3.3182 \(\int (a+b x)^2 (A+B x) (d+e x)^m \, dx\)
Optimal. Leaf size=138 \[ -\frac{(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac{b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac{b^2 B (d+e x)^{m+4}}{e^4 (m+4)} \]
[Out]
-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d +
e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e
*x)^(4 + m))/(e^4*(4 + m))
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Rubi [A] time = 0.0849185, antiderivative size = 138, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{77} \[ -\frac{(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac{b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac{b^2 B (d+e x)^{m+4}}{e^4 (m+4)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
[Out]
-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d +
e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e
*x)^(4 + m))/(e^4*(4 + 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)^2 (A+B x) (d+e x)^m \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^m}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{1+m}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{2+m}}{e^3}+\frac{b^2 B (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac{(b d-a e)^2 (B d-A e) (d+e x)^{1+m}}{e^4 (1+m)}+\frac{(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{2+m}}{e^4 (2+m)}-\frac{b (3 b B d-A b e-2 a B e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac{b^2 B (d+e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}
Mathematica [A] time = 0.111719, size = 122, normalized size = 0.88 \[ \frac{(d+e x)^{m+1} \left (-\frac{b (d+e x)^2 (-2 a B e-A b e+3 b B d)}{m+3}+\frac{(d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)}{m+2}-\frac{(b d-a e)^2 (B d-A e)}{m+1}+\frac{b^2 B (d+e x)^3}{m+4}\right )}{e^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
[Out]
((d + e*x)^(1 + m)*(-(((b*d - a*e)^2*(B*d - A*e))/(1 + m)) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x
))/(2 + m) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2)/(3 + m) + (b^2*B*(d + e*x)^3)/(4 + m)))/e^4
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Maple [B] time = 0.009, size = 576, normalized size = 4.2 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( B{b}^{2}{e}^{3}{m}^{3}{x}^{3}+A{b}^{2}{e}^{3}{m}^{3}{x}^{2}+2\,Bab{e}^{3}{m}^{3}{x}^{2}+6\,B{b}^{2}{e}^{3}{m}^{2}{x}^{3}+2\,Aab{e}^{3}{m}^{3}x+7\,A{b}^{2}{e}^{3}{m}^{2}{x}^{2}+B{a}^{2}{e}^{3}{m}^{3}x+14\,Bab{e}^{3}{m}^{2}{x}^{2}-3\,B{b}^{2}d{e}^{2}{m}^{2}{x}^{2}+11\,B{b}^{2}{e}^{3}m{x}^{3}+A{a}^{2}{e}^{3}{m}^{3}+16\,Aab{e}^{3}{m}^{2}x-2\,A{b}^{2}d{e}^{2}{m}^{2}x+14\,A{b}^{2}{e}^{3}m{x}^{2}+8\,B{a}^{2}{e}^{3}{m}^{2}x-4\,Babd{e}^{2}{m}^{2}x+28\,Bab{e}^{3}m{x}^{2}-9\,B{b}^{2}d{e}^{2}m{x}^{2}+6\,{b}^{2}B{x}^{3}{e}^{3}+9\,A{a}^{2}{e}^{3}{m}^{2}-2\,Aabd{e}^{2}{m}^{2}+38\,Aab{e}^{3}mx-10\,A{b}^{2}d{e}^{2}mx+8\,A{b}^{2}{e}^{3}{x}^{2}-B{a}^{2}d{e}^{2}{m}^{2}+19\,B{a}^{2}{e}^{3}mx-20\,Babd{e}^{2}mx+16\,Bab{e}^{3}{x}^{2}+6\,B{b}^{2}{d}^{2}emx-6\,B{b}^{2}d{e}^{2}{x}^{2}+26\,A{a}^{2}{e}^{3}m-14\,Aabd{e}^{2}m+24\,Aab{e}^{3}x+2\,A{b}^{2}{d}^{2}em-8\,A{b}^{2}d{e}^{2}x-7\,B{a}^{2}d{e}^{2}m+12\,B{a}^{2}{e}^{3}x+4\,Bab{d}^{2}em-16\,Babd{e}^{2}x+6\,B{b}^{2}{d}^{2}ex+24\,A{a}^{2}{e}^{3}-24\,Aabd{e}^{2}+8\,A{b}^{2}{d}^{2}e-12\,B{a}^{2}d{e}^{2}+16\,Bab{d}^{2}e-6\,B{b}^{2}{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2*(B*x+A)*(e*x+d)^m,x)
[Out]
(e*x+d)^(1+m)*(B*b^2*e^3*m^3*x^3+A*b^2*e^3*m^3*x^2+2*B*a*b*e^3*m^3*x^2+6*B*b^2*e^3*m^2*x^3+2*A*a*b*e^3*m^3*x+7
*A*b^2*e^3*m^2*x^2+B*a^2*e^3*m^3*x+14*B*a*b*e^3*m^2*x^2-3*B*b^2*d*e^2*m^2*x^2+11*B*b^2*e^3*m*x^3+A*a^2*e^3*m^3
+16*A*a*b*e^3*m^2*x-2*A*b^2*d*e^2*m^2*x+14*A*b^2*e^3*m*x^2+8*B*a^2*e^3*m^2*x-4*B*a*b*d*e^2*m^2*x+28*B*a*b*e^3*
