\(\int (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4) \, dx\) [36]

Optimal result

Integrand size = 27, antiderivative size = 32 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=4 a c x+\frac {4 c^2 x^3}{3}+c d x^4+\frac {d^2 x^5}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=4 a c x+\frac {4 c^2 x^3}{3}+c d x^4+\frac {d^2 x^5}{5} \]

[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*} \text {integral}& = 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] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=4 a c x+\frac {4 c^2 x^3}{3}+c d x^4+\frac {d^2 x^5}{5} \]

[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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91

method	result	size
gosper	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)
default	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)
norman	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)
risch	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)
parallelrisch	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)
parts	\(4 a c x +\frac {4}{3} c^{2} x^{3}+c d \,x^{4}+\frac {1}{5} d^{2} x^{5}\)	\(29\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=\frac {1}{5} \, d^{2} x^{5} + c d x^{4} + \frac {4}{3} \, c^{2} x^{3} + 4 \, a c x \]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=4 a c x + \frac {4 c^{2} x^{3}}{3} + c d x^{4} + \frac {d^{2} x^{5}}{5} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=\frac {1}{5} \, d^{2} x^{5} + c d x^{4} + \frac {4}{3} \, c^{2} x^{3} + 4 \, a c x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=\frac {1}{5} \, d^{2} x^{5} + c d x^{4} + \frac {4}{3} \, c^{2} x^{3} + 4 \, a c x \]

[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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right ) \, dx=\frac {4\,c^2\,x^3}{3}+c\,d\,x^4+4\,a\,c\,x+\frac {d^2\,x^5}{5} \]

[In]

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

[Out]

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