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

Optimal. Leaf size=92 \[ 16 a^2 c^2 x+\frac{8}{5} c x^5 \left (a d^2+2 c^3\right )+8 a c^2 d x^4+\frac{32}{3} a c^3 x^3+\frac{24}{7} c^2 d^2 x^7+\frac{16}{3} c^3 d x^6+c d^3 x^8+\frac{d^4 x^9}{9} \]

[Out]

16*a^2*c^2*x + (32*a*c^3*x^3)/3 + 8*a*c^2*d*x^4 + (8*c*(2*c^3 + a*d^2)*x^5)/5 + (16*c^3*d*x^6)/3 + (24*c^2*d^2
*x^7)/7 + c*d^3*x^8 + (d^4*x^9)/9

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Rubi [A]  time = 0.0430304, antiderivative size = 92, 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} \[ 16 a^2 c^2 x+\frac{8}{5} c x^5 \left (a d^2+2 c^3\right )+8 a c^2 d x^4+\frac{32}{3} a c^3 x^3+\frac{24}{7} c^2 d^2 x^7+\frac{16}{3} c^3 d x^6+c d^3 x^8+\frac{d^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16*a^2*c^2*x + (32*a*c^3*x^3)/3 + 8*a*c^2*d*x^4 + (8*c*(2*c^3 + a*d^2)*x^5)/5 + (16*c^3*d*x^6)/3 + (24*c^2*d^2
*x^7)/7 + c*d^3*x^8 + (d^4*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 (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2 \, dx &=\int \left (16 a^2 c^2+32 a c^3 x^2+32 a c^2 d x^3+8 c \left (2 c^3+a d^2\right ) x^4+32 c^3 d x^5+24 c^2 d^2 x^6+8 c d^3 x^7+d^4 x^8\right ) \, dx\\ &=16 a^2 c^2 x+\frac{32}{3} a c^3 x^3+8 a c^2 d x^4+\frac{8}{5} c \left (2 c^3+a d^2\right ) x^5+\frac{16}{3} c^3 d x^6+\frac{24}{7} c^2 d^2 x^7+c d^3 x^8+\frac{d^4 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0082144, size = 92, normalized size = 1. \[ 16 a^2 c^2 x+\frac{8}{5} c x^5 \left (a d^2+2 c^3\right )+8 a c^2 d x^4+\frac{32}{3} a c^3 x^3+\frac{24}{7} c^2 d^2 x^7+\frac{16}{3} c^3 d x^6+c d^3 x^8+\frac{d^4 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16*a^2*c^2*x + (32*a*c^3*x^3)/3 + 8*a*c^2*d*x^4 + (8*c*(2*c^3 + a*d^2)*x^5)/5 + (16*c^3*d*x^6)/3 + (24*c^2*d^2
*x^7)/7 + c*d^3*x^8 + (d^4*x^9)/9

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Maple [A]  time = 0., size = 84, normalized size = 0.9 \begin{align*}{\frac{{d}^{4}{x}^{9}}{9}}+c{d}^{3}{x}^{8}+{\frac{24\,{c}^{2}{d}^{2}{x}^{7}}{7}}+{\frac{16\,{c}^{3}d{x}^{6}}{3}}+{\frac{ \left ( 8\,ac{d}^{2}+16\,{c}^{4} \right ){x}^{5}}{5}}+8\,a{c}^{2}d{x}^{4}+{\frac{32\,a{c}^{3}{x}^{3}}{3}}+16\,{a}^{2}{c}^{2}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)^2,x)

[Out]

1/9*d^4*x^9+c*d^3*x^8+24/7*c^2*d^2*x^7+16/3*c^3*d*x^6+1/5*(8*a*c*d^2+16*c^4)*x^5+8*a*c^2*d*x^4+32/3*a*c^3*x^3+
16*a^2*c^2*x

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Maxima [A]  time = 1.09146, size = 127, normalized size = 1.38 \begin{align*} \frac{1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac{16}{7} \, c^{2} d^{2} x^{7} + \frac{16}{5} \, c^{4} x^{5} + 16 \, a^{2} c^{2} x + \frac{8}{15} \,{\left (3 \, d^{2} x^{5} + 15 \, c d x^{4} + 20 \, c^{2} x^{3}\right )} a c + \frac{8}{21} \,{\left (3 \, d^{2} x^{7} + 14 \, c d x^{6}\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)^2,x, algorithm="maxima")

[Out]

1/9*d^4*x^9 + c*d^3*x^8 + 16/7*c^2*d^2*x^7 + 16/5*c^4*x^5 + 16*a^2*c^2*x + 8/15*(3*d^2*x^5 + 15*c*d*x^4 + 20*c
^2*x^3)*a*c + 8/21*(3*d^2*x^7 + 14*c*d*x^6)*c^2

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Fricas [A]  time = 1.12068, size = 192, normalized size = 2.09 \begin{align*} \frac{1}{9} x^{9} d^{4} + x^{8} d^{3} c + \frac{24}{7} x^{7} d^{2} c^{2} + \frac{16}{3} x^{6} d c^{3} + \frac{16}{5} x^{5} c^{4} + \frac{8}{5} x^{5} d^{2} c a + 8 x^{4} d c^{2} a + \frac{32}{3} x^{3} c^{3} a + 16 x c^{2} a^{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)^2,x, algorithm="fricas")

[Out]

1/9*x^9*d^4 + x^8*d^3*c + 24/7*x^7*d^2*c^2 + 16/3*x^6*d*c^3 + 16/5*x^5*c^4 + 8/5*x^5*d^2*c*a + 8*x^4*d*c^2*a +
32/3*x^3*c^3*a + 16*x*c^2*a^2

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Sympy [A]  time = 0.078466, size = 95, normalized size = 1.03 \begin{align*} 16 a^{2} c^{2} x + \frac{32 a c^{3} x^{3}}{3} + 8 a c^{2} d x^{4} + \frac{16 c^{3} d x^{6}}{3} + \frac{24 c^{2} d^{2} x^{7}}{7} + c d^{3} x^{8} + \frac{d^{4} x^{9}}{9} + x^{5} \left (\frac{8 a c d^{2}}{5} + \frac{16 c^{4}}{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)**2,x)

[Out]

16*a**2*c**2*x + 32*a*c**3*x**3/3 + 8*a*c**2*d*x**4 + 16*c**3*d*x**6/3 + 24*c**2*d**2*x**7/7 + c*d**3*x**8 + d
**4*x**9/9 + x**5*(8*a*c*d**2/5 + 16*c**4/5)

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Giac [A]  time = 1.14216, size = 112, normalized size = 1.22 \begin{align*} \frac{1}{9} \, d^{4} x^{9} + c d^{3} x^{8} + \frac{24}{7} \, c^{2} d^{2} x^{7} + \frac{16}{3} \, c^{3} d x^{6} + \frac{16}{5} \, c^{4} x^{5} + \frac{8}{5} \, a c d^{2} x^{5} + 8 \, a c^{2} d x^{4} + \frac{32}{3} \, a c^{3} x^{3} + 16 \, a^{2} c^{2} 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)^2,x, algorithm="giac")

[Out]

1/9*d^4*x^9 + c*d^3*x^8 + 24/7*c^2*d^2*x^7 + 16/3*c^3*d*x^6 + 16/5*c^4*x^5 + 8/5*a*c*d^2*x^5 + 8*a*c^2*d*x^4 +
32/3*a*c^3*x^3 + 16*a^2*c^2*x