\(\int (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4) \, dx\) [42]

Optimal result

Integrand size = 30, antiderivative size = 37 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=8 a e^2 x-\frac {d^3 x^2}{2}+2 d e^2 x^4+\frac {8 e^3 x^5}{5} \]

[Out]

8*a*e^2*x-1/2*d^3*x^2+2*d*e^2*x^4+8/5*e^3*x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, 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 = {} \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=8 a e^2 x-\frac {d^3 x^2}{2}+2 d e^2 x^4+\frac {8 e^3 x^5}{5} \]

[In]

Int[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4,x]

[Out]

8*a*e^2*x - (d^3*x^2)/2 + 2*d*e^2*x^4 + (8*e^3*x^5)/5

Rubi steps \begin{align*} \text {integral}& = 8 a e^2 x-\frac {d^3 x^2}{2}+2 d e^2 x^4+\frac {8 e^3 x^5}{5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=8 a e^2 x-\frac {d^3 x^2}{2}+2 d e^2 x^4+\frac {8 e^3 x^5}{5} \]

[In]

Integrate[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4,x]

[Out]

8*a*e^2*x - (d^3*x^2)/2 + 2*d*e^2*x^4 + (8*e^3*x^5)/5

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92

method	result	size
gosper	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)
default	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)
norman	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)
risch	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)
parallelrisch	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)
parts	\(8 a \,e^{2} x -\frac {1}{2} d^{3} x^{2}+2 d \,e^{2} x^{4}+\frac {8}{5} e^{3} x^{5}\)	\(34\)

[In]

int(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2,x,method=_RETURNVERBOSE)

[Out]

8*a*e^2*x-1/2*d^3*x^2+2*d*e^2*x^4+8/5*e^3*x^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=\frac {8}{5} \, e^{3} x^{5} + 2 \, d e^{2} x^{4} - \frac {1}{2} \, d^{3} x^{2} + 8 \, a e^{2} x \]

[In]

integrate(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2,x, algorithm="fricas")

[Out]

8/5*e^3*x^5 + 2*d*e^2*x^4 - 1/2*d^3*x^2 + 8*a*e^2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=8 a e^{2} x - \frac {d^{3} x^{2}}{2} + 2 d e^{2} x^{4} + \frac {8 e^{3} x^{5}}{5} \]

[In]

integrate(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2,x)

[Out]

8*a*e**2*x - d**3*x**2/2 + 2*d*e**2*x**4 + 8*e**3*x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=\frac {8}{5} \, e^{3} x^{5} + 2 \, d e^{2} x^{4} - \frac {1}{2} \, d^{3} x^{2} + 8 \, a e^{2} x \]

[In]

integrate(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2,x, algorithm="maxima")

[Out]

8/5*e^3*x^5 + 2*d*e^2*x^4 - 1/2*d^3*x^2 + 8*a*e^2*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=\frac {8}{5} \, e^{3} x^{5} + 2 \, d e^{2} x^{4} - \frac {1}{2} \, d^{3} x^{2} + 8 \, a e^{2} x \]

[In]

integrate(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2,x, algorithm="giac")

[Out]

8/5*e^3*x^5 + 2*d*e^2*x^4 - 1/2*d^3*x^2 + 8*a*e^2*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right ) \, dx=-\frac {d^3\,x^2}{2}+2\,d\,e^2\,x^4+\frac {8\,e^3\,x^5}{5}+8\,a\,e^2\,x \]

[In]

int(8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3,x)

[Out]

(8*e^3*x^5)/5 - (d^3*x^2)/2 + 2*d*e^2*x^4 + 8*a*e^2*x