∫ (A + Bx)(d + ex)m(a + bx + cx2)dx

3.2642 \(\int (A+B x) (d+e x)^m (a+b x+c x^2) \, dx\)

Optimal. Leaf size=153 \[ -\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+2} \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{e^4 (m+2)}-\frac{(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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

)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4 + m))

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Rubi [A] time = 0.108617, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ -\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}+\frac{(d+e x)^{m+2} \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{e^4 (m+2)}-\frac{(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(A + B*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

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

*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - ((3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(3 + m))/(e^4*(3 + m

)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4 + m))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In

t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N

eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^3}+\frac{\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (d+e x)^{1+m}}{e^3}+\frac{(-3 B c d+b B e+A c e) (d+e x)^{2+m}}{e^3}+\frac{B c (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac{\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac{(3 B c d-b B e-A c e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac{B c (d+e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}

Mathematica [A] time = 0.381662, size = 181, normalized size = 1.18 \[ \frac{(d+e x)^{m+1} \left (\frac{(d+e x) \left (B \left (c e (2 a e (m+3)+b d (m-2))-b^2 e^2 (m+2)+6 c^2 d^2\right )-A c e (m+4) (2 c d-b e)\right )}{e^2 (m+2)}-\frac{\left (e (a e-b d)+c d^2\right ) (-2 A c e (m+4)+b B e (m+1)+6 B c d)}{e^2 (m+1)}+(a+x (b+c x)) (A c e (m+4)+B (b e-3 c d)+B c e (m+3) x)\right )}{c e^2 (m+3) (m+4)} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(A + B*x)*(d + e*x)^m*(a + b*x + c*x^2),x]

[Out]

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

+ ((-(A*c*e*(2*c*d - b*e)*(4 + m)) + B*(6*c^2*d^2 - b^2*e^2*(2 + m) + c*e*(b*d*(-2 + m) + 2*a*e*(3 + m))))*(d

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

)*(4 + m))

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Maple [B] time = 0.006, size = 498, normalized size = 3.3 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bc{e}^{3}{m}^{3}{x}^{3}+Ac{e}^{3}{m}^{3}{x}^{2}+Bb{e}^{3}{m}^{3}{x}^{2}+6\,Bc{e}^{3}{m}^{2}{x}^{3}+Ab{e}^{3}{m}^{3}x+7\,Ac{e}^{3}{m}^{2}{x}^{2}+Ba{e}^{3}{m}^{3}x+7\,Bb{e}^{3}{m}^{2}{x}^{2}-3\,Bcd{e}^{2}{m}^{2}{x}^{2}+11\,Bc{e}^{3}m{x}^{3}+Aa{e}^{3}{m}^{3}+8\,Ab{e}^{3}{m}^{2}x-2\,Acd{e}^{2}{m}^{2}x+14\,Ac{e}^{3}m{x}^{2}+8\,Ba{e}^{3}{m}^{2}x-2\,Bbd{e}^{2}{m}^{2}x+14\,Bb{e}^{3}m{x}^{2}-9\,Bcd{e}^{2}m{x}^{2}+6\,Bc{x}^{3}{e}^{3}+9\,Aa{e}^{3}{m}^{2}-Abd{e}^{2}{m}^{2}+19\,Ab{e}^{3}mx-10\,Acd{e}^{2}mx+8\,Ac{e}^{3}{x}^{2}-Bad{e}^{2}{m}^{2}+19\,Ba{e}^{3}mx-10\,Bbd{e}^{2}mx+8\,Bb{e}^{3}{x}^{2}+6\,Bc{d}^{2}emx-6\,Bcd{e}^{2}{x}^{2}+26\,Aa{e}^{3}m-7\,Abd{e}^{2}m+12\,Ab{e}^{3}x+2\,Ac{d}^{2}em-8\,Acd{e}^{2}x-7\,Bad{e}^{2}m+12\,Ba{e}^{3}x+2\,Bb{d}^{2}em-8\,Bbd{e}^{2}x+6\,Bc{d}^{2}ex+24\,aA{e}^{3}-12\,Abd{e}^{2}+8\,Ac{d}^{2}e-12\,aBd{e}^{2}+8\,Bb{d}^{2}e-6\,Bc{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)*(e*x+d)^m*(c*x^2+b*x+a),x)