m*x^2-9*B*b^2*d*e^2*m*x^2+6*B*b^2*e^3*x^3+9*A*a^2*e^3*m^2-2*A*a*b*d*e^2*m^2+38*A*a*b*e^3*m*x-10*A*b^2*d*e^2*m*
x+8*A*b^2*e^3*x^2-B*a^2*d*e^2*m^2+19*B*a^2*e^3*m*x-20*B*a*b*d*e^2*m*x+16*B*a*b*e^3*x^2+6*B*b^2*d^2*e*m*x-6*B*b
^2*d*e^2*x^2+26*A*a^2*e^3*m-14*A*a*b*d*e^2*m+24*A*a*b*e^3*x+2*A*b^2*d^2*e*m-8*A*b^2*d*e^2*x-7*B*a^2*d*e^2*m+12
*B*a^2*e^3*x+4*B*a*b*d^2*e*m-16*B*a*b*d*e^2*x+6*B*b^2*d^2*e*x+24*A*a^2*e^3-24*A*a*b*d*e^2+8*A*b^2*d^2*e-12*B*a
^2*d*e^2+16*B*a*b*d^2*e-6*B*b^2*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)
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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)^2*(B*x+A)*(e*x+d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.88807, size = 1386, normalized size = 10.04 \begin{align*} \frac{{\left (A a^{2} d e^{3} m^{3} - 6 \, B b^{2} d^{4} + 24 \, A a^{2} d e^{3} + 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 12 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} +{\left (B b^{2} e^{4} m^{3} + 6 \, B b^{2} e^{4} m^{2} + 11 \, B b^{2} e^{4} m + 6 \, B b^{2} e^{4}\right )} x^{4} +{\left (8 \,{\left (2 \, B a b + A b^{2}\right )} e^{4} +{\left (B b^{2} d e^{3} +{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{3} +{\left (3 \, B b^{2} d e^{3} + 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{2} + 2 \,{\left (B b^{2} d e^{3} + 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m\right )} x^{3} +{\left (9 \, A a^{2} d e^{3} -{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4} +{\left ({\left (2 \, B a b + A b^{2}\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{3} -{\left (3 \, B b^{2} d^{2} e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 8 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{2} -{\left (3 \, B b^{2} d^{2} e^{2} - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 19 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m\right )} x^{2} +{\left (26 \, A a^{2} d e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 7 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m +{\left (24 \, A a^{2} e^{4} +{\left (A a^{2} e^{4} +{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{3} +{\left (9 \, A a^{2} e^{4} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 7 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{2} + 2 \,{\left (3 \, B b^{2} d^{3} e + 13 \, A a^{2} e^{4} - 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="fricas")
[Out]
(A*a^2*d*e^3*m^3 - 6*B*b^2*d^4 + 24*A*a^2*d*e^3 + 8*(2*B*a*b + A*b^2)*d^3*e - 12*(B*a^2 + 2*A*a*b)*d^2*e^2 + (
B*b^2*e^4*m^3 + 6*B*b^2*e^4*m^2 + 11*B*b^2*e^4*m + 6*B*b^2*e^4)*x^4 + (8*(2*B*a*b + A*b^2)*e^4 + (B*b^2*d*e^3
+ (2*B*a*b + A*b^2)*e^4)*m^3 + (3*B*b^2*d*e^3 + 7*(2*B*a*b + A*b^2)*e^4)*m^2 + 2*(B*b^2*d*e^3 + 7*(2*B*a*b + A
*b^2)*e^4)*m)*x^3 + (9*A*a^2*d*e^3 - (B*a^2 + 2*A*a*b)*d^2*e^2)*m^2 + (12*(B*a^2 + 2*A*a*b)*e^4 + ((2*B*a*b +
A*b^2)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)*m^3 - (3*B*b^2*d^2*e^2 - 5*(2*B*a*b + A*b^2)*d*e^3 - 8*(B*a^2 + 2*A*a*b)
*e^4)*m^2 - (3*B*b^2*d^2*e^2 - 4*(2*B*a*b + A*b^2)*d*e^3 - 19*(B*a^2 + 2*A*a*b)*e^4)*m)*x^2 + (26*A*a^2*d*e^3
+ 2*(2*B*a*b + A*b^2)*d^3*e - 7*(B*a^2 + 2*A*a*b)*d^2*e^2)*m + (24*A*a^2*e^4 + (A*a^2*e^4 + (B*a^2 + 2*A*a*b)*
d*e^3)*m^3 + (9*A*a^2*e^4 - 2*(2*B*a*b + A*b^2)*d^2*e^2 + 7*(B*a^2 + 2*A*a*b)*d*e^3)*m^2 + 2*(3*B*b^2*d^3*e +
13*A*a^2*e^4 - 4*(2*B*a*b + A*b^2)*d^2*e^2 + 6*(B*a^2 + 2*A*a*b)*d*e^3)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^
3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)
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Sympy [A] time = 6.51076, size = 6094, normalized size = 44.16 \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)**2*(B*x+A)*(e*x+d)**m,x)
