3.2148 \(\int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac{(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac{3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac{b^3 (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.124512, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{3 b^2 (b d-a e) (d+e x)^{m+3}}{e^4 (m+3)}-\frac{(b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1)}+\frac{3 b (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2)}+\frac{b^3 (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 65.7605, size = 95, normalized size = 0.86 \[ \frac{b^{3} \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{3 b^{2} \left (d + e x\right )^{m + 3} \left (a e - b d\right )}{e^{4} \left (m + 3\right )} + \frac{3 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )^{2}}{e^{4} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{3}}{e^{4} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.190503, size = 178, normalized size = 1.6 \[ \frac{(d+e x)^{m+1} \left (a^3 e^3 \left (m^3+9 m^2+26 m+24\right )-3 a^2 b e^2 \left (m^2+7 m+12\right ) (d-e (m+1) x)+3 a b^2 e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b^3 \left (-\left (6 d^3-6 d^2 e (m+1) x+3 d e^2 \left (m^2+3 m+2\right ) x^2-e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a^3*e^3*(24 + 26*m + 9*m^2 + m^3) - 3*a^2*b*e^2*(12 + 7*m +
m^2)*(d - e*(1 + m)*x) + 3*a*b^2*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3
*m + m^2)*x^2) - b^3*(6*d^3 - 6*d^2*e*(1 + m)*x + 3*d*e^2*(2 + 3*m + m^2)*x^2 -
e^3*(6 + 11*m + 6*m^2 + m^3)*x^3)))/(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))

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Maple [B]  time = 0.013, size = 386, normalized size = 3.5 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({b}^{3}{e}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{e}^{3}{m}^{3}{x}^{2}+6\,{b}^{3}{e}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}b{e}^{3}{m}^{3}x+21\,a{b}^{2}{e}^{3}{m}^{2}{x}^{2}-3\,{b}^{3}d{e}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}{e}^{3}m{x}^{3}+{a}^{3}{e}^{3}{m}^{3}+24\,{a}^{2}b{e}^{3}{m}^{2}x-6\,a{b}^{2}d{e}^{2}{m}^{2}x+42\,a{b}^{2}{e}^{3}m{x}^{2}-9\,{b}^{3}d{e}^{2}m{x}^{2}+6\,{x}^{3}{b}^{3}{e}^{3}+9\,{a}^{3}{e}^{3}{m}^{2}-3\,{a}^{2}bd{e}^{2}{m}^{2}+57\,{a}^{2}b{e}^{3}mx-30\,a{b}^{2}d{e}^{2}mx+24\,{x}^{2}a{b}^{2}{e}^{3}+6\,{b}^{3}{d}^{2}emx-6\,{x}^{2}{b}^{3}d{e}^{2}+26\,{a}^{3}{e}^{3}m-21\,{a}^{2}bd{e}^{2}m+36\,x{a}^{2}b{e}^{3}+6\,a{b}^{2}{d}^{2}em-24\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e+24\,{a}^{3}{e}^{3}-36\,{a}^{2}bd{e}^{2}+24\,a{b}^{2}{d}^{2}e-6\,{b}^{3}{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

(e*x+d)^(1+m)*(b^3*e^3*m^3*x^3+3*a*b^2*e^3*m^3*x^2+6*b^3*e^3*m^2*x^3+3*a^2*b*e^3
*m^3*x+21*a*b^2*e^3*m^2*x^2-3*b^3*d*e^2*m^2*x^2+11*b^3*e^3*m*x^3+a^3*e^3*m^3+24*
a^2*b*e^3*m^2*x-6*a*b^2*d*e^2*m^2*x+42*a*b^2*e^3*m*x^2-9*b^3*d*e^2*m*x^2+6*b^3*e
^3*x^3+9*a^3*e^3*m^2-3*a^2*b*d*e^2*m^2+57*a^2*b*e^3*m*x-30*a*b^2*d*e^2*m*x+24*a*
b^2*e^3*x^2+6*b^3*d^2*e*m*x-6*b^3*d*e^2*x^2+26*a^3*e^3*m-21*a^2*b*d*e^2*m+36*a^2
*b*e^3*x+6*a*b^2*d^2*e*m-24*a*b^2*d*e^2*x+6*b^3*d^2*e*x+24*a^3*e^3-36*a^2*b*d*e^
2+24*a*b^2*d^2*e-6*b^3*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.33053, size = 670, normalized size = 6.04 \[ \frac{{\left (a^{3} d e^{3} m^{3} - 6 \, b^{3} d^{4} + 24 \, a b^{2} d^{3} e - 36 \, a^{2} b d^{2} e^{2} + 24 \, a^{3} d e^{3} +{\left (b^{3} e^{4} m^{3} + 6 \, b^{3} e^{4} m^{2} + 11 \, b^{3} e^{4} m + 6 \, b^{3} e^{4}\right )} x^{4} +{\left (24 \, a b^{2} e^{4} +{\left (b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} m^{3} + 3 \,{\left (b^{3} d e^{3} + 7 \, a b^{2} e^{4}\right )} m^{2} + 2 \,{\left (b^{3} d e^{3} + 21 \, a b^{2} e^{4}\right )} m\right )} x^{3} - 3 \,{\left (a^{2} b d^{2} e^{2} - 3 \, a^{3} d e^{3}\right )} m^{2} + 3 \,{\left (12 \, a^{2} b e^{4} +{\left (a b^{2} d e^{3} + a^{2} b e^{4}\right )} m^{3} -{\left (b^{3} d^{2} e^{2} - 5 \, a b^{2} d e^{3} - 8 \, a^{2} b e^{4}\right )} m^{2} -{\left (b^{3} d^{2} e^{2} - 4 \, a b^{2} d e^{3} - 19 \, a^{2} b e^{4}\right )} m\right )} x^{2} +{\left (6 \, a b^{2} d^{3} e - 21 \, a^{2} b d^{2} e^{2} + 26 \, a^{3} d e^{3}\right )} m +{\left (24 \, a^{3} e^{4} +{\left (3 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} m^{3} - 3 \,{\left (2 \, a b^{2} d^{2} e^{2} - 7 \, a^{2} b d e^{3} - 3 \, a^{3} e^{4}\right )} m^{2} + 2 \,{\left (3 \, b^{3} d^{3} e - 12 \, a b^{2} d^{2} e^{2} + 18 \, a^{2} b d e^{3} + 13 \, a^{3} e^{4}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(a^3*d*e^3*m^3 - 6*b^3*d^4 + 24*a*b^2*d^3*e - 36*a^2*b*d^2*e^2 + 24*a^3*d*e^3 +
(b^3*e^4*m^3 + 6*b^3*e^4*m^2 + 11*b^3*e^4*m + 6*b^3*e^4)*x^4 + (24*a*b^2*e^4 + (
b^3*d*e^3 + 3*a*b^2*e^4)*m^3 + 3*(b^3*d*e^3 + 7*a*b^2*e^4)*m^2 + 2*(b^3*d*e^3 +
21*a*b^2*e^4)*m)*x^3 - 3*(a^2*b*d^2*e^2 - 3*a^3*d*e^3)*m^2 + 3*(12*a^2*b*e^4 + (