[Out]

(e*x+d)^(1+m)*(B*c*e^3*m^3*x^3+A*c*e^3*m^3*x^2+B*b*e^3*m^3*x^2+6*B*c*e^3*m^2*x^3+A*b*e^3*m^3*x+7*A*c*e^3*m^2*x

^2+B*a*e^3*m^3*x+7*B*b*e^3*m^2*x^2-3*B*c*d*e^2*m^2*x^2+11*B*c*e^3*m*x^3+A*a*e^3*m^3+8*A*b*e^3*m^2*x-2*A*c*d*e^

2*m^2*x+14*A*c*e^3*m*x^2+8*B*a*e^3*m^2*x-2*B*b*d*e^2*m^2*x+14*B*b*e^3*m*x^2-9*B*c*d*e^2*m*x^2+6*B*c*e^3*x^3+9*

A*a*e^3*m^2-A*b*d*e^2*m^2+19*A*b*e^3*m*x-10*A*c*d*e^2*m*x+8*A*c*e^3*x^2-B*a*d*e^2*m^2+19*B*a*e^3*m*x-10*B*b*d*

e^2*m*x+8*B*b*e^3*x^2+6*B*c*d^2*e*m*x-6*B*c*d*e^2*x^2+26*A*a*e^3*m-7*A*b*d*e^2*m+12*A*b*e^3*x+2*A*c*d^2*e*m-8*

A*c*d*e^2*x-7*B*a*d*e^2*m+12*B*a*e^3*x+2*B*b*d^2*e*m-8*B*b*d*e^2*x+6*B*c*d^2*e*x+24*A*a*e^3-12*A*b*d*e^2+8*A*c

*d^2*e-12*B*a*d*e^2+8*B*b*d^2*e-6*B*c*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)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B] time = 1.63352, size = 1165, normalized size = 7.61 \begin{align*} \frac{{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 24 \, A a d e^{3} + 8 \,{\left (B b + A c\right )} d^{3} e - 12 \,{\left (B a + A b\right )} d^{2} e^{2} +{\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} +{\left (8 \,{\left (B b + A c\right )} e^{4} +{\left (B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} m^{3} +{\left (3 \, B c d e^{3} + 7 \,{\left (B b + A c\right )} e^{4}\right )} m^{2} + 2 \,{\left (B c d e^{3} + 7 \,{\left (B b + A c\right )} e^{4}\right )} m\right )} x^{3} +{\left (9 \, A a d e^{3} -{\left (B a + A b\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (B a + A b\right )} e^{4} +{\left ({\left (B b + A c\right )} d e^{3} +{\left (B a + A b\right )} e^{4}\right )} m^{3} -{\left (3 \, B c d^{2} e^{2} - 5 \,{\left (B b + A c\right )} d e^{3} - 8 \,{\left (B a + A b\right )} e^{4}\right )} m^{2} -{\left (3 \, B c d^{2} e^{2} - 4 \,{\left (B b + A c\right )} d e^{3} - 19 \,{\left (B a + A b\right )} e^{4}\right )} m\right )} x^{2} +{\left (26 \, A a d e^{3} + 2 \,{\left (B b + A c\right )} d^{3} e - 7 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} m +{\left (24 \, A a e^{4} +{\left (A a e^{4} +{\left (B a + A b\right )} d e^{3}\right )} m^{3} +{\left (9 \, A a e^{4} - 2 \,{\left (B b + A c\right )} d^{2} e^{2} + 7 \,{\left (B a + A b\right )} d e^{3}\right )} m^{2} + 2 \,{\left (3 \, B c d^{3} e + 13 \, A a e^{4} - 4 \,{\left (B b + A c\right )} d^{2} e^{2} + 6 \,{\left (B 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)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(A*a*d*e^3*m^3 - 6*B*c*d^4 + 24*A*a*d*e^3 + 8*(B*b + A*c)*d^3*e - 12*(B*a + A*b)*d^2*e^2 + (B*c*e^4*m^3 + 6*B*

c*e^4*m^2 + 11*B*c*e^4*m + 6*B*c*e^4)*x^4 + (8*(B*b + A*c)*e^4 + (B*c*d*e^3 + (B*b + A*c)*e^4)*m^3 + (3*B*c*d*

e^3 + 7*(B*b + A*c)*e^4)*m^2 + 2*(B*c*d*e^3 + 7*(B*b + A*c)*e^4)*m)*x^3 + (9*A*a*d*e^3 - (B*a + A*b)*d^2*e^2)*

m^2 + (12*(B*a + A*b)*e^4 + ((B*b + A*c)*d*e^3 + (B*a + A*b)*e^4)*m^3 - (3*B*c*d^2*e^2 - 5*(B*b + A*c)*d*e^3 -

