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

Optimal. Leaf size=203 \[ -\frac{1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+512 a^3 e^6 x-\frac{128}{3} e^5 x^9 \left (d^4-4 a e^3\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )+\frac{24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )-\frac{384}{5} a e^4 x^5 \left (d^4-4 a e^3\right )+8 a d^6 e^2 x^3+\frac{1536}{11} d^2 e^7 x^{11}+32 d^3 e^6 x^{10}+128 d e^8 x^{12}+\frac{512 e^9 x^{13}}{13} \]

[Out]

512*a^3*e^6*x - 96*a^2*d^3*e^4*x^2 + 8*a*d^6*e^2*x^3 - (d*(d^8 - 1536*a^2*e^6)*x^4)/4 - (384*a*e^4*(d^4 - 4*a*
e^3)*x^5)/5 + 4*d^3*e^2*(d^4 - 16*a*e^3)*x^6 + (24*d^2*e^3*(d^4 + 64*a*e^3)*x^7)/7 - 24*d*e^4*(d^4 - 16*a*e^3)
*x^8 - (128*e^5*(d^4 - 4*a*e^3)*x^9)/3 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 128*d*e^8*x^12 + (512*e^9*
x^13)/13

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Rubi [A]  time = 0.122958, antiderivative size = 203, 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} \[ -\frac{1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+512 a^3 e^6 x-\frac{128}{3} e^5 x^9 \left (d^4-4 a e^3\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )+\frac{24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )-\frac{384}{5} a e^4 x^5 \left (d^4-4 a e^3\right )+8 a d^6 e^2 x^3+\frac{1536}{11} d^2 e^7 x^{11}+32 d^3 e^6 x^{10}+128 d e^8 x^{12}+\frac{512 e^9 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

512*a^3*e^6*x - 96*a^2*d^3*e^4*x^2 + 8*a*d^6*e^2*x^3 - (d*(d^8 - 1536*a^2*e^6)*x^4)/4 - (384*a*e^4*(d^4 - 4*a*
e^3)*x^5)/5 + 4*d^3*e^2*(d^4 - 16*a*e^3)*x^6 + (24*d^2*e^3*(d^4 + 64*a*e^3)*x^7)/7 - 24*d*e^4*(d^4 - 16*a*e^3)
*x^8 - (128*e^5*(d^4 - 4*a*e^3)*x^9)/3 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 128*d*e^8*x^12 + (512*e^9*
x^13)/13

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 )^3 \, dx &=\int \left (512 a^3 e^6-192 a^2 d^3 e^4 x+24 a d^6 e^2 x^2-d \left (d^8-1536 a^2 e^6\right ) x^3-384 a e^4 \left (d^4-4 a e^3\right ) x^4+24 d^3 e^2 \left (d^4-16 a e^3\right ) x^5+24 d^2 e^3 \left (d^4+64 a e^3\right ) x^6-192 d e^4 \left (d^4-16 a e^3\right ) x^7-384 e^5 \left (d^4-4 a e^3\right ) x^8+320 d^3 e^6 x^9+1536 d^2 e^7 x^{10}+1536 d e^8 x^{11}+512 e^9 x^{12}\right ) \, dx\\ &=512 a^3 e^6 x-96 a^2 d^3 e^4 x^2+8 a d^6 e^2 x^3-\frac{1}{4} d \left (d^8-1536 a^2 e^6\right ) x^4-\frac{384}{5} a e^4 \left (d^4-4 a e^3\right ) x^5+4 d^3 e^2 \left (d^4-16 a e^3\right ) x^6+\frac{24}{7} d^2 e^3 \left (d^4+64 a e^3\right ) x^7-24 d e^4 \left (d^4-16 a e^3\right ) x^8-\frac{128}{3} e^5 \left (d^4-4 a e^3\right ) x^9+32 d^3 e^6 x^{10}+\frac{1536}{11} d^2 e^7 x^{11}+128 d e^8 x^{12}+\frac{512 e^9 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0248907, size = 207, normalized size = 1.02 \[ -\frac{1}{4} d x^4 \left (d^8-1536 a^2 e^6\right )-96 a^2 d^3 e^4 x^2+512 a^3 e^6 x+\frac{128}{3} e^5 x^9 \left (4 a e^3-d^4\right )-24 d e^4 x^8 \left (d^4-16 a e^3\right )+\frac{24}{7} d^2 e^3 x^7 \left (64 a e^3+d^4\right )+4 d^3 e^2 x^6 \left (d^4-16 a e^3\right )+\frac{384}{5} a e^4 x^5 \left (4 a e^3-d^4\right )+8 a d^6 e^2 x^3+\frac{1536}{11} d^2 e^7 x^{11}+32 d^3 e^6 x^{10}+128 d e^8 x^{12}+\frac{512 e^9 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

