3.1656 \(\int (b+2 c x) (d+e x)^m (a+b x+c x^2) \, dx\)

Optimal. Leaf size=143 \[ \frac{(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*c*(2*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (2*c^2*(d
+ e*x)^(4 + m))/(e^4*(4 + m))

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Rubi [A]  time = 0.0918753, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*c*(2*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (2*c^2*(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 (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^3}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{1+m}}{e^3}-\frac{3 c (2 c d-b e) (d+e x)^{2+m}}{e^3}+\frac{2 c^2 (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac{3 c (2 c d-b e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac{2 c^2 (d+e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.221926, size = 134, normalized size = 0.94 \[ \frac{(d+e x)^{m+1} \left (\frac{(d+e x) \left (4 c e (a e (m+3)-3 b d)-b^2 e^2 m+12 c^2 d^2\right )}{e^2 (m+2)}+\frac{6 (b e-2 c d) \left (e (a e-b d)+c d^2\right )}{e^2 (m+1)}+(a+x (b+c x)) (b e (m+6)-6 c d+2 c e (m+3) x)\right )}{e^2 (m+3) (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((6*(-2*c*d + b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(e^2*(1 + m)) + ((12*c^2*d^2 - b^2*e^2*m + 4
*c*e*(-3*b*d + a*e*(3 + m)))*(d + e*x))/(e^2*(2 + m)) + (-6*c*d + b*e*(6 + m) + 2*c*e*(3 + m)*x)*(a + x*(b + c
*x))))/(e^2*(3 + m)*(4 + m))

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Maple [B]  time = 0.006, size = 424, normalized size = 3. \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m} \left ( 2\,{c}^{2}{e}^{3}{m}^{3}{x}^{3}+3\,bc{e}^{3}{m}^{3}{x}^{2}+12\,{c}^{2}{e}^{3}{m}^{2}{x}^{3}+2\,ac{e}^{3}{m}^{3}x+{b}^{2}{e}^{3}{m}^{3}x+21\,bc{e}^{3}{m}^{2}{x}^{2}-6\,{c}^{2}d{e}^{2}{m}^{2}{x}^{2}+22\,{c}^{2}{e}^{3}m{x}^{3}+ab{e}^{3}{m}^{3}+16\,ac{e}^{3}{m}^{2}x+8\,{b}^{2}{e}^{3}{m}^{2}x-6\,bcd{e}^{2}{m}^{2}x+42\,bc{e}^{3}m{x}^{2}-18\,{c}^{2}d{e}^{2}m{x}^{2}+12\,{x}^{3}{c}^{2}{e}^{3}+9\,ab{e}^{3}{m}^{2}-2\,acd{e}^{2}{m}^{2}+38\,ac{e}^{3}mx-{b}^{2}d{e}^{2}{m}^{2}+19\,{b}^{2}{e}^{3}mx-30\,bcd{e}^{2}mx+24\,{x}^{2}bc{e}^{3}+12\,{c}^{2}{d}^{2}emx-12\,{x}^{2}{c}^{2}d{e}^{2}+26\,ab{e}^{3}m-14\,acd{e}^{2}m+24\,xac{e}^{3}-7\,{b}^{2}d{e}^{2}m+12\,x{b}^{2}{e}^{3}+6\,bc{d}^{2}em-24\,xbcd{e}^{2}+12\,e{d}^{2}x{c}^{2}+24\,ab{e}^{3}-24\,acd{e}^{2}-12\,{b}^{2}d{e}^{2}+24\,b{d}^{2}ce-12\,{c}^{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((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x)

[Out]

