3.16.18 \(\int (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\)

Optimal. Leaf size=34 \[ \frac {1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac {1}{3} c d e x^3 \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Antiderivative was successfully verified.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 38, normalized size = 1.12 \begin {gather*} a d e x+\frac {1}{2} a e^2 x^2+\frac {1}{2} c d^2 x^2+\frac {1}{3} c d e x^3 \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.35, size = 32, normalized size = 0.94 \begin {gather*} \frac {1}{3} x^{3} e d c + \frac {1}{2} x^{2} d^{2} c + \frac {1}{2} x^{2} e^{2} a + x e d a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 31, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, c d x^{3} e + a d x e + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 31, normalized size = 0.91 \begin {gather*} \frac {c d e \,x^{3}}{3}+a d e x +\frac {\left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 30, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 31, normalized size = 0.91 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 32, normalized size = 0.94 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________