\(\int (d+e x) (a+b x+c x^2) \, dx\) [2110]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=a d x+\frac {1}{2} (b d+a e) x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4 \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {645} \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=\frac {1}{2} x^2 (a e+b d)+a d x+\frac {1}{3} x^3 (b e+c d)+\frac {1}{4} c e x^4 \]

[In]

Int[(d + e*x)*(a + b*x + c*x^2),x]

[Out]

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

Rule 645

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=a d x+\frac {1}{2} (b d+a e) x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88

method	result	size
default	\(a d x +\frac {\left (a e +b d \right ) x^{2}}{2}+\frac {\left (b e +c d \right ) x^{3}}{3}+\frac {c e \,x^{4}}{4}\)	\(37\)
norman	\(\frac {c e \,x^{4}}{4}+\left (\frac {b e}{3}+\frac {c d}{3}\right ) x^{3}+\left (\frac {a e}{2}+\frac {b d}{2}\right ) x^{2}+a d x\)	\(39\)
gosper	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+\frac {1}{2} b d \,x^{2}+a d x\)	\(41\)
risch	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+\frac {1}{2} b d \,x^{2}+a d x\)	\(41\)
parallelrisch	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} b e \,x^{3}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+\frac {1}{2} b d \,x^{2}+a d x\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} + a d x + \frac {1}{2} \, {\left (b d + a e\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=a d x + \frac {c e x^{4}}{4} + x^{3} \left (\frac {b e}{3} + \frac {c d}{3}\right ) + x^{2} \left (\frac {a e}{2} + \frac {b d}{2}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} + a d x + \frac {1}{2} \, {\left (b d + a e\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c e x^{4} + \frac {1}{3} \, c d x^{3} + \frac {1}{3} \, b e x^{3} + \frac {1}{2} \, b d x^{2} + \frac {1}{2} \, a e x^{2} + a d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int (d+e x) \left (a+b x+c x^2\right ) \, dx=\frac {c\,e\,x^4}{4}+\left (\frac {b\,e}{3}+\frac {c\,d}{3}\right )\,x^3+\left (\frac {a\,e}{2}+\frac {b\,d}{2}\right )\,x^2+a\,d\,x \]

[In]

int((d + e*x)*(a + b*x + c*x^2),x)

[Out]

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