3.3184 \(\int (A+B x) (d+e x)^m \, dx\)

Optimal. Leaf size=47 \[ \frac{B (d+e x)^{m+2}}{e^2 (m+2)}-\frac{(B d-A e) (d+e x)^{m+1}}{e^2 (m+1)} \]

[Out]

-(((B*d - A*e)*(d + e*x)^(1 + m))/(e^2*(1 + m))) + (B*(d + e*x)^(2 + m))/(e^2*(2 + m))

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Rubi [A]  time = 0.0216293, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{B (d+e x)^{m+2}}{e^2 (m+2)}-\frac{(B d-A e) (d+e x)^{m+1}}{e^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^m,x]

[Out]

-(((B*d - A*e)*(d + e*x)^(1 + m))/(e^2*(1 + m))) + (B*(d + e*x)^(2 + m))/(e^2*(2 + m))

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

Mathematica [A]  time = 0.0273005, size = 41, normalized size = 0.87 \[ \frac{(d+e x)^{m+1} (A e (m+2)-B d+B e (m+1) x)}{e^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^m,x]

[Out]

((d + e*x)^(1 + m)*(-(B*d) + A*e*(2 + m) + B*e*(1 + m)*x))/(e^2*(1 + m)*(2 + m))

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Maple [A]  time = 0.004, size = 46, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bemx+Aem+Bex+2\,Ae-Bd \right ) }{{e}^{2} \left ({m}^{2}+3\,m+2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^m,x)

[Out]

(e*x+d)^(1+m)*(B*e*m*x+A*e*m+B*e*x+2*A*e-B*d)/e^2/(m^2+3*m+2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36135, size = 171, normalized size = 3.64 \begin{align*} \frac{{\left (A d e m - B d^{2} + 2 \, A d e +{\left (B e^{2} m + B e^{2}\right )} x^{2} +{\left (2 \, A e^{2} +{\left (B d e + A e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 3 \, e^{2} m + 2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.732246, size = 401, normalized size = 8.53 \begin{align*} \begin{cases} d^{m} \left (A x + \frac{B x^{2}}{2}\right ) & \text{for}\: e = 0 \\\frac{A e^{2} x}{d^{2} e^{2} + d e^{3} x} + \frac{B d^{2} \log{\left (\frac{d}{e} + x \right )}}{d^{2} e^{2} + d e^{3} x} + \frac{B d e x \log{\left (\frac{d}{e} + x \right )}}{d^{2} e^{2} + d e^{3} x} - \frac{B d e x}{d^{2} e^{2} + d e^{3} x} & \text{for}\: m = -2 \\\frac{A \log{\left (\frac{d}{e} + x \right )}}{e} - \frac{B d \log{\left (\frac{d}{e} + x \right )}}{e^{2}} + \frac{B x}{e} & \text{for}\: m = -1 \\\frac{A d e m \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{2 A d e \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{A e^{2} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{2 A e^{2} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} - \frac{B d^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B d e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**m,x)

[Out]

Piecewise((d**m*(A*x + B*x**2/2), Eq(e, 0)), (A*e**2*x/(d**2*e**2 + d*e**3*x) + B*d**2*log(d/e + x)/(d**2*e**2
+ d*e**3*x) + B*d*e*x*log(d/e + x)/(d**2*e**2 + d*e**3*x) - B*d*e*x/(d**2*e**2 + d*e**3*x), Eq(m, -2)), (A*lo
g(d/e + x)/e - B*d*log(d/e + x)/e**2 + B*x/e, Eq(m, -1)), (A*d*e*m*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2
) + 2*A*d*e*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2) + A*e**2*m*x*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e
**2) + 2*A*e**2*x*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2) - B*d**2*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2
*e**2) + B*d*e*m*x*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2) + B*e**2*m*x**2*(d + e*x)**m/(e**2*m**2 + 3*e*
*2*m + 2*e**2) + B*e**2*x**2*(d + e*x)**m/(e**2*m**2 + 3*e**2*m + 2*e**2), True))

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Giac [B]  time = 2.15477, size = 184, normalized size = 3.91 \begin{align*} \frac{{\left (x e + d\right )}^{m} B m x^{2} e^{2} +{\left (x e + d\right )}^{m} B d m x e +{\left (x e + d\right )}^{m} A m x e^{2} +{\left (x e + d\right )}^{m} B x^{2} e^{2} +{\left (x e + d\right )}^{m} A d m e -{\left (x e + d\right )}^{m} B d^{2} + 2 \,{\left (x e + d\right )}^{m} A x e^{2} + 2 \,{\left (x e + d\right )}^{m} A d e}{m^{2} e^{2} + 3 \, m e^{2} + 2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m,x, algorithm="giac")

[Out]

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