3.3156 \(\int (A+B x) (d+e x)^m \, dx\)

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 = 0.059418, 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 \[ \frac{B (d+e x)^{m+2}}{e^2 (m+2)}-\frac{(B d-A e) (d+e x)^{m+1}}{e^2 (m+1)} \]

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))

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Rubi in Sympy [A]  time = 10.2659, size = 37, normalized size = 0.79 \[ \frac{B \left (d + e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right )}{e^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m,x)

[Out]

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

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Mathematica [A]  time = 0.0362918, 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. \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bemx+Aem+Bex+2\,Ae-Bd \right ) }{{e}^{2} \left ({m}^{2}+3\,m+2 \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.307022, size = 112, normalized size = 2.38 \[ \frac{{\left (A d e m - B d^{2} + 2 \, A d e +{\left (B e^{2} m + B e^{2}\right )} x^{2} +{\left (2 \, A e^{2} +{\left (B d e + A e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{2} m^{2} + 3 \, e^{2} m + 2 \, e^{2}} \]

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 = 1.20854, size = 377, normalized size = 8.02 \[ \begin{cases} d^{m} \left (A x + \frac{B x^{2}}{2}\right ) & \text{for}\: e = 0 \\- \frac{A e}{d e^{2} + e^{3} x} + \frac{B d \log{\left (\frac{d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac{B d}{d e^{2} + e^{3} x} + \frac{B e x \log{\left (\frac{d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text{for}\: m = -2 \\\frac{A \log{\left (\frac{d}{e} + x \right )}}{e} - \frac{B d \log{\left (\frac{d}{e} + x \right )}}{e^{2}} + \frac{B x}{e} & \text{for}\: m = -1 \\\frac{A d e m \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{2 A d e \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{A e^{2} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{2 A e^{2} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} - \frac{B d^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B d e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac{B e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} & \text{otherwise} \end{cases} \]

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/(d*e**2 + e**3*x) + B*d*log(d
/e + x)/(d*e**2 + e**3*x) + B*d/(d*e**2 + e**3*x) + B*e*x*log(d/e + x)/(d*e**2 +
e**3*x), Eq(m, -2)), (A*log(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/XCAS [A]  time = 0.229354, size = 205, normalized size = 4.36 \[ \frac{B m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + B d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + A m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + B x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + A d m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} - B d^{2} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 2 \, A x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 2 \, A d e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )}}{m^{2} e^{2} + 3 \, m e^{2} + 2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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