### 3.436 $$\int (d+e x)^m (c d x+c e x^2) \, dx$$

Optimal. Leaf size=41 $\frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)}$

[Out]

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

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Rubi [A]  time = 0.0221012, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {626, 12, 43} $\frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^m*(c*d*x + c*e*x^2),x]

[Out]

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

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]

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])

Rubi steps

\begin{align*} \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx &=\int c x (d+e x)^{1+m} \, dx\\ &=c \int x (d+e x)^{1+m} \, dx\\ &=c \int \left (-\frac{d (d+e x)^{1+m}}{e}+\frac{(d+e x)^{2+m}}{e}\right ) \, dx\\ &=-\frac{c d (d+e x)^{2+m}}{e^2 (2+m)}+\frac{c (d+e x)^{3+m}}{e^2 (3+m)}\\ \end{align*}

Mathematica [A]  time = 0.0199035, size = 34, normalized size = 0.83 $\frac{c (d+e x)^{m+2} (e (m+2) x-d)}{e^2 (m+2) (m+3)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 37, normalized size = 0.9 \begin{align*} -{\frac{c \left ( ex+d \right ) ^{2+m} \left ( -mex-2\,ex+d \right ) }{{e}^{2} \left ({m}^{2}+5\,m+6 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(e*x+d)^(2+m)*(-e*m*x-2*e*x+d)/e^2/(m^2+5*m+6)

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Maxima [B]  time = 1.23387, size = 154, normalized size = 3.76 \begin{align*} \frac{{\left (e^{2}{\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )}{\left (e x + d\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \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}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*c*d/((m^2 + 3*m + 2)*e^2) + ((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/((m^3 + 6*m^2 + 11*m + 6)*e^2)

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Fricas [A]  time = 2.08918, size = 163, normalized size = 3.98 \begin{align*} \frac{{\left (c d^{2} e m x - c d^{3} +{\left (c e^{3} m + 2 \, c e^{3}\right )} x^{3} +{\left (2 \, c d e^{2} m + 3 \, c d e^{2}\right )} x^{2}\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d^2*e*m*x - c*d^3 + (c*e^3*m + 2*c*e^3)*x^3 + (2*c*d*e^2*m + 3*c*d*e^2)*x^2)*(e*x + d)^m/(e^2*m^2 + 5*e^2*m
+ 6*e^2)

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Sympy [A]  time = 1.50261, size = 299, normalized size = 7.29 \begin{align*} \begin{cases} \frac{c d d^{m} x^{2}}{2} & \text{for}\: e = 0 \\\frac{c d \log{\left (\frac{d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac{c d}{d e^{2} + e^{3} x} + \frac{c e x \log{\left (\frac{d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text{for}\: m = -3 \\- \frac{c d \log{\left (\frac{d}{e} + x \right )}}{e^{2}} + \frac{c x}{e} & \text{for}\: m = -2 \\- \frac{c d^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c d^{2} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac{2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c*d*d**m*x**2/2, Eq(e, 0)), (c*d*log(d/e + x)/(d*e**2 + e**3*x) + c*d/(d*e**2 + e**3*x) + c*e*x*log
(d/e + x)/(d*e**2 + e**3*x), Eq(m, -3)), (-c*d*log(d/e + x)/e**2 + c*x/e, Eq(m, -2)), (-c*d**3*(d + e*x)**m/(e
**2*m**2 + 5*e**2*m + 6*e**2) + c*d**2*e*m*x*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 2*c*d*e**2*m*x**2*
(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 3*c*d*e**2*x**2*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) +
c*e**3*m*x**3*(d + e*x)**m/(e**2*m**2 + 5*e**2*m + 6*e**2) + 2*c*e**3*x**3*(d + e*x)**m/(e**2*m**2 + 5*e**2*m
+ 6*e**2), True))

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Giac [B]  time = 1.27748, size = 159, normalized size = 3.88 \begin{align*} \frac{{\left (x e + d\right )}^{m} c m x^{3} e^{3} + 2 \,{\left (x e + d\right )}^{m} c d m x^{2} e^{2} +{\left (x e + d\right )}^{m} c d^{2} m x e + 2 \,{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d x^{2} e^{2} -{\left (x e + d\right )}^{m} c d^{3}}{m^{2} e^{2} + 5 \, m e^{2} + 6 \, e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c*m*x^3*e^3 + 2*(x*e + d)^m*c*d*m*x^2*e^2 + (x*e + d)^m*c*d^2*m*x*e + 2*(x*e + d)^m*c*x^3*e^3 + 3
*(x*e + d)^m*c*d*x^2*e^2 - (x*e + d)^m*c*d^3)/(m^2*e^2 + 5*m*e^2 + 6*e^2)