(e*x+d)^(1+m)*(2*c^2*e^3*m^3*x^3+3*b*c*e^3*m^3*x^2+12*c^2*e^3*m^2*x^3+2*a*c*e^3*m^3*x+b^2*e^3*m^3*x+21*b*c*e^3
*m^2*x^2-6*c^2*d*e^2*m^2*x^2+22*c^2*e^3*m*x^3+a*b*e^3*m^3+16*a*c*e^3*m^2*x+8*b^2*e^3*m^2*x-6*b*c*d*e^2*m^2*x+4
2*b*c*e^3*m*x^2-18*c^2*d*e^2*m*x^2+12*c^2*e^3*x^3+9*a*b*e^3*m^2-2*a*c*d*e^2*m^2+38*a*c*e^3*m*x-b^2*d*e^2*m^2+1
9*b^2*e^3*m*x-30*b*c*d*e^2*m*x+24*b*c*e^3*x^2+12*c^2*d^2*e*m*x-12*c^2*d*e^2*x^2+26*a*b*e^3*m-14*a*c*d*e^2*m+24
*a*c*e^3*x-7*b^2*d*e^2*m+12*b^2*e^3*x+6*b*c*d^2*e*m-24*b*c*d*e^2*x+12*c^2*d^2*e*x+24*a*b*e^3-24*a*c*d*e^2-12*b
^2*d*e^2+24*b*c*d^2*e-12*c^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((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.97945, size = 1099, normalized size = 7.69 \begin{align*} \frac{{\left (a b d e^{3} m^{3} - 12 \, c^{2} d^{4} + 24 \, b c d^{3} e + 24 \, a b d e^{3} - 12 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 2 \,{\left (c^{2} e^{4} m^{3} + 6 \, c^{2} e^{4} m^{2} + 11 \, c^{2} e^{4} m + 6 \, c^{2} e^{4}\right )} x^{4} +{\left (24 \, b c e^{4} +{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} m^{3} + 3 \,{\left (2 \, c^{2} d e^{3} + 7 \, b c e^{4}\right )} m^{2} + 2 \,{\left (2 \, c^{2} d e^{3} + 21 \, b c e^{4}\right )} m\right )} x^{3} +{\left (9 \, a b d e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (b^{2} + 2 \, a c\right )} e^{4} +{\left (3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{3} -{\left (6 \, c^{2} d^{2} e^{2} - 15 \, b c d e^{3} - 8 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{2} -{\left (6 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} - 19 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m\right )} x^{2} +{\left (6 \, b c d^{3} e + 26 \, a b d e^{3} - 7 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m +{\left (24 \, a b e^{4} +{\left (a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{3} -{\left (6 \, b c d^{2} e^{2} - 9 \, a b e^{4} - 7 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + 13 \, a b e^{4} + 6 \,{\left (b^{2} + 2 \, a c\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((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(a*b*d*e^3*m^3 - 12*c^2*d^4 + 24*b*c*d^3*e + 24*a*b*d*e^3 - 12*(b^2 + 2*a*c)*d^2*e^2 + 2*(c^2*e^4*m^3 + 6*c^2*
e^4*m^2 + 11*c^2*e^4*m + 6*c^2*e^4)*x^4 + (24*b*c*e^4 + (2*c^2*d*e^3 + 3*b*c*e^4)*m^3 + 3*(2*c^2*d*e^3 + 7*b*c
*e^4)*m^2 + 2*(2*c^2*d*e^3 + 21*b*c*e^4)*m)*x^3 + (9*a*b*d*e^3 - (b^2 + 2*a*c)*d^2*e^2)*m^2 + (12*(b^2 + 2*a*c
)*e^4 + (3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*m^3 - (6*c^2*d^2*e^2 - 15*b*c*d*e^3 - 8*(b^2 + 2*a*c)*e^4)*m^2 - (6*
c^2*d^2*e^2 - 12*b*c*d*e^3 - 19*(b^2 + 2*a*c)*e^4)*m)*x^2 + (6*b*c*d^3*e + 26*a*b*d*e^3 - 7*(b^2 + 2*a*c)*d^2*
e^2)*m + (24*a*b*e^4 + (a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*m^3 - (6*b*c*d^2*e^2 - 9*a*b*e^4 - 7*(b^2 + 2*a*c)*d*e^
3)*m^2 + 2*(6*c^2*d^3*e - 12*b*c*d^2*e^2 + 13*a*b*e^4 + 6*(b^2 + 2*a*c)*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 = 5.47352, size = 4760, normalized size = 33.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**m*(c*x**2+b*x+a),x)