[Out]
Piecewise((d**m*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4), Eq(e
, 0)), (-2*A*a**2*d**2*e**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*A*a*b*d*
e**4*x**2/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 2*A*a*b*e**5*x**3/(6*d**5*e*
*4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 2*A*b**2*d*e**4*x**3/(6*d**5*e**4 + 18*d**4*e**5
*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 3*B*a**2*d*e**4*x**2/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6
*x**2 + 6*d**2*e**7*x**3) + B*a**2*e**5*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x
**3) + 4*B*a*b*d*e**4*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*B*b**2*d*
*5*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 2*B*b**2*d**5/(6*d**5*
e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 18*B*b**2*d**4*e*x*log(d/e + x)/(6*d**5*e**4 +
18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 18*B*b**2*d**3*e**2*x**2*log(d/e + x)/(6*d**5*e**4 +
18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) - 9*B*b**2*d**3*e**2*x**2/(6*d**5*e**4 + 18*d**4*e**5*
x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*B*b**2*d**2*e**3*x**3*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x
+ 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) - 9*B*b**2*d**2*e**3*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**
6*x**2 + 6*d**2*e**7*x**3), Eq(m, -4)), (-A*a**2*d*e**3/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*A*a*
b*e**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*A*b**2*d**3*e*log(d/e + x)/(2*d**3*e**4 + 4*d**2
*e**5*x + 2*d*e**6*x**2) + A*b**2*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*A*b**2*d**2*e**2*x*
log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*A*b**2*d*e**3*x**2*log(d/e + x)/(2*d**3*e**4 +
4*d**2*e**5*x + 2*d*e**6*x**2) - 2*A*b**2*d*e**3*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + B*a**2*e
**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*B*a*b*d**3*e*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**
5*x + 2*d*e**6*x**2) + 2*B*a*b*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 8*B*a*b*d**2*e**2*x*log(
d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*B*a*b*d*e**3*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**
2*e**5*x + 2*d*e**6*x**2) - 4*B*a*b*d*e**3*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*B*b**2*d**4*
log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 3*B*b**2*d**4/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*
e**6*x**2) - 12*B*b**2*d**3*e*x*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*B*b**2*d**2*e**
2*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*B*b**2*d**2*e**2*x**2/(2*d**3*e**4 + 4*d
**2*e**5*x + 2*d*e**6*x**2) + 2*B*b**2*d*e**3*x**3/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2), Eq(m, -3)),
(-2*A*a**2*e**3/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*d*e**2/(2*
d*e**4 + 2*e**5*x) + 4*A*a*b*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*b**2*d**2*e*log(d/e + x)/(2*d*e**
4 + 2*e**5*x) - 4*A*b**2*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*b**2*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) +
2*A*b**2*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*d*e*
*2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 8*B*a*b*d**2*e*log(d/e + x)/(2
*d*e**4 + 2*e**5*x) - 8*B*a*b*d**2*e/(2*d*e**4 + 2*e**5*x) - 8*B*a*b*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*
x) + 4*B*a*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**
3/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*B*b**2*d*e**2*x**2/(2*d*e**
4 + 2*e**5*x) + B*b**2*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (A*a**2*log(d/e + x)/e - 2*A*a*b*d*log(d/e
