\(\int (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3) \, dx\) [17]

Optimal result

Integrand size = 44, antiderivative size = 56 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=a c e x+\frac {1}{2} (b c e+a d e+a c f) x^2+\frac {1}{3} (b d e+b c f+a d f) x^3+\frac {1}{4} b d f x^4 \]

[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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, 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 = {} \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=\frac {1}{3} x^3 (a d f+b c f+b d e)+\frac {1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac {1}{4} b d f x^4 \]

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

Time = 0.00 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=a c e x+\frac {1}{2} b c e x^2+\frac {1}{2} a d e x^2+\frac {1}{2} a c f x^2+\frac {1}{3} b d e x^3+\frac {1}{3} b c f x^3+\frac {1}{3} a d f x^3+\frac {1}{4} b d f x^4 \]

[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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {b d f \,x^{4}}{4}+\frac {\left (\left (c f +e d \right ) b +a d f \right ) x^{3}}{3}+\frac {\left (e b c +\left (c f +e d \right ) a \right ) x^{2}}{2}+a c e x\)	\(53\)
norman	\(\frac {b d f \,x^{4}}{4}+\left (\frac {1}{3} a d f +\frac {1}{3} f b c +\frac {1}{3} b d e \right ) x^{3}+\left (\frac {1}{2} a c f +\frac {1}{2} e d a +\frac {1}{2} e b c \right ) x^{2}+a c e x\)	\(55\)
gosper	\(\frac {x \left (3 b d f \,x^{3}+4 a d f \,x^{2}+4 b c f \,x^{2}+4 b d e \,x^{2}+6 a c f x +6 e d a x +6 b c e x +12 a c e \right )}{12}\)	\(60\)
risch	\(a c e x +\frac {1}{2} a c f \,x^{2}+\frac {1}{2} a d e \,x^{2}+\frac {1}{2} b c e \,x^{2}+\frac {1}{3} a d f \,x^{3}+\frac {1}{3} b c f \,x^{3}+\frac {1}{3} e b \,x^{3} d +\frac {1}{4} b d f \,x^{4}\)	\(63\)
parallelrisch	\(a c e x +\frac {1}{2} a c f \,x^{2}+\frac {1}{2} a d e \,x^{2}+\frac {1}{2} b c e \,x^{2}+\frac {1}{3} a d f \,x^{3}+\frac {1}{3} b c f \,x^{3}+\frac {1}{3} e b \,x^{3} d +\frac {1}{4} b d f \,x^{4}\)	\(63\)
parts	\(a c e x +\frac {1}{2} a c f \,x^{2}+\frac {1}{2} a d e \,x^{2}+\frac {1}{2} b c e \,x^{2}+\frac {1}{3} a d f \,x^{3}+\frac {1}{3} b c f \,x^{3}+\frac {1}{3} e b \,x^{3} d +\frac {1}{4} b d f \,x^{4}\)	\(63\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=\frac {1}{4} \, b d f x^{4} + a c e x + \frac {1}{3} \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{3} + \frac {1}{2} \, {\left (a c f + {\left (b c + a d\right )} e\right )} x^{2} \]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=a c e x + \frac {b d f x^{4}}{4} + x^{3} \left (\frac {a d f}{3} + \frac {b c f}{3} + \frac {b d e}{3}\right ) + x^{2} \left (\frac {a c f}{2} + \frac {a d e}{2} + \frac {b c e}{2}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=\frac {1}{4} \, b d f x^{4} + a c e x + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} x^{3} + \frac {1}{2} \, {\left (b c e + a d e + a c f\right )} x^{2} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=\frac {1}{4} \, b d f x^{4} + a c e x + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} x^{3} + \frac {1}{2} \, {\left (b c e + a d e + a c f\right )} x^{2} \]

[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 + 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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right ) \, dx=\frac {b\,d\,f\,x^4}{4}+\left (\frac {a\,d\,f}{3}+\frac {b\,c\,f}{3}+\frac {b\,d\,e}{3}\right )\,x^3+\left (\frac {a\,c\,f}{2}+\frac {a\,d\,e}{2}+\frac {b\,c\,e}{2}\right )\,x^2+a\,c\,e\,x \]

[In]

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

[Out]

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