512*a^3*e^6*x - 96*a^2*d^3*e^4*x^2 + 8*a*d^6*e^2*x^3 - (d*(d^8 - 1536*a^2*e^6)*x^4)/4 + (384*a*e^4*(-d^4 + 4*a
*e^3)*x^5)/5 + 4*d^3*e^2*(d^4 - 16*a*e^3)*x^6 + (24*d^2*e^3*(d^4 + 64*a*e^3)*x^7)/7 - 24*d*e^4*(d^4 - 16*a*e^3
)*x^8 + (128*e^5*(-d^4 + 4*a*e^3)*x^9)/3 + 32*d^3*e^6*x^10 + (1536*d^2*e^7*x^11)/11 + 128*d*e^8*x^12 + (512*e^
9*x^13)/13

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Maple [A]  time = 0.001, size = 288, normalized size = 1.4 \begin{align*}{\frac{512\,{e}^{9}{x}^{13}}{13}}+128\,d{e}^{8}{x}^{12}+{\frac{1536\,{d}^{2}{e}^{7}{x}^{11}}{11}}+32\,{d}^{3}{e}^{6}{x}^{10}+{\frac{ \left ( 512\,a{e}^{8}-256\,{d}^{4}{e}^{5}+8\,{e}^{3} \left ( 128\,a{e}^{5}-16\,{d}^{4}{e}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 2048\,a{e}^{7}d-64\,{d}^{5}{e}^{4}+8\,d{e}^{2} \left ( 128\,a{e}^{5}-16\,{d}^{4}{e}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 1536\,a{e}^{6}{d}^{2}+24\,{d}^{6}{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( -256\,a{e}^{5}{d}^{3}-{d}^{3} \left ( 128\,a{e}^{5}-16\,{d}^{4}{e}^{2} \right ) +8\,{d}^{7}{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 8\,a{e}^{2} \left ( 128\,a{e}^{5}-16\,{d}^{4}{e}^{2} \right ) -256\,{d}^{4}a{e}^{4}+512\,{e}^{7}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 1536\,{a}^{2}{e}^{6}d-{d}^{9} \right ){x}^{4}}{4}}+8\,a{d}^{6}{e}^{2}{x}^{3}-96\,{a}^{2}{d}^{3}{e}^{4}{x}^{2}+512\,{a}^{3}{e}^{6}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)^3,x)

[Out]

512/13*e^9*x^13+128*d*e^8*x^12+1536/11*d^2*e^7*x^11+32*d^3*e^6*x^10+1/9*(512*a*e^8-256*d^4*e^5+8*e^3*(128*a*e^
5-16*d^4*e^2))*x^9+1/8*(2048*a*e^7*d-64*d^5*e^4+8*d*e^2*(128*a*e^5-16*d^4*e^2))*x^8+1/7*(1536*a*d^2*e^6+24*d^6
*e^3)*x^7+1/6*(-256*a*e^5*d^3-d^3*(128*a*e^5-16*d^4*e^2)+8*d^7*e^2)*x^6+1/5*(8*a*e^2*(128*a*e^5-16*d^4*e^2)-25
6*d^4*a*e^4+512*e^7*a^2)*x^5+1/4*(1536*a^2*d*e^6-d^9)*x^4+8*a*d^6*e^2*x^3-96*a^2*d^3*e^4*x^2+512*a^3*e^6*x