+ x)/e**2 + 2*A*a*b*x/e + A*b**2*d**2*log(d/e + x)/e**3 - A*b**2*d*x/e**2 + A*b**2*x**2/(2*e) - B*a**2*d*log(
d/e + x)/e**2 + B*a**2*x/e + 2*B*a*b*d**2*log(d/e + x)/e**3 - 2*B*a*b*d*x/e**2 + B*a*b*x**2/e - B*b**2*d**3*lo
g(d/e + x)/e**4 + B*b**2*d**2*x/e**3 - B*b**2*d*x**2/(2*e**2) + B*b**2*x**3/(3*e), Eq(m, -1)), (A*a**2*d*e**3*
m**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a**2*d*e**3*m**2*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a**2*d*e**3*m*(d + e*x)**m/(e**
4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a**2*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e*
*4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*a**2*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a**2*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2
+ 50*e**4*m + 24*e**4) + 26*A*a**2*e**4*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) + 24*A*a**2*e**4*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) -
2*A*a*b*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 14*A*a*b
*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 24*A*a*b*d**2*e**2
*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*A*a*b*d*e**3*m**3*x*(d + e*x
)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*A*a*b*d*e**3*m**2*x*(d + e*x)**m/(e*
*4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*b*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*A*a*b*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m
**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 16*A*a*b*e**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) + 38*A*a*b*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 24*A*a*b*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*
m + 24*e**4) + 2*A*b**2*d**3*e*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
+ 8*A*b**2*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*A*b**2*d**2
*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*A*b**2*d**2*e**2
*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*b**2*d*e**3*m**3*x**2*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 5*A*b**2*d*e**3*m**2*x**2*(d + e*
x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 4*A*b**2*d*e**3*m*x**2*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*b**2*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4
+ 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*A*b**2*e**4*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e*
*4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*A*b**2*e**4*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*b**2*e**4*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) - B*a**2*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e*
*4*m + 24*e**4) - 7*B*a**2*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*
e**4) - 12*B*a**2*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*a
**2*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*B*a**2*d*e*
*3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a**2*d*e**3*m*x*
(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*a**2*e**4*m**3*x**2*(d + e*x)
**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*B*a**2*e**4*m**2*x**2*(d + e*x)**m/(e*
*4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 19*B*a**2*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a**2*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m*
*3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 4*B*a*b*d**3*e*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m
**2 + 50*e**4*m + 24*e**4) + 16*B*a*b*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) - 4*B*a*b*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*
e**4) - 16*B*a*b*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