8*(B*a + A*b)*e^4)*m^2 - (3*B*c*d^2*e^2 - 4*(B*b + A*c)*d*e^3 - 19*(B*a + A*b)*e^4)*m)*x^2 + (26*A*a*d*e^3 +

2*(B*b + A*c)*d^3*e - 7*(B*a + A*b)*d^2*e^2)*m + (24*A*a*e^4 + (A*a*e^4 + (B*a + A*b)*d*e^3)*m^3 + (9*A*a*e^4

- 2*(B*b + A*c)*d^2*e^2 + 7*(B*a + A*b)*d*e^3)*m^2 + 2*(3*B*c*d^3*e + 13*A*a*e^4 - 4*(B*b + A*c)*d^2*e^2 + 6*(

B*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 = 7.64629, size = 5930, normalized size = 38.76 \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)*(e*x+d)**m*(c*x**2+b*x+a),x)

[Out]

Piecewise((d**m*(A*a*x + A*b*x**2/2 + A*c*x**3/3 + B*a*x**2/2 + B*b*x**3/3 + B*c*x**4/4), Eq(e, 0)), (-2*A*a*e

**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - A*b*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x +

18*d*e**6*x**2 + 6*e**7*x**3) - 3*A*b*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) -

2*A*c*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*c*d*e**2*x/(6*d**3*e**4 + 18*

d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*c*e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 +

6*e**7*x**3) - B*a*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 3*B*a*e**3*x/(6*d**

3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*B*b*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e*

*6*x**2 + 6*e**7*x**3) - 6*B*b*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*B*b*

e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*d**3*log(d/e + x)/(6*d**3*e**4

+ 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*c*d**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2

+ 6*e**7*x**3) + 18*B*c*d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) +

27*B*c*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d*e**2*x**2*log(d/e +

x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e

**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6

*x**2 + 6*e**7*x**3), Eq(m, -4)), (-A*a*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - A*b*d*e**2/(2*d**2*e**

4 + 4*d*e**5*x + 2*e**6*x**2) - 2*A*b*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*d**2*e*log(d/e +

x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*A*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*c*d

*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*c*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e

**6*x**2) + 2*A*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - B*a*d*e**2/(2*d**2*e**4 +

4*d*e**5*x + 2*e**6*x**2) - 2*B*a*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*b*d**2*e*log(d/e + x)/

(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*B*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*B*b*d*e**

2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*B*b*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*

x**2) + 2*B*b*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*B*c*d**3*log(d/e + x)/(2*d**

2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*B*c*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*c*d**2*e*x*log

(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*c*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)

- 6*B*c*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*c*e**3*x**3/(2*d**2*e**4 + 4*d

*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-2*A*a*e**3/(2*d*e**4 + 2*e**5*x) + 2*A*b*d*e**2*log(d/e + x)/(2*d*e**4 +

2*e**5*x) + 2*A*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*A*b*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2

*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*c*d*e**2*x*log(d/e + x)/(2*d*

e**4 + 2*e**5*x) + 2*A*c*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2

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

)/(2*d*e**4 + 2*e**5*x) - 4*B*b*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*B*b*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*

x) + 2*B*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3/(2*d*e

**4 + 2*e**5*x) + 6*B*c*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*B*c*d*e**2*x**2/(2*d*e**4 + 2*e**5*x)

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

*c*d**2*log(d/e + x)/e**3 - A*c*d*x/e**2 + A*c*x**2/(2*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) - B*c*d**3*log(d/e + x)/e**4 + B*c*d**2*x/e**3 - B*c*d*x**2/(2*e*

*2) + B*c*x**3/(3*e), Eq(m, -1)), (A*a*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*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*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*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*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*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*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*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) - 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) - 7*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) -

