3.3183 \(\int (a+b x) (A+B x) (d+e x)^m \, dx\)

Optimal. Leaf size=90 \[ \frac{(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac{b B (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.0477808, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac{b B (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x)^m,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0960917, size = 79, normalized size = 0.88 \[ \frac{(d+e x)^{m+1} \left (-\frac{(d+e x) (-a B e-A b e+2 b B d)}{m+2}+\frac{(b d-a e) (B d-A e)}{m+1}+\frac{b B (d+e x)^2}{m+3}\right )}{e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x)^m,x]

[Out]

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

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Maple [B]  time = 0.003, size = 189, normalized size = 2.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bb{e}^{2}{m}^{2}{x}^{2}+Ab{e}^{2}{m}^{2}x+Ba{e}^{2}{m}^{2}x+3\,Bb{e}^{2}m{x}^{2}+Aa{e}^{2}{m}^{2}+4\,Ab{e}^{2}mx+4\,Ba{e}^{2}mx-2\,Bbdemx+2\,bB{x}^{2}{e}^{2}+5\,Aa{e}^{2}m-Abdem+3\,Ab{e}^{2}x-Badem+3\,Ba{e}^{2}x-2\,Bbdex+6\,Aa{e}^{2}-3\,Abde-3\,Bade+2\,Bb{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1+m)*(B*b*e^2*m^2*x^2+A*b*e^2*m^2*x+B*a*e^2*m^2*x+3*B*b*e^2*m*x^2+A*a*e^2*m^2+4*A*b*e^2*m*x+4*B*a*e^2
*m*x-2*B*b*d*e*m*x+2*B*b*e^2*x^2+5*A*a*e^2*m-A*b*d*e*m+3*A*b*e^2*x-B*a*d*e*m+3*B*a*e^2*x-2*B*b*d*e*x+6*A*a*e^2
-3*A*b*d*e-3*B*a*d*e+2*B*b*d^2)/e^3/(m^3+6*m^2+11*m+6)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.53898, size = 545, normalized size = 6.06 \begin{align*} \frac{{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \,{\left (B a + A b\right )} d^{2} e +{\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} +{\left (3 \,{\left (B a + A b\right )} e^{3} +{\left (B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} m^{2} +{\left (B b d e^{2} + 4 \,{\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} +{\left (5 \, A a d e^{2} -{\left (B a + A b\right )} d^{2} e\right )} m +{\left (6 \, A a e^{3} +{\left (A a e^{3} +{\left (B a + A b\right )} d e^{2}\right )} m^{2} -{\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \,{\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*a*d*e^2*m^2 + 2*B*b*d^3 + 6*A*a*d*e^2 - 3*(B*a + A*b)*d^2*e + (B*b*e^3*m^2 + 3*B*b*e^3*m + 2*B*b*e^3)*x^3 +
(3*(B*a + A*b)*e^3 + (B*b*d*e^2 + (B*a + A*b)*e^3)*m^2 + (B*b*d*e^2 + 4*(B*a + A*b)*e^3)*m)*x^2 + (5*A*a*d*e^
2 - (B*a + A*b)*d^2*e)*m + (6*A*a*e^3 + (A*a*e^3 + (B*a + A*b)*d*e^2)*m^2 - (2*B*b*d^2*e - 5*A*a*e^3 - 3*(B*a
+ A*b)*d*e^2)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^3)

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Sympy [A]  time = 2.57726, size = 2011, normalized size = 22.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/3), Eq(e, 0)), (-A*a*d*e**2/(2*d**3*e**3 + 4*d**2*
e**4*x + 2*d*e**5*x**2) + A*b*e**3*x**2/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + B*a*e**3*x**2/(2*d**3*
e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 2*B*b*d**3*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2)
+ B*b*d**3/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 4*B*b*d**2*e*x*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e
**4*x + 2*d*e**5*x**2) + 2*B*b*d*e**2*x**2*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) - 2*B*b*
d*e**2*x**2/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2), Eq(m, -3)), (A*a*e**3*x/(d**2*e**3 + d*e**4*x) + A*
b*d**2*e*log(d/e + x)/(d**2*e**3 + d*e**4*x) + A*b*d*e**2*x*log(d/e + x)/(d**2*e**3 + d*e**4*x) - A*b*d*e**2*x
/(d**2*e**3 + d*e**4*x) + B*a*d**2*e*log(d/e + x)/(d**2*e**3 + d*e**4*x) + B*a*d*e**2*x*log(d/e + x)/(d**2*e**
3 + d*e**4*x) - B*a*d*e**2*x/(d**2*e**3 + d*e**4*x) - 2*B*b*d**3*log(d/e + x)/(d**2*e**3 + d*e**4*x) - 2*B*b*d
**2*e*x*log(d/e + x)/(d**2*e**3 + d*e**4*x) + 2*B*b*d**2*e*x/(d**2*e**3 + d*e**4*x) + B*b*d*e**2*x**2/(d**2*e*
*3 + d*e**4*x), Eq(m, -2)), (A*a*log(d/e + x)/e - A*b*d*log(d/e + x)/e**2 + A*b*x/e - B*a*d*log(d/e + x)/e**2
+ B*a*x/e + B*b*d**2*log(d/e + x)/e**3 - B*b*d*x/e**2 + B*b*x**2/(2*e), Eq(m, -1)), (A*a*d*e**2*m**2*(d + e*x)
**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*A*a*d*e**2*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11
*e**3*m + 6*e**3) + 6*A*a*d*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + A*a*e**3*m**2*x
*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*A*a*e**3*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3
*m**2 + 11*e**3*m + 6*e**3) + 6*A*a*e**3*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - A*b*d
**2*e*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3*A*b*d**2*e*(d + e*x)**m/(e**3*m**3 + 6
*e**3*m**2 + 11*e**3*m + 6*e**3) + A*b*d*e**2*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**
3) + 3*A*b*d*e**2*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + A*b*e**3*m**2*x**2*(d + e*
x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 4*A*b*e**3*m*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2
+ 11*e**3*m + 6*e**3) + 3*A*b*e**3*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - B*a*d**
2*e*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3*B*a*d**2*e*(d + e*x)**m/(e**3*m**3 + 6*e
**3*m**2 + 11*e**3*m + 6*e**3) + B*a*d*e**2*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3)
+ 3*B*a*d*e**2*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*a*e**3*m**2*x**2*(d + e*x)
**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 4*B*a*e**3*m*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 +
11*e**3*m + 6*e**3) + 3*B*a*e**3*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*B*b*d**
3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*B*b*d**2*e*m*x*(d + e*x)**m/(e**3*m**3 + 6*e
**3*m**2 + 11*e**3*m + 6*e**3) + B*b*d*e**2*m**2*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e*
*3) + B*b*d*e**2*m*x**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*b*e**3*m**2*x**3*(d +
e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*B*b*e**3*m*x**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m*
*2 + 11*e**3*m + 6*e**3) + 2*B*b*e**3*x**3*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3), True))