a*b^2*d*e^3 + a^2*b*e^4)*m^3 - (b^3*d^2*e^2 - 5*a*b^2*d*e^3 - 8*a^2*b*e^4)*m^2 -
(b^3*d^2*e^2 - 4*a*b^2*d*e^3 - 19*a^2*b*e^4)*m)*x^2 + (6*a*b^2*d^3*e - 21*a^2*b
*d^2*e^2 + 26*a^3*d*e^3)*m + (24*a^3*e^4 + (3*a^2*b*d*e^3 + a^3*e^4)*m^3 - 3*(2*
a*b^2*d^2*e^2 - 7*a^2*b*d*e^3 - 3*a^3*e^4)*m^2 + 2*(3*b^3*d^3*e - 12*a*b^2*d^2*e
^2 + 18*a^2*b*d*e^3 + 13*a^3*e^4)*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 = 12.3668, size = 4056, normalized size = 36.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((d**m*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(e, 0)
), (-2*a**3*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) + 9*a**2*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) + 3*a**2*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) + 6*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) + 6*b**3*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**3*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**3*
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**3*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**3*d**3*e**2*x**2/(6*d**5*e**4 + 1
8*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*b**3*d**2*e**3*x**3*lo
g(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**3*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**3*d*e**3/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**
6*x**2) + 3*a**2*b*e**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*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) + 3*a*b*
*2*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 12*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) + 6*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) - 6*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) - 6*b**3*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**3*d**4/(2*d**3*e**4 + 4*d**2*e
**5*x + 2*d*e**6*x**2) - 12*b**3*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**3*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**3*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**3*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**3*e**3/(2*d*e**4 + 2*e**5*x) + 6*a**2*b*d*e**2*log(d/e + x)
/(2*d*e**4 + 2*e**5*x) + 6*a**2*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 6*a**2*b*e**3*x
*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 12*a*b**2*d**2*e*log(d/e + x)/(2*d*e**4 +
2*e**5*x) - 12*a*b**2*d**2*e/(2*d*e**4 + 2*e**5*x) - 12*a*b**2*d*e**2*x*log(d/e
+ x)/(2*d*e**4 + 2*e**5*x) + 6*a*b**2*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*b**3*d
**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*b**3*d**3/(2*d*e**4 + 2*e**5*x) + 6*b
**3*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*b**3*d*e**2*x**2/(2*d*e**4 +
2*e**5*x) + b**3*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (a**3*log(d/e + x
)/e - 3*a**2*b*d*log(d/e + x)/e**2 + 3*a**2*b*x/e + 3*a*b**2*d**2*log(d/e + x)/e
**3 - 3*a*b**2*d*x/e**2 + 3*a*b**2*x**2/(2*e) - b**3*d**3*log(d/e + x)/e**4 + b*
*3*d**2*x/e**3 - b**3*d*x**2/(2*e**2) + b**3*x**3/(3*e), Eq(m, -1)), (a**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*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) - 3*a**2*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) - 21*a**2*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)
- 36*a**2*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) + 3*a**2*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) + 21*a**2*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) + 36*a**2*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) + 3*a**2*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) + 24*a**2*b*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) + 57*a**2*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) + 36*a**
2*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) + 6*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) + 24*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) - 6*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) - 24*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) + 3*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) + 15*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) + 12*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) + 3*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) + 21*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) + 42*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) + 24*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) - 6*b**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3
*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**3*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**3*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**3*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/XCAS [A]  time = 0.304986, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done