3.11.89 \(\int (d+e x)^m (c d^2+2 c d e x+c e^2 x^2)^2 \, dx\) [1089]

Optimal. Leaf size=21 \[ \frac {c^2 (d+e x)^{5+m}}{e (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \begin {gather*} \frac {c^2 (d+e x)^{m+5}}{e (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^(5 + m))/(e*(5 + m))

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)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^{4+m} \, dx\\ &=c^2 \int (d+e x)^{4+m} \, dx\\ &=\frac {c^2 (d+e x)^{5+m}}{e (5+m)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.05 \begin {gather*} \frac {c^2 (d+e x)^{5+m}}{5 e+e m} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^(5 + m))/(5*e + e*m)

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Maple [A]
time = 0.58, size = 40, normalized size = 1.90

method	result	size
gosper	\(\frac {\left (e x +d \right )^{1+m} c^{2} \left (e^{2} x^{2}+2 d x e +d^{2}\right )^{2}}{e \left (5+m \right )}\)	\(40\)
risch	\(\frac {c^{2} \left (e^{5} x^{5}+5 d \,e^{4} x^{4}+10 d^{2} e^{3} x^{3}+10 d^{3} e^{2} x^{2}+5 d^{4} e x +d^{5}\right ) \left (e x +d \right )^{m}}{e \left (5+m \right )}\)	\(69\)
norman	\(\frac {c^{2} d^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (5+m \right )}+\frac {e^{4} c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {5 c^{2} d^{4} x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {5 c^{2} d \,e^{3} x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {10 c^{2} d^{2} e^{2} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {10 c^{2} d^{3} e \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1+m)*c^2*(e^2*x^2+2*d*e*x+d^2)^2/e/(5+m)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (21) = 42\).
time = 0.29, size = 399, normalized size = 19.00 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} c^{2} d^{4} e^{\left (-1\right )}}{m + 1} + \frac {4 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} c^{2} d^{3} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{2} + 3 \, m + 2} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} c^{2} d^{2} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} c^{2} d e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} c^{2} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)^(m + 1)*c^2*d^4*e^(-1)/(m + 1) + 4*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*c^2*d^3*e^(m*log(x*e + d) - 1)/
(m^2 + 3*m + 2) + 6*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*c^2*d^2*e^(m*log(x*e
+ d) - 1)/(m^3 + 6*m^2 + 11*m + 6) + 4*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*
(m^2 + m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*c^2*d*e^(m*log(x*e + d) - 1)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)
+ ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m
)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*c^2*e^(m*log(x*e + d) - 1)/(m^5 + 15*m^4 + 8
5*m^3 + 225*m^2 + 274*m + 120)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
time = 2.90, size = 81, normalized size = 3.86 \begin {gather*} \frac {{\left (c^{2} x^{5} e^{5} + 5 \, c^{2} d x^{4} e^{4} + 10 \, c^{2} d^{2} x^{3} e^{3} + 10 \, c^{2} d^{3} x^{2} e^{2} + 5 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} {\left (x e + d\right )}^{m} e^{\left (-1\right )}}{m + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (15) = 30\).
time = 0.48, size = 185, normalized size = 8.81 \begin {gather*} \begin {cases} \frac {c^{2} x}{d} & \text {for}\: e = 0 \wedge m = -5 \\c^{2} d^{4} d^{m} x & \text {for}\: e = 0 \\\frac {c^{2} \log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: m = -5 \\\frac {c^{2} d^{5} \left (d + e x\right )^{m}}{e m + 5 e} + \frac {5 c^{2} d^{4} e x \left (d + e x\right )^{m}}{e m + 5 e} + \frac {10 c^{2} d^{3} e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 5 e} + \frac {10 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 5 e} + \frac {5 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e m + 5 e} + \frac {c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e m + 5 e} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**2*x/d, Eq(e, 0) & Eq(m, -5)), (c**2*d**4*d**m*x, Eq(e, 0)), (c**2*log(d/e + x)/e, Eq(m, -5)), (c
**2*d**5*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d**4*e*x*(d + e*x)**m/(e*m + 5*e) + 10*c**2*d**3*e**2*x**2*(d + e*x
)**m/(e*m + 5*e) + 10*c**2*d**2*e**3*x**3*(d + e*x)**m/(e*m + 5*e) + 5*c**2*d*e**4*x**4*(d + e*x)**m/(e*m + 5*
e) + c**2*e**5*x**5*(d + e*x)**m/(e*m + 5*e), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (21) = 42\).
time = 1.00, size = 125, normalized size = 5.95 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 5 \, {\left (x e + d\right )}^{m} c^{2} d x^{4} e^{4} + 10 \, {\left (x e + d\right )}^{m} c^{2} d^{2} x^{3} e^{3} + 10 \, {\left (x e + d\right )}^{m} c^{2} d^{3} x^{2} e^{2} + 5 \, {\left (x e + d\right )}^{m} c^{2} d^{4} x e + {\left (x e + d\right )}^{m} c^{2} d^{5}}{m e + 5 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.51, size = 106, normalized size = 5.05 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {5\,c^2\,d^4\,x}{m+5}+\frac {c^2\,d^5}{e\,\left (m+5\right )}+\frac {c^2\,e^4\,x^5}{m+5}+\frac {10\,c^2\,d^3\,e\,x^2}{m+5}+\frac {5\,c^2\,d\,e^3\,x^4}{m+5}+\frac {10\,c^2\,d^2\,e^2\,x^3}{m+5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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