3.34 \(\int (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4)^3 \, dx\)

Optimal. Leaf size=171 \[ 48 a^2 c^3 d x^4+64 a^2 c^4 x^3+64 a^3 c^3 x+\frac{4}{3} c d^2 x^9 \left (a d^2+20 c^3\right )+12 c^2 d x^8 \left (a d^2+2 c^3\right )+\frac{32}{7} c^3 x^7 \left (9 a d^2+2 c^3\right )+\frac{48}{5} a c^2 x^5 \left (a d^2+4 c^3\right )+64 a c^4 d x^6+\frac{60}{11} c^2 d^4 x^{11}+16 c^3 d^3 x^{10}+c d^5 x^{12}+\frac{d^6 x^{13}}{13} \]

[Out]

64*a^3*c^3*x + 64*a^2*c^4*x^3 + 48*a^2*c^3*d*x^4 + (48*a*c^2*(4*c^3 + a*d^2)*x^5)/5 + 64*a*c^4*d*x^6 + (32*c^3
*(2*c^3 + 9*a*d^2)*x^7)/7 + 12*c^2*d*(2*c^3 + a*d^2)*x^8 + (4*c*d^2*(20*c^3 + a*d^2)*x^9)/3 + 16*c^3*d^3*x^10
+ (60*c^2*d^4*x^11)/11 + c*d^5*x^12 + (d^6*x^13)/13

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Rubi [A]  time = 0.0919228, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2061} \[ 48 a^2 c^3 d x^4+64 a^2 c^4 x^3+64 a^3 c^3 x+\frac{4}{3} c d^2 x^9 \left (a d^2+20 c^3\right )+12 c^2 d x^8 \left (a d^2+2 c^3\right )+\frac{32}{7} c^3 x^7 \left (9 a d^2+2 c^3\right )+\frac{48}{5} a c^2 x^5 \left (a d^2+4 c^3\right )+64 a c^4 d x^6+\frac{60}{11} c^2 d^4 x^{11}+16 c^3 d^3 x^{10}+c d^5 x^{12}+\frac{d^6 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^3,x]

[Out]

64*a^3*c^3*x + 64*a^2*c^4*x^3 + 48*a^2*c^3*d*x^4 + (48*a*c^2*(4*c^3 + a*d^2)*x^5)/5 + 64*a*c^4*d*x^6 + (32*c^3
*(2*c^3 + 9*a*d^2)*x^7)/7 + 12*c^2*d*(2*c^3 + a*d^2)*x^8 + (4*c*d^2*(20*c^3 + a*d^2)*x^9)/3 + 16*c^3*d^3*x^10
+ (60*c^2*d^4*x^11)/11 + c*d^5*x^12 + (d^6*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 (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^3 \, dx &=\int \left (64 a^3 c^3+192 a^2 c^4 x^2+192 a^2 c^3 d x^3+48 a c^2 \left (4 c^3+a d^2\right ) x^4+384 a c^4 d x^5+32 c^3 \left (2 c^3+9 a d^2\right ) x^6+96 c^2 d \left (2 c^3+a d^2\right ) x^7+12 c d^2 \left (20 c^3+a d^2\right ) x^8+160 c^3 d^3 x^9+60 c^2 d^4 x^{10}+12 c d^5 x^{11}+d^6 x^{12}\right ) \, dx\\ &=64 a^3 c^3 x+64 a^2 c^4 x^3+48 a^2 c^3 d x^4+\frac{48}{5} a c^2 \left (4 c^3+a d^2\right ) x^5+64 a c^4 d x^6+\frac{32}{7} c^3 \left (2 c^3+9 a d^2\right ) x^7+12 c^2 d \left (2 c^3+a d^2\right ) x^8+\frac{4}{3} c d^2 \left (20 c^3+a d^2\right ) x^9+16 c^3 d^3 x^{10}+\frac{60}{11} c^2 d^4 x^{11}+c d^5 x^{12}+\frac{d^6 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0171551, size = 171, normalized size = 1. \[ 48 a^2 c^3 d x^4+64 a^2 c^4 x^3+64 a^3 c^3 x+\frac{4}{3} c d^2 x^9 \left (a d^2+20 c^3\right )+12 c^2 d x^8 \left (a d^2+2 c^3\right )+\frac{32}{7} c^3 x^7 \left (9 a d^2+2 c^3\right )+\frac{48}{5} a c^2 x^5 \left (a d^2+4 c^3\right )+64 a c^4 d x^6+\frac{60}{11} c^2 d^4 x^{11}+16 c^3 d^3 x^{10}+c d^5 x^{12}+\frac{d^6 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^3,x]

[Out]