[Out]

Piecewise((d**m*(a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2), Eq(e, 0)), (-2*a*b*e**3/(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*e**2/(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/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - b**2*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**2*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 1
8*d*e**6*x**2 + 6*e**7*x**3) - 6*b*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) - 18
*b*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) - 18*b*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) + 12*c**2*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) + 22*c**2*d**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 36*c
**2*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) + 54*c**2*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) + 36*c**2*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 1
8*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 36*c**2*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) + 12*c**2*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*b*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*a*c*d*e**2/(2*d**2*e**4 + 4*d*e**5*x
+ 2*e**6*x**2) - 4*a*c*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - b**2*d*e**2/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) - 2*b**2*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*b*c*d**2*e*log(d/e + x)/(2*d**2*
e**4 + 4*d*e**5*x + 2*e**6*x**2) + 9*b*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*b*c*d*e**2*x*log
(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 12*b*c*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)
+ 6*b*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c**2*d**3*log(d/e + x)/(2*d**2*e*
*4 + 4*d*e**5*x + 2*e**6*x**2) - 18*c**2*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**2*d**2*e*x*log(
d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 24*c**2*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2)
- 12*c**2*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*c**2*e**3*x**3/(2*d**2*e**4 +
4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-a*b*e**3/(d*e**4 + e**5*x) + 2*a*c*d*e**2*log(d/e + x)/(d*e**4 + e**5
*x) + 2*a*c*d*e**2/(d*e**4 + e**5*x) + 2*a*c*e**3*x*log(d/e + x)/(d*e**4 + e**5*x) + b**2*d*e**2*log(d/e + x)/
(d*e**4 + e**5*x) + b**2*d*e**2/(d*e**4 + e**5*x) + b**2*e**3*x*log(d/e + x)/(d*e**4 + e**5*x) - 6*b*c*d**2*e*
log(d/e + x)/(d*e**4 + e**5*x) - 6*b*c*d**2*e/(d*e**4 + e**5*x) - 6*b*c*d*e**2*x*log(d/e + x)/(d*e**4 + e**5*x
) + 3*b*c*e**3*x**2/(d*e**4 + e**5*x) + 6*c**2*d**3*log(d/e + x)/(d*e**4 + e**5*x) + 6*c**2*d**3/(d*e**4 + e**
5*x) + 6*c**2*d**2*e*x*log(d/e + x)/(d*e**4 + e**5*x) - 3*c**2*d*e**2*x**2/(d*e**4 + e**5*x) + c**2*e**3*x**3/
(d*e**4 + e**5*x), Eq(m, -2)), (a*b*log(d/e + x)/e - 2*a*c*d*log(d/e + x)/e**2 + 2*a*c*x/e - b**2*d*log(d/e +
x)/e**2 + b**2*x/e + 3*b*c*d**2*log(d/e + x)/e**3 - 3*b*c*d*x/e**2 + 3*b*c*x**2/(2*e) - 2*c**2*d**3*log(d/e +
x)/e**4 + 2*c**2*d**2*x/e**3 - c**2*d*x**2/e**2 + 2*c**2*x**3/(3*e), Eq(m, -1)), (a*b*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*b*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*b*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*b*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + a*b*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*b*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*b
*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*b*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*c*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*c*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*c*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*c*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*c*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*c*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)
+ 2*a*c*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*c*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) + 38*a*c*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*c*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) - b**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**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**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**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**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**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**
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**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**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**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) + 6*b*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) + 24*b*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) - 6*b*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 + 5
0*e**4*m + 24*e**4) - 24*b*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) + 3*b*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
) + 15*b*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) + 12*
b*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) + 3*b*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) + 21*b*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) + 42*b*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) + 24*b*c*e**4*x**3*(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) - 12*c**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) + 12*c**2*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) - 6*c**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) - 6*c**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) + 2*c**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
) + 6*c**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) + 4*c
**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) + 2*c**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) + 12*c**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) + 22*c**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) + 12*c**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 = 1.66405, size = 1323, normalized size = 9.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

