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

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

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Rubi [A]  time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.10, size = 79, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (A+B x) (d+e x)^m \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.10, size = 256, normalized size = 2.84 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.00, size = 497, normalized size = 5.52 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.01, size = 189, normalized size = 2.10 \begin {gather*} \frac {\left (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 \,x^{2} e^{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}\right ) \left (e x +d \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \end {gather*}

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)^(m+1)*(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 [A]  time = 0.50, size = 179, normalized size = 1.99 \begin {gather*} \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \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} B b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \end {gather*}

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]

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

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mupad [B]  time = 2.56, size = 259, normalized size = 2.88 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {x\,\left (6\,A\,a\,e^3+5\,A\,a\,e^3\,m+A\,a\,e^3\,m^2+3\,A\,b\,d\,e^2\,m+3\,B\,a\,d\,e^2\,m-2\,B\,b\,d^2\,e\,m+A\,b\,d\,e^2\,m^2+B\,a\,d\,e^2\,m^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,\left (6\,A\,a\,e^2+2\,B\,b\,d^2+5\,A\,a\,e^2\,m+A\,a\,e^2\,m^2-3\,A\,b\,d\,e-3\,B\,a\,d\,e-A\,b\,d\,e\,m-B\,a\,d\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {B\,b\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x^2\,\left (m+1\right )\,\left (3\,A\,b\,e+3\,B\,a\,e+A\,b\,e\,m+B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.79, size = 1982, normalized size = 22.02

result too large to display

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*e**2/(2*d**2*e**3 + 4*d*e**4*
x + 2*e**5*x**2) - A*b*d*e/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) - 2*A*b*e**2*x/(2*d**2*e**3 + 4*d*e**4*x +
2*e**5*x**2) - B*a*d*e/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) - 2*B*a*e**2*x/(2*d**2*e**3 + 4*d*e**4*x + 2*
e**5*x**2) + 2*B*b*d**2*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 3*B*b*d**2/(2*d**2*e**3 + 4*d*
e**4*x + 2*e**5*x**2) + 4*B*b*d*e*x*log(d/e + x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 4*B*b*d*e*x/(2*d**
2*e**3 + 4*d*e**4*x + 2*e**5*x**2) + 2*B*b*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, -3)), (-A*a*e**2/(d*e**3 + e**4*x) + A*b*d*e*log(d/e + x)/(d*e**3 + e**4*x) + A*b*d*e/(d*e**3 + e**4*x) +
A*b*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) + B*a*d*e*log(d/e + x)/(d*e**3 + e**4*x) + B*a*d*e/(d*e**3 + e**4*x)
+ B*a*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*B*b*d**2*log(d/e + x)/(d*e**3 + e**4*x) - 2*B*b*d**2/(d*e**3
+ e**4*x) - 2*B*b*d*e*x*log(d/e + x)/(d*e**3 + e**4*x) + B*b*e**2*x**2/(d*e**3 + 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 + 1
1*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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