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

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

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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 = 0.02, size = 22, normalized size = 1.05 \begin {gather*} \frac {c^2 (d+e x)^{m+5}}{e m+5 e} \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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IntegrateAlgebraic [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 85, normalized size = 4.05 \begin {gather*} \frac {{\left (c^{2} e^{5} x^{5} + 5 \, c^{2} d e^{4} x^{4} + 10 \, c^{2} d^{2} e^{3} x^{3} + 10 \, c^{2} d^{3} e^{2} x^{2} + 5 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} {\left (e x + d\right )}^{m}}{e m + 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="fricas")

[Out]

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

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giac [B]  time = 0.25, 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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maple [A]  time = 0.05, size = 40, normalized size = 1.90 \begin {gather*} \frac {\left (e^{2} x^{2}+2 d x e +d^{2}\right )^{2} c^{2} \left (e x +d \right )^{m +1}}{\left (m +5\right ) e} \end {gather*}

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)

[Out]

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

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maxima [B]  time = 1.65, size = 397, normalized size = 18.90 \begin {gather*} \frac {4 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c^{2} d^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {{\left (e x + d\right )}^{m + 1} c^{2} d^{4}}{e {\left (m + 1\right )}} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2} d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} 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="maxima")

[Out]

4*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*c^2*d^3/((m^2 + 3*m + 2)*e) + (e*x + d)^(m + 1)*c^2*d^4/(e*(m
+ 1)) + 6*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*c^2*d^2/((m^3 + 6*
m^2 + 11*m + 6)*e) + 4*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2
*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*c^2*d/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e) + ((m^4 + 10*m^3 + 35*m^
2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2
+ m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*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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sympy [A]  time = 1.75, 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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