64*a^3*c^3*x + 64*a^2*c^4*x^3 + 48*a^2*c^3*d*x^4 + (48*a*c^2*(4*c^3 + a*d^2)*x^5)/5 + 64*a*c^4*d*x^6 + (32*c^3
*(2*c^3 + 9*a*d^2)*x^7)/7 + 12*c^2*d*(2*c^3 + a*d^2)*x^8 + (4*c*d^2*(20*c^3 + a*d^2)*x^9)/3 + 16*c^3*d^3*x^10
+ (60*c^2*d^4*x^11)/11 + c*d^5*x^12 + (d^6*x^13)/13

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Maple [A]  time = 0.001, size = 231, normalized size = 1.4 \begin{align*}{\frac{{d}^{6}{x}^{13}}{13}}+c{d}^{5}{x}^{12}+{\frac{60\,{c}^{2}{d}^{4}{x}^{11}}{11}}+16\,{c}^{3}{d}^{3}{x}^{10}+{\frac{ \left ( 4\,ac{d}^{4}+224\,{c}^{4}{d}^{2}+{d}^{2} \left ( 8\,ac{d}^{2}+16\,{c}^{4} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 64\,a{c}^{2}{d}^{3}+128\,{c}^{5}d+4\,cd \left ( 8\,ac{d}^{2}+16\,{c}^{4} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 256\,a{c}^{3}{d}^{2}+4\,{c}^{2} \left ( 8\,ac{d}^{2}+16\,{c}^{4} \right ) \right ){x}^{7}}{7}}+64\,a{c}^{4}d{x}^{6}+{\frac{ \left ( 4\,ac \left ( 8\,ac{d}^{2}+16\,{c}^{4} \right ) +128\,{c}^{5}a+16\,{a}^{2}{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+48\,{a}^{2}{c}^{3}d{x}^{4}+64\,{a}^{2}{c}^{4}{x}^{3}+64\,{a}^{3}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^3,x)

[Out]

1/13*d^6*x^13+c*d^5*x^12+60/11*c^2*d^4*x^11+16*c^3*d^3*x^10+1/9*(4*a*c*d^4+224*c^4*d^2+d^2*(8*a*c*d^2+16*c^4))
*x^9+1/8*(64*a*c^2*d^3+128*c^5*d+4*c*d*(8*a*c*d^2+16*c^4))*x^8+1/7*(256*a*c^3*d^2+4*c^2*(8*a*c*d^2+16*c^4))*x^
7+64*a*c^4*d*x^6+1/5*(4*a*c*(8*a*c*d^2+16*c^4)+128*c^5*a+16*a^2*c^2*d^2)*x^5+48*a^2*c^3*d*x^4+64*a^2*c^4*x^3+6
4*a^3*c^3*x

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Maxima [A]  time = 1.11101, size = 277, normalized size = 1.62 \begin{align*} \frac{1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac{48}{11} \, c^{2} d^{4} x^{11} + \frac{32}{5} \, c^{3} d^{3} x^{10} + \frac{64}{7} \, c^{6} x^{7} + 64 \, a^{3} c^{3} x + \frac{16}{5} \,{\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a^{2} c^{2} + \frac{8}{3} \,{\left (2 \, d^{2} x^{9} + 9 \, c d x^{8}\right )} c^{4} + \frac{4}{105} \,{\left (35 \, d^{4} x^{9} + 315 \, c d^{3} x^{8} + 720 \, c^{2} d^{2} x^{7} + 1008 \, c^{4} x^{5} + 120 \,{\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\right )} c^{2}\right )} a c + \frac{4}{165} \,{\left (45 \, d^{4} x^{11} + 396 \, c d^{3} x^{10} + 880 \, c^{2} d^{2} x^{9}\right )} c^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*d^6*x^13 + c*d^5*x^12 + 48/11*c^2*d^4*x^11 + 32/5*c^3*d^3*x^10 + 64/7*c^6*x^7 + 64*a^3*c^3*x + 16/5*(3*d^
2*x^5 + 15*c*d*x^4 + 20*c^2*x^3)*a^2*c^2 + 8/3*(2*d^2*x^9 + 9*c*d*x^8)*c^4 + 4/105*(35*d^4*x^9 + 315*c*d^3*x^8
+ 720*c^2*d^2*x^7 + 1008*c^4*x^5 + 120*(3*d^2*x^7 + 14*c*d*x^6)*c^2)*a*c + 4/165*(45*d^4*x^11 + 396*c*d^3*x^1
0 + 880*c^2*d^2*x^9)*c^2