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Maxima [A]  time = 1.16221, size = 289, normalized size = 1.42 \begin{align*} \frac{512}{13} \, e^{9} x^{13} + 128 \, d e^{8} x^{12} + \frac{1536}{11} \, d^{2} e^{7} x^{11} + \frac{256}{5} \, d^{3} e^{6} x^{10} - \frac{1}{4} \, d^{9} x^{4} + 512 \, a^{3} e^{6} x + \frac{4}{7} \,{\left (6 \, e^{3} x^{7} + 7 \, d e^{2} x^{6}\right )} d^{6} + \frac{96}{5} \,{\left (16 \, e^{3} x^{5} + 20 \, d e^{2} x^{4} - 5 \, d^{3} x^{2}\right )} a^{2} e^{4} - \frac{8}{15} \,{\left (36 \, e^{6} x^{10} + 80 \, d e^{5} x^{9} + 45 \, d^{2} e^{4} x^{8}\right )} d^{3} + \frac{8}{105} \,{\left (2240 \, e^{6} x^{9} + 5040 \, d e^{5} x^{8} + 2880 \, d^{2} e^{4} x^{7} + 105 \, d^{6} x^{3} - 168 \,{\left (5 \, e^{3} x^{6} + 6 \, d e^{2} x^{5}\right )} d^{3}\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)^3,x, algorithm="maxima")

[Out]

512/13*e^9*x^13 + 128*d*e^8*x^12 + 1536/11*d^2*e^7*x^11 + 256/5*d^3*e^6*x^10 - 1/4*d^9*x^4 + 512*a^3*e^6*x + 4
/7*(6*e^3*x^7 + 7*d*e^2*x^6)*d^6 + 96/5*(16*e^3*x^5 + 20*d*e^2*x^4 - 5*d^3*x^2)*a^2*e^4 - 8/15*(36*e^6*x^10 +
80*d*e^5*x^9 + 45*d^2*e^4*x^8)*d^3 + 8/105*(2240*e^6*x^9 + 5040*d*e^5*x^8 + 2880*d^2*e^4*x^7 + 105*d^6*x^3 - 1
68*(5*e^3*x^6 + 6*d*e^2*x^5)*d^3)*a*e^2

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Fricas [A]  time = 1.1556, size = 482, normalized size = 2.37 \begin{align*} \frac{512}{13} x^{13} e^{9} + 128 x^{12} e^{8} d + \frac{1536}{11} x^{11} e^{7} d^{2} + 32 x^{10} e^{6} d^{3} - \frac{128}{3} x^{9} e^{5} d^{4} + \frac{512}{3} x^{9} e^{8} a - 24 x^{8} e^{4} d^{5} + 384 x^{8} e^{7} d a + \frac{24}{7} x^{7} e^{3} d^{6} + \frac{1536}{7} x^{7} e^{6} d^{2} a + 4 x^{6} e^{2} d^{7} - 64 x^{6} e^{5} d^{3} a - \frac{384}{5} x^{5} e^{4} d^{4} a + \frac{1536}{5} x^{5} e^{7} a^{2} - \frac{1}{4} x^{4} d^{9} + 384 x^{4} e^{6} d a^{2} + 8 x^{3} e^{2} d^{6} a - 96 x^{2} e^{4} d^{3} a^{2} + 512 x e^{6} a^{3} \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)^3,x, algorithm="fricas")

[Out]

512/13*x^13*e^9 + 128*x^12*e^8*d + 1536/11*x^11*e^7*d^2 + 32*x^10*e^6*d^3 - 128/3*x^9*e^5*d^4 + 512/3*x^9*e^8*
a - 24*x^8*e^4*d^5 + 384*x^8*e^7*d*a + 24/7*x^7*e^3*d^6 + 1536/7*x^7*e^6*d^2*a + 4*x^6*e^2*d^7 - 64*x^6*e^5*d^
3*a - 384/5*x^5*e^4*d^4*a + 1536/5*x^5*e^7*a^2 - 1/4*x^4*d^9 + 384*x^4*e^6*d*a^2 + 8*x^3*e^2*d^6*a - 96*x^2*e^
4*d^3*a^2 + 512*x*e^6*a^3

