\(\int (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1832]

Optimal result

Integrand size = 25, antiderivative size = 34 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d e x+\frac {1}{2} \left (c d^2+a e^2\right ) x^2+\frac {1}{3} c d e x^3 \]

[Out]

a*d*e*x+1/2*(a*e^2+c*d^2)*x^2+1/3*c*d*e*x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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 = {} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac {1}{3} c d e x^3 \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = a d e x+\frac {1}{2} \left (c d^2+a e^2\right ) x^2+\frac {1}{3} c d e x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d e x+\frac {1}{2} c d^2 x^2+\frac {1}{2} a e^2 x^2+\frac {1}{3} c d e x^3 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91

method	result	size
default	\(a d e x +\frac {\left (e^{2} a +c \,d^{2}\right ) x^{2}}{2}+\frac {c d e \,x^{3}}{3}\)	\(31\)
gosper	\(\frac {x \left (2 c d e \,x^{2}+3 a \,e^{2} x +3 c \,d^{2} x +6 a d e \right )}{6}\)	\(32\)
norman	\(\frac {c d e \,x^{3}}{3}+\left (\frac {e^{2} a}{2}+\frac {c \,d^{2}}{2}\right ) x^{2}+a d e x\)	\(32\)
risch	\(a d e x +\frac {1}{2} x^{2} e^{2} a +\frac {1}{2} c \,d^{2} x^{2}+\frac {1}{3} c d e \,x^{3}\)	\(33\)
parallelrisch	\(a d e x +\frac {1}{2} x^{2} e^{2} a +\frac {1}{2} c \,d^{2} x^{2}+\frac {1}{3} c d e \,x^{3}\)	\(33\)
parts	\(a d e x +\frac {1}{2} x^{2} e^{2} a +\frac {1}{2} c \,d^{2} x^{2}+\frac {1}{3} c d e \,x^{3}\)	\(33\)

[In]

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

[Out]

a*d*e*x+1/2*(a*e^2+c*d^2)*x^2+1/3*c*d*e*x^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} \, c d e x^{3} + a d e x + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} x^{2} \]

[In]

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

[Out]

1/3*c*d*e*x^3 + a*d*e*x + 1/2*(c*d^2 + a*e^2)*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d e x + \frac {c d e x^{3}}{3} + x^{2} \left (\frac {a e^{2}}{2} + \frac {c d^{2}}{2}\right ) \]

[In]

integrate(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2,x)

[Out]

a*d*e*x + c*d*e*x**3/3 + x**2*(a*e**2/2 + c*d**2/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} \, c d e x^{3} + a d e x + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} x^{2} \]

[In]

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

[Out]

1/3*c*d*e*x^3 + a*d*e*x + 1/2*(c*d^2 + a*e^2)*x^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} \, c d e x^{3} + a d e x + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} x^{2} \]

[In]

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

[Out]

1/3*c*d*e*x^3 + a*d*e*x + 1/2*(c*d^2 + a*e^2)*x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {c\,d\,e\,x^3}{3}+\left (\frac {c\,d^2}{2}+\frac {a\,e^2}{2}\right )\,x^2+a\,d\,e\,x \]

[In]

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

[Out]

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