(2*(x*e + d)^m*c^2*m^3*x^4*e^4 + 2*(x*e + d)^m*c^2*d*m^3*x^3*e^3 + 3*(x*e + d)^m*b*c*m^3*x^3*e^4 + 12*(x*e + d
)^m*c^2*m^2*x^4*e^4 + 3*(x*e + d)^m*b*c*d*m^3*x^2*e^3 + 6*(x*e + d)^m*c^2*d*m^2*x^3*e^3 - 6*(x*e + d)^m*c^2*d^
2*m^2*x^2*e^2 + (x*e + d)^m*b^2*m^3*x^2*e^4 + 2*(x*e + d)^m*a*c*m^3*x^2*e^4 + 21*(x*e + d)^m*b*c*m^2*x^3*e^4 +
22*(x*e + d)^m*c^2*m*x^4*e^4 + (x*e + d)^m*b^2*d*m^3*x*e^3 + 2*(x*e + d)^m*a*c*d*m^3*x*e^3 + 15*(x*e + d)^m*b
*c*d*m^2*x^2*e^3 + 4*(x*e + d)^m*c^2*d*m*x^3*e^3 - 6*(x*e + d)^m*b*c*d^2*m^2*x*e^2 - 6*(x*e + d)^m*c^2*d^2*m*x
^2*e^2 + 12*(x*e + d)^m*c^2*d^3*m*x*e + (x*e + d)^m*a*b*m^3*x*e^4 + 8*(x*e + d)^m*b^2*m^2*x^2*e^4 + 16*(x*e +
d)^m*a*c*m^2*x^2*e^4 + 42*(x*e + d)^m*b*c*m*x^3*e^4 + 12*(x*e + d)^m*c^2*x^4*e^4 + (x*e + d)^m*a*b*d*m^3*e^3 +
7*(x*e + d)^m*b^2*d*m^2*x*e^3 + 14*(x*e + d)^m*a*c*d*m^2*x*e^3 + 12*(x*e + d)^m*b*c*d*m*x^2*e^3 - (x*e + d)^m
*b^2*d^2*m^2*e^2 - 2*(x*e + d)^m*a*c*d^2*m^2*e^2 - 24*(x*e + d)^m*b*c*d^2*m*x*e^2 + 6*(x*e + d)^m*b*c*d^3*m*e
- 12*(x*e + d)^m*c^2*d^4 + 9*(x*e + d)^m*a*b*m^2*x*e^4 + 19*(x*e + d)^m*b^2*m*x^2*e^4 + 38*(x*e + d)^m*a*c*m*x
^2*e^4 + 24*(x*e + d)^m*b*c*x^3*e^4 + 9*(x*e + d)^m*a*b*d*m^2*e^3 + 12*(x*e + d)^m*b^2*d*m*x*e^3 + 24*(x*e + d
)^m*a*c*d*m*x*e^3 - 7*(x*e + d)^m*b^2*d^2*m*e^2 - 14*(x*e + d)^m*a*c*d^2*m*e^2 + 24*(x*e + d)^m*b*c*d^3*e + 26
*(x*e + d)^m*a*b*m*x*e^4 + 12*(x*e + d)^m*b^2*x^2*e^4 + 24*(x*e + d)^m*a*c*x^2*e^4 + 26*(x*e + d)^m*a*b*d*m*e^
3 - 12*(x*e + d)^m*b^2*d^2*e^2 - 24*(x*e + d)^m*a*c*d^2*e^2 + 24*(x*e + d)^m*a*b*x*e^4 + 24*(x*e + d)^m*a*b*d*
e^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)