2*B*a*b*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 10*B*a
*b*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*B*a*b*d*e
**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*B*a*b*e**4*m**3*x*
*3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*B*a*b*e**4*m**2*x**3*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 28*B*a*b*e**4*m*x**3*(d + e*x)**m/(
e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 16*B*a*b*e**4*x**3*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*B*b**2*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*b**2*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2
+ 50*e**4*m + 24*e**4) - 3*B*b**2*d**2*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 +
50*e**4*m + 24*e**4) - 3*B*b**2*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e*
*4*m + 24*e**4) + B*b**2*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 3*B*b**2*d*e**3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e*
*4) + 2*B*b**2*d*e**3*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*
b**2*e**4*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*b**2*e*
*4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 11*B*b**2*e**4*m*x
**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*b**2*e**4*x**4*(d + e*x
)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4), True))
________________________________________________________________________________________
Giac [B] time = 2.39395, size = 1710, normalized size = 12.39 \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)^2*(B*x+A)*(e*x+d)^m,x, algorithm="giac")
[Out]
((x*e + d)^m*B*b^2*m^3*x^4*e^4 + (x*e + d)^m*B*b^2*d*m^3*x^3*e^3 + 2*(x*e + d)^m*B*a*b*m^3*x^3*e^4 + (x*e + d)
^m*A*b^2*m^3*x^3*e^4 + 6*(x*e + d)^m*B*b^2*m^2*x^4*e^4 + 2*(x*e + d)^m*B*a*b*d*m^3*x^2*e^3 + (x*e + d)^m*A*b^2
*d*m^3*x^2*e^3 + 3*(x*e + d)^m*B*b^2*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*b^2*d^2*m^2*x^2*e^2 + (x*e + d)^m*B*a^2*m
^3*x^2*e^4 + 2*(x*e + d)^m*A*a*b*m^3*x^2*e^4 + 14*(x*e + d)^m*B*a*b*m^2*x^3*e^4 + 7*(x*e + d)^m*A*b^2*m^2*x^3*
e^4 + 11*(x*e + d)^m*B*b^2*m*x^4*e^4 + (x*e + d)^m*B*a^2*d*m^3*x*e^3 + 2*(x*e + d)^m*A*a*b*d*m^3*x*e^3 + 10*(x
*e + d)^m*B*a*b*d*m^2*x^2*e^3 + 5*(x*e + d)^m*A*b^2*d*m^2*x^2*e^3 + 2*(x*e + d)^m*B*b^2*d*m*x^3*e^3 - 4*(x*e +
d)^m*B*a*b*d^2*m^2*x*e^2 - 2*(x*e + d)^m*A*b^2*d^2*m^2*x*e^2 - 3*(x*e + d)^m*B*b^2*d^2*m*x^2*e^2 + 6*(x*e + d
)^m*B*b^2*d^3*m*x*e + (x*e + d)^m*A*a^2*m^3*x*e^4 + 8*(x*e + d)^m*B*a^2*m^2*x^2*e^4 + 16*(x*e + d)^m*A*a*b*m^2
*x^2*e^4 + 28*(x*e + d)^m*B*a*b*m*x^3*e^4 + 14*(x*e + d)^m*A*b^2*m*x^3*e^4 + 6*(x*e + d)^m*B*b^2*x^4*e^4 + (x*
e + d)^m*A*a^2*d*m^3*e^3 + 7*(x*e + d)^m*B*a^2*d*m^2*x*e^3 + 14*(x*e + d)^m*A*a*b*d*m^2*x*e^3 + 8*(x*e + d)^m*
B*a*b*d*m*x^2*e^3 + 4*(x*e + d)^m*A*b^2*d*m*x^2*e^3 - (x*e + d)^m*B*a^2*d^2*m^2*e^2 - 2*(x*e + d)^m*A*a*b*d^2*
m^2*e^2 - 16*(x*e + d)^m*B*a*b*d^2*m*x*e^2 - 8*(x*e + d)^m*A*b^2*d^2*m*x*e^2 + 4*(x*e + d)^m*B*a*b*d^3*m*e + 2
*(x*e + d)^m*A*b^2*d^3*m*e - 6*(x*e + d)^m*B*b^2*d^4 + 9*(x*e + d)^m*A*a^2*m^2*x*e^4 + 19*(x*e + d)^m*B*a^2*m*
x^2*e^4 + 38*(x*e + d)^m*A*a*b*m*x^2*e^4 + 16*(x*e + d)^m*B*a*b*x^3*e^4 + 8*(x*e + d)^m*A*b^2*x^3*e^4 + 9*(x*e
+ d)^m*A*a^2*d*m^2*e^3 + 12*(x*e + d)^m*B*a^2*d*m*x*e^3 + 24*(x*e + d)^m*A*a*b*d*m*x*e^3 - 7*(x*e + d)^m*B*a^
2*d^2*m*e^2 - 14*(x*e + d)^m*A*a*b*d^2*m*e^2 + 16*(x*e + d)^m*B*a*b*d^3*e + 8*(x*e + d)^m*A*b^2*d^3*e + 26*(x*
e + d)^m*A*a^2*m*x*e^4 + 12*(x*e + d)^m*B*a^2*x^2*e^4 + 24*(x*e + d)^m*A*a*b*x^2*e^4 + 26*(x*e + d)^m*A*a^2*d*
m*e^3 - 12*(x*e + d)^m*B*a^2*d^2*e^2 - 24*(x*e + d)^m*A*a*b*d^2*e^2 + 24*(x*e + d)^m*A*a^2*x*e^4 + 24*(x*e + d
)^m*A*a^2*d*e^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)