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Giac [B]  time = 1.85865, size = 671, normalized size = 7.46 \begin{align*} \frac{{\left (x e + d\right )}^{m} B b m^{2} x^{3} e^{3} +{\left (x e + d\right )}^{m} B b d m^{2} x^{2} e^{2} +{\left (x e + d\right )}^{m} B a m^{2} x^{2} e^{3} +{\left (x e + d\right )}^{m} A b m^{2} x^{2} e^{3} + 3 \,{\left (x e + d\right )}^{m} B b m x^{3} e^{3} +{\left (x e + d\right )}^{m} B a d m^{2} x e^{2} +{\left (x e + d\right )}^{m} A b d m^{2} x e^{2} +{\left (x e + d\right )}^{m} B b d m x^{2} e^{2} - 2 \,{\left (x e + d\right )}^{m} B b d^{2} m x e +{\left (x e + d\right )}^{m} A a m^{2} x e^{3} + 4 \,{\left (x e + d\right )}^{m} B a m x^{2} e^{3} + 4 \,{\left (x e + d\right )}^{m} A b m x^{2} e^{3} + 2 \,{\left (x e + d\right )}^{m} B b x^{3} e^{3} +{\left (x e + d\right )}^{m} A a d m^{2} e^{2} + 3 \,{\left (x e + d\right )}^{m} B a d m x e^{2} + 3 \,{\left (x e + d\right )}^{m} A b d m x e^{2} -{\left (x e + d\right )}^{m} B a d^{2} m e -{\left (x e + d\right )}^{m} A b d^{2} m e + 2 \,{\left (x e + d\right )}^{m} B b d^{3} + 5 \,{\left (x e + d\right )}^{m} A a m x e^{3} + 3 \,{\left (x e + d\right )}^{m} B a x^{2} e^{3} + 3 \,{\left (x e + d\right )}^{m} A b x^{2} e^{3} + 5 \,{\left (x e + d\right )}^{m} A a d m e^{2} - 3 \,{\left (x e + d\right )}^{m} B a d^{2} e - 3 \,{\left (x e + d\right )}^{m} A b d^{2} e + 6 \,{\left (x e + d\right )}^{m} A a x e^{3} + 6 \,{\left (x e + d\right )}^{m} A a d e^{2}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*B*b*m^2*x^3*e^3 + (x*e + d)^m*B*b*d*m^2*x^2*e^2 + (x*e + d)^m*B*a*m^2*x^2*e^3 + (x*e + d)^m*A*b*m
^2*x^2*e^3 + 3*(x*e + d)^m*B*b*m*x^3*e^3 + (x*e + d)^m*B*a*d*m^2*x*e^2 + (x*e + d)^m*A*b*d*m^2*x*e^2 + (x*e +
d)^m*B*b*d*m*x^2*e^2 - 2*(x*e + d)^m*B*b*d^2*m*x*e + (x*e + d)^m*A*a*m^2*x*e^3 + 4*(x*e + d)^m*B*a*m*x^2*e^3 +
4*(x*e + d)^m*A*b*m*x^2*e^3 + 2*(x*e + d)^m*B*b*x^3*e^3 + (x*e + d)^m*A*a*d*m^2*e^2 + 3*(x*e + d)^m*B*a*d*m*x
*e^2 + 3*(x*e + d)^m*A*b*d*m*x*e^2 - (x*e + d)^m*B*a*d^2*m*e - (x*e + d)^m*A*b*d^2*m*e + 2*(x*e + d)^m*B*b*d^3
+ 5*(x*e + d)^m*A*a*m*x*e^3 + 3*(x*e + d)^m*B*a*x^2*e^3 + 3*(x*e + d)^m*A*b*x^2*e^3 + 5*(x*e + d)^m*A*a*d*m*e
^2 - 3*(x*e + d)^m*B*a*d^2*e - 3*(x*e + d)^m*A*b*d^2*e + 6*(x*e + d)^m*A*a*x*e^3 + 6*(x*e + d)^m*A*a*d*e^2)/(m
^3*e^3 + 6*m^2*e^3 + 11*m*e^3 + 6*e^3)