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

Optimal. Leaf size=32 \[ 4 a c x+\frac{4 c^2 x^3}{3}+c d x^4+\frac{d^2 x^5}{5} \]

[Out]

4*a*c*x + (4*c^2*x^3)/3 + c*d*x^4 + (d^2*x^5)/5

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Rubi [A]  time = 0.0058112, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ 4 a c x+\frac{4 c^2 x^3}{3}+c d x^4+\frac{d^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*c*x + (4*c^2*x^3)/3 + c*d*x^4 + (d^2*x^5)/5

Rubi steps

\begin{align*} \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx &=4 a c x+\frac{4 c^2 x^3}{3}+c d x^4+\frac{d^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0000498, size = 32, normalized size = 1. \[ 4 a c x+\frac{4 c^2 x^3}{3}+c d x^4+\frac{d^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*c*x + (4*c^2*x^3)/3 + c*d*x^4 + (d^2*x^5)/5

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Maple [A]  time = 0., size = 29, normalized size = 0.9 \begin{align*} 4\,acx+{\frac{4\,{c}^{2}{x}^{3}}{3}}+cd{x}^{4}+{\frac{{d}^{2}{x}^{5}}{5}} \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,x)

[Out]

4*a*c*x+4/3*c^2*x^3+c*d*x^4+1/5*d^2*x^5

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Maxima [A]  time = 1.04914, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{5} \, d^{2} x^{5} + c d x^{4} + \frac{4}{3} \, c^{2} x^{3} + 4 \, a c 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,x, algorithm="maxima")

[Out]

1/5*d^2*x^5 + c*d*x^4 + 4/3*c^2*x^3 + 4*a*c*x

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Fricas [A]  time = 1.10546, size = 63, normalized size = 1.97 \begin{align*} \frac{1}{5} x^{5} d^{2} + x^{4} d c + \frac{4}{3} x^{3} c^{2} + 4 x c a \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,x, algorithm="fricas")

[Out]

1/5*x^5*d^2 + x^4*d*c + 4/3*x^3*c^2 + 4*x*c*a

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Sympy [A]  time = 0.066172, size = 31, normalized size = 0.97 \begin{align*} 4 a c x + \frac{4 c^{2} x^{3}}{3} + c d x^{4} + \frac{d^{2} x^{5}}{5} \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,x)

[Out]

4*a*c*x + 4*c**2*x**3/3 + c*d*x**4 + d**2*x**5/5

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Giac [A]  time = 1.11857, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{5} \, d^{2} x^{5} + c d x^{4} + \frac{4}{3} \, c^{2} x^{3} + 4 \, a c 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,x, algorithm="giac")

[Out]

1/5*d^2*x^5 + c*d*x^4 + 4/3*c^2*x^3 + 4*a*c*x