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

Optimal. Leaf size=69 \[ \frac {c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {626, 12, 43} \[ \frac {c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*d^2*(d + e*x)^(3 + m))/(e^3*(3 + m)) - (2*c^2*d*(d + e*x)^(4 + m))/(e^3*(4 + m)) + (c^2*(d + e*x)^(5 + m)
)/(e^3*(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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx &=\int c^2 x^2 (d+e x)^{2+m} \, dx\\ &=c^2 \int x^2 (d+e x)^{2+m} \, dx\\ &=c^2 \int \left (\frac {d^2 (d+e x)^{2+m}}{e^2}-\frac {2 d (d+e x)^{3+m}}{e^2}+\frac {(d+e x)^{4+m}}{e^2}\right ) \, dx\\ &=\frac {c^2 d^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c^2 d (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^3 (5+m)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 60, normalized size = 0.87 \[ \frac {c^2 (d+e x)^{m+3} \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )}{e^3 (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^(3 + m)*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2))/(e^3*(3 + m)*(4 + m)*(5 + m))

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fricas [B]  time = 1.13, size = 201, normalized size = 2.91 \[ -\frac {{\left (2 \, c^{2} d^{4} e m x - 2 \, c^{2} d^{5} - {\left (c^{2} e^{5} m^{2} + 7 \, c^{2} e^{5} m + 12 \, c^{2} e^{5}\right )} x^{5} - {\left (3 \, c^{2} d e^{4} m^{2} + 19 \, c^{2} d e^{4} m + 30 \, c^{2} d e^{4}\right )} x^{4} - {\left (3 \, c^{2} d^{2} e^{3} m^{2} + 15 \, c^{2} d^{2} e^{3} m + 20 \, c^{2} d^{2} e^{3}\right )} x^{3} - {\left (c^{2} d^{3} e^{2} m^{2} + c^{2} d^{3} e^{2} m\right )} x^{2}\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c^2*d^4*e*m*x - 2*c^2*d^5 - (c^2*e^5*m^2 + 7*c^2*e^5*m + 12*c^2*e^5)*x^5 - (3*c^2*d*e^4*m^2 + 19*c^2*d*e^4
*m + 30*c^2*d*e^4)*x^4 - (3*c^2*d^2*e^3*m^2 + 15*c^2*d^2*e^3*m + 20*c^2*d^2*e^3)*x^3 - (c^2*d^3*e^2*m^2 + c^2*
d^3*e^2*m)*x^2)*(e*x + d)^m/(e^3*m^3 + 12*e^3*m^2 + 47*e^3*m + 60*e^3)

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giac [B]  time = 0.22, size = 292, normalized size = 4.23 \[ \frac {{\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + 3 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 7 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 19 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} + 15 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + 12 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 30 \, {\left (x e + d\right )}^{m} c^{2} d x^{4} e^{4} + 20 \, {\left (x e + d\right )}^{m} c^{2} d^{2} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} c^{2} d^{5}}{m^{3} e^{3} + 12 \, m^{2} e^{3} + 47 \, m e^{3} + 60 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^2*m^2*x^5*e^5 + 3*(x*e + d)^m*c^2*d*m^2*x^4*e^4 + 3*(x*e + d)^m*c^2*d^2*m^2*x^3*e^3 + (x*e + d)
^m*c^2*d^3*m^2*x^2*e^2 + 7*(x*e + d)^m*c^2*m*x^5*e^5 + 19*(x*e + d)^m*c^2*d*m*x^4*e^4 + 15*(x*e + d)^m*c^2*d^2
*m*x^3*e^3 + (x*e + d)^m*c^2*d^3*m*x^2*e^2 - 2*(x*e + d)^m*c^2*d^4*m*x*e + 12*(x*e + d)^m*c^2*x^5*e^5 + 30*(x*
e + d)^m*c^2*d*x^4*e^4 + 20*(x*e + d)^m*c^2*d^2*x^3*e^3 + 2*(x*e + d)^m*c^2*d^5)/(m^3*e^3 + 12*m^2*e^3 + 47*m*
e^3 + 60*e^3)

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maple [A]  time = 0.04, size = 76, normalized size = 1.10 \[ \frac {\left (e^{2} m^{2} x^{2}+7 e^{2} m \,x^{2}-2 d e m x +12 x^{2} e^{2}-6 e x d +2 d^{2}\right ) c^{2} \left (e x +d \right )^{m +3}}{\left (m^{3}+12 m^{2}+47 m +60\right ) e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(m+3)*(e^2*m^2*x^2+7*e^2*m*x^2-2*d*e*m*x+12*e^2*x^2-6*d*e*x+2*d^2)*c^2/e^3/(m^3+12*m^2+47*m+60)

