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

Optimal. Leaf size=15 \[ \frac {c (d+e x)^5}{5 e} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32} \begin {gather*} \frac {c (d+e x)^5}{5 e} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\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]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {c (d+e x)^5}{5 e} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.34, size = 47, normalized size = 3.13 \begin {gather*} \frac {1}{5} x^{5} e^{4} c + x^{4} e^{3} d c + 2 x^{3} e^{2} d^{2} c + 2 x^{2} e d^{3} c + x d^{4} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*x^5*e^4*c + x^4*e^3*d*c + 2*x^3*e^2*d^2*c + 2*x^2*e*d^3*c + x*d^4*c

________________________________________________________________________________________

giac [B]  time = 0.16, size = 45, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, c x^{5} e^{4} + c d x^{4} e^{3} + 2 \, c d^{2} x^{3} e^{2} + 2 \, c d^{3} x^{2} e + c d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*x^5*e^4 + c*d*x^4*e^3 + 2*c*d^2*x^3*e^2 + 2*c*d^3*x^2*e + c*d^4*x

________________________________________________________________________________________

maple [B]  time = 0.04, size = 48, normalized size = 3.20 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.28, size = 47, normalized size = 3.13 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 0.42, size = 47, normalized size = 3.13 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [B]  time = 0.08, size = 51, normalized size = 3.40 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________