12*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) + 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) + 7*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) + 12*A*b*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) + 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) + 8*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) + 19*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) + 12*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*c*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*c

*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*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*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*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*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*d*e**3*m**2*x*(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) + 12*B*a*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*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*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*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*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*B*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) + 8*B*b*d**3*e*(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*B*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) - 8*B*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) + B*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) + 5*B*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) + 4*B*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

) + B*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) + 7*B*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) + 14*B*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) + 8*B*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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*c*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 = 1.30099, size = 1569, normalized size = 10.25 \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)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

((x*e + d)^m*B*c*m^3*x^4*e^4 + (x*e + d)^m*B*c*d*m^3*x^3*e^3 + (x*e + d)^m*B*b*m^3*x^3*e^4 + (x*e + d)^m*A*c*m

^3*x^3*e^4 + 6*(x*e + d)^m*B*c*m^2*x^4*e^4 + (x*e + d)^m*B*b*d*m^3*x^2*e^3 + (x*e + d)^m*A*c*d*m^3*x^2*e^3 + 3

*(x*e + d)^m*B*c*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*c*d^2*m^2*x^2*e^2 + (x*e + d)^m*B*a*m^3*x^2*e^4 + (x*e + d)^m

*A*b*m^3*x^2*e^4 + 7*(x*e + d)^m*B*b*m^2*x^3*e^4 + 7*(x*e + d)^m*A*c*m^2*x^3*e^4 + 11*(x*e + d)^m*B*c*m*x^4*e^

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

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

m^2*x*e^2 - 3*(x*e + d)^m*B*c*d^2*m*x^2*e^2 + 6*(x*e + d)^m*B*c*d^3*m*x*e + (x*e + d)^m*A*a*m^3*x*e^4 + 8*(x*e

+ d)^m*B*a*m^2*x^2*e^4 + 8*(x*e + d)^m*A*b*m^2*x^2*e^4 + 14*(x*e + d)^m*B*b*m*x^3*e^4 + 14*(x*e + d)^m*A*c*m*

x^3*e^4 + 6*(x*e + d)^m*B*c*x^4*e^4 + (x*e + d)^m*A*a*d*m^3*e^3 + 7*(x*e + d)^m*B*a*d*m^2*x*e^3 + 7*(x*e + d)^

m*A*b*d*m^2*x*e^3 + 4*(x*e + d)^m*B*b*d*m*x^2*e^3 + 4*(x*e + d)^m*A*c*d*m*x^2*e^3 - (x*e + d)^m*B*a*d^2*m^2*e^

2 - (x*e + d)^m*A*b*d^2*m^2*e^2 - 8*(x*e + d)^m*B*b*d^2*m*x*e^2 - 8*(x*e + d)^m*A*c*d^2*m*x*e^2 + 2*(x*e + d)^

m*B*b*d^3*m*e + 2*(x*e + d)^m*A*c*d^3*m*e - 6*(x*e + d)^m*B*c*d^4 + 9*(x*e + d)^m*A*a*m^2*x*e^4 + 19*(x*e + d)

^m*B*a*m*x^2*e^4 + 19*(x*e + d)^m*A*b*m*x^2*e^4 + 8*(x*e + d)^m*B*b*x^3*e^4 + 8*(x*e + d)^m*A*c*x^3*e^4 + 9*(x

*e + d)^m*A*a*d*m^2*e^3 + 12*(x*e + d)^m*B*a*d*m*x*e^3 + 12*(x*e + d)^m*A*b*d*m*x*e^3 - 7*(x*e + d)^m*B*a*d^2*

m*e^2 - 7*(x*e + d)^m*A*b*d^2*m*e^2 + 8*(x*e + d)^m*B*b*d^3*e + 8*(x*e + d)^m*A*c*d^3*e + 26*(x*e + d)^m*A*a*m

*x*e^4 + 12*(x*e + d)^m*B*a*x^2*e^4 + 12*(x*e + d)^m*A*b*x^2*e^4 + 26*(x*e + d)^m*A*a*d*m*e^3 - 12*(x*e + d)^m

*B*a*d^2*e^2 - 12*(x*e + d)^m*A*b*d^2*e^2 + 24*(x*e + d)^m*A*a*x*e^4 + 24*(x*e + d)^m*A*a*d*e^3)/(m^4*e^4 + 10

*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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