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maxima [B]  time = 1.59, size = 323, normalized size = 4.68 \[ \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2} d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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^3) + 2*((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^3) + ((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^3)

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mupad [B]  time = 0.42, size = 197, normalized size = 2.86 \[ {\left (d+e\,x\right )}^m\,\left (\frac {2\,c^2\,d^5}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,d^2\,x^3\,\left (3\,m^2+15\,m+20\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,c^2\,d^4\,m\,x}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d\,e\,x^4\,\left (3\,m^2+19\,m+30\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,d^3\,m\,x^2\,\left (m+1\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d + e*x)^m*((2*c^2*d^5)/(e^3*(47*m + 12*m^2 + m^3 + 60)) + (c^2*e^2*x^5*(7*m + m^2 + 12))/(47*m + 12*m^2 + m^
3 + 60) + (c^2*d^2*x^3*(15*m + 3*m^2 + 20))/(47*m + 12*m^2 + m^3 + 60) - (2*c^2*d^4*m*x)/(e^2*(47*m + 12*m^2 +
m^3 + 60)) + (c^2*d*e*x^4*(19*m + 3*m^2 + 30))/(47*m + 12*m^2 + m^3 + 60) + (c^2*d^3*m*x^2*(m + 1))/(e*(47*m
+ 12*m^2 + m^3 + 60)))

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sympy [A]  time = 2.99, size = 993, normalized size = 14.39 \[ \begin {cases} \frac {c^{2} d^{2} d^{m} x^{3}}{3} & \text {for}\: e = 0 \\\frac {2 c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {3 c^{2} d^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c^{2} d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} & \text {for}\: m = -5 \\- \frac {2 c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {4 c^{2} d^{2}}{d e^{3} + e^{4} x} - \frac {2 c^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 c^{2} d e x}{d e^{3} + e^{4} x} + \frac {c^{2} e^{2} x^{2}}{d e^{3} + e^{4} x} & \text {for}\: m = -4 \\\frac {c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{3}} - \frac {c^{2} d x}{e^{2}} + \frac {c^{2} x^{2}}{2 e} & \text {for}\: m = -3 \\\frac {2 c^{2} d^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} - \frac {2 c^{2} d^{4} e m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} d^{3} e^{2} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} d^{3} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {3 c^{2} d^{2} e^{3} m^{2} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {15 c^{2} d^{2} e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {20 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {3 c^{2} d e^{4} m^{2} x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {19 c^{2} d e^{4} m x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {30 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} e^{5} m^{2} x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {7 c^{2} e^{5} m x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {12 c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**2*d**2*d**m*x**3/3, Eq(e, 0)), (2*c**2*d**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2
) + 3*c**2*d**2/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*c**2*d*e*x*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x
+ 2*e**5*x**2) + 4*c**2*d*e*x/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*c**2*e**2*x**2*log(d/e + x)/(2*d**
2*e**3 + 4*d*e**4*x + 2*e**5*x**2), Eq(m, -5)), (-2*c**2*d**2*log(d/e + x)/(d*e**3 + e**4*x) - 4*c**2*d**2/(d*
e**3 + e**4*x) - 2*c**2*d*e*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*c**2*d*e*x/(d*e**3 + e**4*x) + c**2*e**2*x**2
/(d*e**3 + e**4*x), Eq(m, -4)), (c**2*d**2*log(d/e + x)/e**3 - c**2*d*x/e**2 + c**2*x**2/(2*e), Eq(m, -3)), (2
*c**2*d**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) - 2*c**2*d**4*e*m*x*(d + e*x)**m/(e**
3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*d**3*e**2*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2
+ 47*e**3*m + 60*e**3) + c**2*d**3*e**2*m*x**2*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3)
+ 3*c**2*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 15*c**2*d**2*e**3
*m*x**3*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 20*c**2*d**2*e**3*x**3*(d + e*x)**m/(e
**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 3*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**
2 + 47*e**3*m + 60*e**3) + 19*c**2*d*e**4*m*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3)
+ 30*c**2*d*e**4*x**4*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + c**2*e**5*m**2*x**5*(d
+ e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e**3) + 7*c**2*e**5*m*x**5*(d + e*x)**m/(e**3*m**3 + 12*e
**3*m**2 + 47*e**3*m + 60*e**3) + 12*c**2*e**5*x**5*(d + e*x)**m/(e**3*m**3 + 12*e**3*m**2 + 47*e**3*m + 60*e*
*3), True))

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