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Sympy [A]  time = 0.099933, size = 218, normalized size = 1.07 \begin{align*} 512 a^{3} e^{6} x - 96 a^{2} d^{3} e^{4} x^{2} + 8 a d^{6} e^{2} x^{3} + 32 d^{3} e^{6} x^{10} + \frac{1536 d^{2} e^{7} x^{11}}{11} + 128 d e^{8} x^{12} + \frac{512 e^{9} x^{13}}{13} + x^{9} \left (\frac{512 a e^{8}}{3} - \frac{128 d^{4} e^{5}}{3}\right ) + x^{8} \left (384 a d e^{7} - 24 d^{5} e^{4}\right ) + x^{7} \left (\frac{1536 a d^{2} e^{6}}{7} + \frac{24 d^{6} e^{3}}{7}\right ) + x^{6} \left (- 64 a d^{3} e^{5} + 4 d^{7} e^{2}\right ) + x^{5} \left (\frac{1536 a^{2} e^{7}}{5} - \frac{384 a d^{4} e^{4}}{5}\right ) + x^{4} \left (384 a^{2} d e^{6} - \frac{d^{9}}{4}\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)**3,x)

[Out]

512*a**3*e**6*x - 96*a**2*d**3*e**4*x**2 + 8*a*d**6*e**2*x**3 + 32*d**3*e**6*x**10 + 1536*d**2*e**7*x**11/11 +
128*d*e**8*x**12 + 512*e**9*x**13/13 + x**9*(512*a*e**8/3 - 128*d**4*e**5/3) + x**8*(384*a*d*e**7 - 24*d**5*e
**4) + x**7*(1536*a*d**2*e**6/7 + 24*d**6*e**3/7) + x**6*(-64*a*d**3*e**5 + 4*d**7*e**2) + x**5*(1536*a**2*e**
7/5 - 384*a*d**4*e**4/5) + x**4*(384*a**2*d*e**6 - d**9/4)

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Giac [A]  time = 1.14532, size = 252, normalized size = 1.24 \begin{align*} \frac{512}{13} \, x^{13} e^{9} + 128 \, d x^{12} e^{8} + \frac{1536}{11} \, d^{2} x^{11} e^{7} + 32 \, d^{3} x^{10} e^{6} - \frac{128}{3} \, d^{4} x^{9} e^{5} - 24 \, d^{5} x^{8} e^{4} + \frac{24}{7} \, d^{6} x^{7} e^{3} + 4 \, d^{7} x^{6} e^{2} - \frac{1}{4} \, d^{9} x^{4} + \frac{512}{3} \, a x^{9} e^{8} + 384 \, a d x^{8} e^{7} + \frac{1536}{7} \, a d^{2} x^{7} e^{6} - 64 \, a d^{3} x^{6} e^{5} - \frac{384}{5} \, a d^{4} x^{5} e^{4} + 8 \, a d^{6} x^{3} e^{2} + \frac{1536}{5} \, a^{2} x^{5} e^{7} + 384 \, a^{2} d x^{4} e^{6} - 96 \, a^{2} d^{3} x^{2} e^{4} + 512 \, a^{3} x e^{6} \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)^3,x, algorithm="giac")

[Out]

512/13*x^13*e^9 + 128*d*x^12*e^8 + 1536/11*d^2*x^11*e^7 + 32*d^3*x^10*e^6 - 128/3*d^4*x^9*e^5 - 24*d^5*x^8*e^4
+ 24/7*d^6*x^7*e^3 + 4*d^7*x^6*e^2 - 1/4*d^9*x^4 + 512/3*a*x^9*e^8 + 384*a*d*x^8*e^7 + 1536/7*a*d^2*x^7*e^6 -
64*a*d^3*x^6*e^5 - 384/5*a*d^4*x^5*e^4 + 8*a*d^6*x^3*e^2 + 1536/5*a^2*x^5*e^7 + 384*a^2*d*x^4*e^6 - 96*a^2*d^
3*x^2*e^4 + 512*a^3*x*e^6