8 a d^3 e^2 x^2+32 a d e^4 x^4+\frac{64}{7} d^2 e^4 x^7-\frac{8}{3} d^3 e^3 x^6+\frac{d^6 x^3}{3}+16 d e^5 x^8+\frac{64 e^6 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^2,x]

[Out]

64*a^2*e^4*x - 8*a*d^3*e^2*x^2 + (d^6*x^3)/3 + 32*a*d*e^4*x^4 + (16*e^2*(-d^4 + 8*a*e^3)*x^5)/5 - (8*d^3*e^3*x
^6)/3 + (64*d^2*e^4*x^7)/7 + 16*d*e^5*x^8 + (64*e^6*x^9)/9

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Maple [A]  time = 0., size = 100, normalized size = 0.9 \begin{align*}{\frac{64\,{e}^{6}{x}^{9}}{9}}+16\,d{e}^{5}{x}^{8}+{\frac{64\,{d}^{2}{e}^{4}{x}^{7}}{7}}-{\frac{8\,{d}^{3}{e}^{3}{x}^{6}}{3}}+{\frac{ \left ( 128\,a{e}^{5}-16\,{d}^{4}{e}^{2} \right ){x}^{5}}{5}}+32\,ad{e}^{4}{x}^{4}+{\frac{{d}^{6}{x}^{3}}{3}}-8\,a{d}^{3}{e}^{2}{x}^{2}+64\,{a}^{2}{e}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x)

[Out]

64/9*e^6*x^9+16*d*e^5*x^8+64/7*d^2*e^4*x^7-8/3*d^3*e^3*x^6+1/5*(128*a*e^5-16*d^4*e^2)*x^5+32*a*d*e^4*x^4+1/3*d
^6*x^3-8*a*d^3*e^2*x^2+64*a^2*e^4*x

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Maxima [A]  time = 1.1223, size = 136, normalized size = 1.27 \begin{align*} \frac{64}{9} \, e^{6} x^{9} + 16 \, d e^{5} x^{8} + \frac{64}{7} \, d^{2} e^{4} x^{7} + \frac{1}{3} \, d^{6} x^{3} + 64 \, a^{2} e^{4} x - \frac{8}{15} \,{\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3} + \frac{8}{5} \,{\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="maxima")

[Out]

64/9*e^6*x^9 + 16*d*e^5*x^8 + 64/7*d^2*e^4*x^7 + 1/3*d^6*x^3 + 64*a^2*e^4*x - 8/15*(5*e^3*x^6 + 6*d*e^2*x^5)*d
^3 + 8/5*(16*e^3*x^5 + 20*d*e^2*x^4 - 5*d^3*x^2)*a*e^2

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Fricas [A]  time = 1.3873, size = 225, normalized size = 2.1 \begin{align*} \frac{64}{9} x^{9} e^{6} + 16 x^{8} e^{5} d + \frac{64}{7} x^{7} e^{4} d^{2} - \frac{8}{3} x^{6} e^{3} d^{3} - \frac{16}{5} x^{5} e^{2} d^{4} + \frac{128}{5} x^{5} e^{5} a + 32 x^{4} e^{4} d a + \frac{1}{3} x^{3} d^{6} - 8 x^{2} e^{2} d^{3} a + 64 x e^{4} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="fricas")

[Out]

64/9*x^9*e^6 + 16*x^8*e^5*d + 64/7*x^7*e^4*d^2 - 8/3*x^6*e^3*d^3 - 16/5*x^5*e^2*d^4 + 128/5*x^5*e^5*a + 32*x^4
*e^4*d*a + 1/3*x^3*d^6 - 8*x^2*e^2*d^3*a + 64*x*e^4*a^2

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Sympy [A]  time = 0.080088, size = 112, normalized size = 1.05 \begin{align*} 64 a^{2} e^{4} x - 8 a d^{3} e^{2} x^{2} + 32 a d e^{4} x^{4} + \frac{d^{6} x^{3}}{3} - \frac{8 d^{3} e^{3} x^{6}}{3} + \frac{64 d^{2} e^{4} x^{7}}{7} + 16 d e^{5} x^{8} + \frac{64 e^{6} x^{9}}{9} + x^{5} \left (\frac{128 a e^{5}}{5} - \frac{16 d^{4} e^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**2,x)

[Out]

64*a**2*e**4*x - 8*a*d**3*e**2*x**2 + 32*a*d*e**4*x**4 + d**6*x**3/3 - 8*d**3*e**3*x**6/3 + 64*d**2*e**4*x**7/
7 + 16*d*e**5*x**8 + 64*e**6*x**9/9 + x**5*(128*a*e**5/5 - 16*d**4*e**2/5)

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Giac [A]  time = 1.12963, size = 122, normalized size = 1.14 \begin{align*} \frac{64}{9} \, x^{9} e^{6} + 16 \, d x^{8} e^{5} + \frac{64}{7} \, d^{2} x^{7} e^{4} - \frac{8}{3} \, d^{3} x^{6} e^{3} - \frac{16}{5} \, d^{4} x^{5} e^{2} + \frac{1}{3} \, d^{6} x^{3} + \frac{128}{5} \, a x^{5} e^{5} + 32 \, a d x^{4} e^{4} - 8 \, a d^{3} x^{2} e^{2} + 64 \, a^{2} x e^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^2,x, algorithm="giac")

[Out]

64/9*x^9*e^6 + 16*d*x^8*e^5 + 64/7*d^2*x^7*e^4 - 8/3*d^3*x^6*e^3 - 16/5*d^4*x^5*e^2 + 1/3*d^6*x^3 + 128/5*a*x^
5*e^5 + 32*a*d*x^4*e^4 - 8*a*d^3*x^2*e^2 + 64*a^2*x*e^4