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Fricas [A]  time = 1.17696, size = 383, normalized size = 2.24 \begin{align*} \frac{1}{13} x^{13} d^{6} + x^{12} d^{5} c + \frac{60}{11} x^{11} d^{4} c^{2} + 16 x^{10} d^{3} c^{3} + \frac{80}{3} x^{9} d^{2} c^{4} + \frac{4}{3} x^{9} d^{4} c a + 24 x^{8} d c^{5} + 12 x^{8} d^{3} c^{2} a + \frac{64}{7} x^{7} c^{6} + \frac{288}{7} x^{7} d^{2} c^{3} a + 64 x^{6} d c^{4} a + \frac{192}{5} x^{5} c^{5} a + \frac{48}{5} x^{5} d^{2} c^{2} a^{2} + 48 x^{4} d c^{3} a^{2} + 64 x^{3} c^{4} a^{2} + 64 x c^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*d^6 + x^12*d^5*c + 60/11*x^11*d^4*c^2 + 16*x^10*d^3*c^3 + 80/3*x^9*d^2*c^4 + 4/3*x^9*d^4*c*a + 24*x^
8*d*c^5 + 12*x^8*d^3*c^2*a + 64/7*x^7*c^6 + 288/7*x^7*d^2*c^3*a + 64*x^6*d*c^4*a + 192/5*x^5*c^5*a + 48/5*x^5*
d^2*c^2*a^2 + 48*x^4*d*c^3*a^2 + 64*x^3*c^4*a^2 + 64*x*c^3*a^3

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Sympy [A]  time = 0.092107, size = 180, normalized size = 1.05 \begin{align*} 64 a^{3} c^{3} x + 64 a^{2} c^{4} x^{3} + 48 a^{2} c^{3} d x^{4} + 64 a c^{4} d x^{6} + 16 c^{3} d^{3} x^{10} + \frac{60 c^{2} d^{4} x^{11}}{11} + c d^{5} x^{12} + \frac{d^{6} x^{13}}{13} + x^{9} \left (\frac{4 a c d^{4}}{3} + \frac{80 c^{4} d^{2}}{3}\right ) + x^{8} \left (12 a c^{2} d^{3} + 24 c^{5} d\right ) + x^{7} \left (\frac{288 a c^{3} d^{2}}{7} + \frac{64 c^{6}}{7}\right ) + x^{5} \left (\frac{48 a^{2} c^{2} d^{2}}{5} + \frac{192 a c^{5}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**3,x)

[Out]

64*a**3*c**3*x + 64*a**2*c**4*x**3 + 48*a**2*c**3*d*x**4 + 64*a*c**4*d*x**6 + 16*c**3*d**3*x**10 + 60*c**2*d**
4*x**11/11 + c*d**5*x**12 + d**6*x**13/13 + x**9*(4*a*c*d**4/3 + 80*c**4*d**2/3) + x**8*(12*a*c**2*d**3 + 24*c
**5*d) + x**7*(288*a*c**3*d**2/7 + 64*c**6/7) + x**5*(48*a**2*c**2*d**2/5 + 192*a*c**5/5)

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Giac [A]  time = 1.1121, size = 224, normalized size = 1.31 \begin{align*} \frac{1}{13} \, d^{6} x^{13} + c d^{5} x^{12} + \frac{60}{11} \, c^{2} d^{4} x^{11} + 16 \, c^{3} d^{3} x^{10} + \frac{80}{3} \, c^{4} d^{2} x^{9} + \frac{4}{3} \, a c d^{4} x^{9} + 24 \, c^{5} d x^{8} + 12 \, a c^{2} d^{3} x^{8} + \frac{64}{7} \, c^{6} x^{7} + \frac{288}{7} \, a c^{3} d^{2} x^{7} + 64 \, a c^{4} d x^{6} + \frac{192}{5} \, a c^{5} x^{5} + \frac{48}{5} \, a^{2} c^{2} d^{2} x^{5} + 48 \, a^{2} c^{3} d x^{4} + 64 \, a^{2} c^{4} x^{3} + 64 \, a^{3} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*d^6*x^13 + c*d^5*x^12 + 60/11*c^2*d^4*x^11 + 16*c^3*d^3*x^10 + 80/3*c^4*d^2*x^9 + 4/3*a*c*d^4*x^9 + 24*c^
5*d*x^8 + 12*a*c^2*d^3*x^8 + 64/7*c^6*x^7 + 288/7*a*c^3*d^2*x^7 + 64*a*c^4*d*x^6 + 192/5*a*c^5*x^5 + 48/5*a^2*
c^2*d^2*x^5 + 48*a^2*c^3*d*x^4 + 64*a^2*c^4*x^3 + 64*a^3*c^3*x