\(\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2) \, dx\) [978]

Optimal result

Integrand size = 28, antiderivative size = 15 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^5}{5 e} \]

[Out]

1/5*c*(e*x+d)^5/e

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^5}{5 e} \]

[In]

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

[Out]

(c*(d + e*x)^5)/(5*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int c (d+e x)^4 \, dx \\ & = c \int (d+e x)^4 \, dx \\ & = \frac {c (d+e x)^5}{5 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^5}{5 e} \]

[In]

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

[Out]

(c*(d + e*x)^5)/(5*e)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(13)=26\).

Time = 2.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 3.00

method	result	size
gosper	\(\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) c}{5}\)	\(45\)
default	\(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\)	\(48\)
norman	\(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\)	\(48\)
risch	\(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\)	\(48\)
parallelrisch	\(\frac {1}{5} e^{4} c \,x^{5}+d \,e^{3} c \,x^{4}+2 d^{2} e^{2} c \,x^{3}+2 c \,d^{3} e \,x^{2}+c \,d^{4} x\)	\(48\)

[In]

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

[Out]

1/5*x*(e^4*x^4+5*d*e^3*x^3+10*d^2*e^2*x^2+10*d^3*e*x+5*d^4)*c

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (13) = 26\).

Time = 0.37 (sec) , antiderivative size = 47, normalized size of antiderivative = 3.13 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {1}{5} \, c e^{4} x^{5} + c d e^{3} x^{4} + 2 \, c d^{2} e^{2} x^{3} + 2 \, c d^{3} e x^{2} + c d^{4} x \]

[In]

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

[Out]

1/5*c*e^4*x^5 + c*d*e^3*x^4 + 2*c*d^2*e^2*x^3 + 2*c*d^3*e*x^2 + c*d^4*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (10) = 20\).

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.40 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=c d^{4} x + 2 c d^{3} e x^{2} + 2 c d^{2} e^{2} x^{3} + c d e^{3} x^{4} + \frac {c e^{4} x^{5}}{5} \]

[In]

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

[Out]

c*d**4*x + 2*c*d**3*e*x**2 + 2*c*d**2*e**2*x**3 + c*d*e**3*x**4 + c*e**4*x**5/5

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (13) = 26\).

Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 3.13 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {1}{5} \, c e^{4} x^{5} + c d e^{3} x^{4} + 2 \, c d^{2} e^{2} x^{3} + 2 \, c d^{3} e x^{2} + c d^{4} x \]

[In]

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

[Out]

1/5*c*e^4*x^5 + c*d*e^3*x^4 + 2*c*d^2*e^2*x^3 + 2*c*d^3*e*x^2 + c*d^4*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 3.13 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {1}{5} \, c e^{4} x^{5} + c d e^{3} x^{4} + 2 \, c d^{2} e^{2} x^{3} + 2 \, c d^{3} e x^{2} + c d^{4} x \]

[In]

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

[Out]

1/5*c*e^4*x^5 + c*d*e^3*x^4 + 2*c*d^2*e^2*x^3 + 2*c*d^3*e*x^2 + c*d^4*x

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 47, normalized size of antiderivative = 3.13 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=c\,d^4\,x+2\,c\,d^3\,e\,x^2+2\,c\,d^2\,e^2\,x^3+c\,d\,e^3\,x^4+\frac {c\,e^4\,x^5}{5} \]

[In]

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

[Out]

(c*e^4*x^5)/5 + c*d^4*x + 2*c*d^2*e^2*x^3 + 2*c*d^3*e*x^2 + c*d*e^3*x^4