3.10.88 \(\int (d+e x) (c d^2+2 c d e x+c e^2 x^2)^2 \, dx\) [988]

Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^6}{6 e} \]

[Out]

1/6*c^2*(e*x+d)^6/e

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Rubi [A]
time = 0.00, antiderivative size = 17, 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^2 (d+e x)^6}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^6)/(6*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) \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^5 \, dx\\ &=c^2 \int (d+e x)^5 \, dx\\ &=\frac {c^2 (d+e x)^6}{6 e}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^6}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Maple [A]
time = 0.67, size = 31, normalized size = 1.82

method	result	size
default	\(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{3}}{6 c e}\)	\(31\)
gosper	\(\frac {x \left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) c^{2}}{6}\)	\(58\)
norman	\(c^{2} d^{5} x +c^{2} d \,x^{5} e^{4}+\frac {1}{6} c^{2} x^{6} e^{5}+\frac {5}{2} c^{2} d^{2} e^{3} x^{4}+\frac {10}{3} c^{2} d^{3} e^{2} x^{3}+\frac {5}{2} c^{2} d^{4} e \,x^{2}\)	\(72\)
risch	\(\frac {c^{2} x^{6} e^{5}}{6}+c^{2} d \,x^{5} e^{4}+\frac {5 c^{2} d^{2} e^{3} x^{4}}{2}+\frac {10 c^{2} d^{3} e^{2} x^{3}}{3}+\frac {5 c^{2} d^{4} e \,x^{2}}{2}+c^{2} d^{5} x +\frac {c^{2} d^{6}}{6 e}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*e^2*x^2+2*c*d*e*x+c*d^2)^3/c/e

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Maxima [A]
time = 0.28, size = 29, normalized size = 1.71 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{3} e^{\left (-1\right )}}{6 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^3*e^(-1)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (15) = 30\).
time = 2.11, size = 68, normalized size = 4.00 \begin {gather*} \frac {1}{6} \, c^{2} x^{6} e^{5} + c^{2} d x^{5} e^{4} + \frac {5}{2} \, c^{2} d^{2} x^{4} e^{3} + \frac {10}{3} \, c^{2} d^{3} x^{3} e^{2} + \frac {5}{2} \, c^{2} d^{4} x^{2} e + c^{2} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*x^6*e^5 + c^2*d*x^5*e^4 + 5/2*c^2*d^2*x^4*e^3 + 10/3*c^2*d^3*x^3*e^2 + 5/2*c^2*d^4*x^2*e + c^2*d^5*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (12) = 24\).
time = 0.02, size = 80, normalized size = 4.71 \begin {gather*} c^{2} d^{5} x + \frac {5 c^{2} d^{4} e x^{2}}{2} + \frac {10 c^{2} d^{3} e^{2} x^{3}}{3} + \frac {5 c^{2} d^{2} e^{3} x^{4}}{2} + c^{2} d e^{4} x^{5} + \frac {c^{2} e^{5} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*d**5*x + 5*c**2*d**4*e*x**2/2 + 10*c**2*d**3*e**2*x**3/3 + 5*c**2*d**2*e**3*x**4/2 + c**2*d*e**4*x**5 + c
**2*e**5*x**6/6

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (15) = 30\).
time = 3.48, size = 63, normalized size = 3.71 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} c^{2} d^{4} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{2} c^{2} d^{2} e + \frac {1}{6} \, {\left (x^{2} e + 2 \, d x\right )}^{3} c^{2} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2*e + 2*d*x)*c^2*d^4 + 1/2*(x^2*e + 2*d*x)^2*c^2*d^2*e + 1/6*(x^2*e + 2*d*x)^3*c^2*e^2

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Mupad [B]
time = 0.03, size = 71, normalized size = 4.18 \begin {gather*} c^2\,d^5\,x+\frac {5\,c^2\,d^4\,e\,x^2}{2}+\frac {10\,c^2\,d^3\,e^2\,x^3}{3}+\frac {5\,c^2\,d^2\,e^3\,x^4}{2}+c^2\,d\,e^4\,x^5+\frac {c^2\,e^5\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*d^5*x + (c^2*e^5*x^6)/6 + (5*c^2*d^4*e*x^2)/2 + c^2*d*e^4*x^5 + (10*c^2*d^3*e^2*x^3)/3 + (5*c^2*d^2*e^3*x^
4)/2

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