3.17 \(\int (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3) \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{3} x^3 (a d f+b c f+b d e)+\frac{1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac{1}{4} b d f x^4 \]

[Out]

a*c*e*x + ((b*c*e + a*d*e + a*c*f)*x^2)/2 + ((b*d*e + b*c*f + a*d*f)*x^3)/3 + (b*d*f*x^4)/4

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Rubi [A]  time = 0.0164289, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{1}{3} x^3 (a d f+b c f+b d e)+\frac{1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac{1}{4} b d f x^4 \]

Antiderivative was successfully verified.

[In]

Int[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3,x]

[Out]

a*c*e*x + ((b*c*e + a*d*e + a*c*f)*x^2)/2 + ((b*d*e + b*c*f + a*d*f)*x^3)/3 + (b*d*f*x^4)/4

Rubi steps

\begin{align*} \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx &=a c e x+\frac{1}{2} (b c e+a d e+a c f) x^2+\frac{1}{3} (b d e+b c f+a d f) x^3+\frac{1}{4} b d f x^4\\ \end{align*}

Mathematica [A]  time = 0.0000704, size = 76, normalized size = 1.36 \[ a c e x+\frac{1}{2} a c f x^2+\frac{1}{2} a d e x^2+\frac{1}{3} a d f x^3+\frac{1}{2} b c e x^2+\frac{1}{3} b c f x^3+\frac{1}{3} b d e x^3+\frac{1}{4} b d f x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3,x]

[Out]

a*c*e*x + (b*c*e*x^2)/2 + (a*d*e*x^2)/2 + (a*c*f*x^2)/2 + (b*d*e*x^3)/3 + (b*c*f*x^3)/3 + (a*d*f*x^3)/3 + (b*d
*f*x^4)/4

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Maple [A]  time = 0., size = 51, normalized size = 0.9 \begin{align*} acex+{\frac{ \left ( acf+ade+bce \right ){x}^{2}}{2}}+{\frac{ \left ( adf+bcf+bde \right ){x}^{3}}{3}}+{\frac{bdf{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3,x)

[Out]

a*c*e*x+1/2*(a*c*f+a*d*e+b*c*e)*x^2+1/3*(a*d*f+b*c*f+b*d*e)*x^3+1/4*b*d*f*x^4

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Maxima [A]  time = 1.22255, size = 68, normalized size = 1.21 \begin{align*} \frac{1}{4} \, b d f x^{4} + a c e x + \frac{1}{3} \,{\left (b d e + b c f + a d f\right )} x^{3} + \frac{1}{2} \,{\left (b c e + a d e + a c f\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3,x, algorithm="maxima")

[Out]

1/4*b*d*f*x^4 + a*c*e*x + 1/3*(b*d*e + b*c*f + a*d*f)*x^3 + 1/2*(b*c*e + a*d*e + a*c*f)*x^2

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Fricas [A]  time = 1.08366, size = 163, normalized size = 2.91 \begin{align*} \frac{1}{4} x^{4} f d b + \frac{1}{3} x^{3} e d b + \frac{1}{3} x^{3} f c b + \frac{1}{3} x^{3} f d a + \frac{1}{2} x^{2} e c b + \frac{1}{2} x^{2} e d a + \frac{1}{2} x^{2} f c a + x e c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3,x, algorithm="fricas")

[Out]

1/4*x^4*f*d*b + 1/3*x^3*e*d*b + 1/3*x^3*f*c*b + 1/3*x^3*f*d*a + 1/2*x^2*e*c*b + 1/2*x^2*e*d*a + 1/2*x^2*f*c*a
+ x*e*c*a

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Sympy [A]  time = 0.067543, size = 63, normalized size = 1.12 \begin{align*} a c e x + \frac{b d f x^{4}}{4} + x^{3} \left (\frac{a d f}{3} + \frac{b c f}{3} + \frac{b d e}{3}\right ) + x^{2} \left (\frac{a c f}{2} + \frac{a d e}{2} + \frac{b c e}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3,x)

[Out]

a*c*e*x + b*d*f*x**4/4 + x**3*(a*d*f/3 + b*c*f/3 + b*d*e/3) + x**2*(a*c*f/2 + a*d*e/2 + b*c*e/2)

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Giac [A]  time = 1.06873, size = 73, normalized size = 1.3 \begin{align*} \frac{1}{4} \, b d f x^{4} + \frac{1}{3} \,{\left (b c f + a d f + b d e\right )} x^{3} + a c x e + \frac{1}{2} \,{\left (a c f + b c e + a d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3,x, algorithm="giac")

[Out]

1/4*b*d*f*x^4 + 1/3*(b*c*f + a*d*f + b*d*e)*x^3 + a*c*x*e + 1/2*(a*c*f + b*c*e + a*d*e)*x^2