3.1721 \(\int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=142 \[ -\frac{4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac{(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac{b^4 (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

((b*d - a*e)^4*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*b*(b*d - a*e)^3*(d + e*x)^(
2 + m))/(e^5*(2 + m)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 + m))/(e^5*(3 + m)) -
(4*b^3*(b*d - a*e)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (b^4*(d + e*x)^(5 + m))/(e
^5*(5 + m))

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Rubi [A]  time = 0.180641, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac{(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac{b^4 (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)^4*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*b*(b*d - a*e)^3*(d + e*x)^(
2 + m))/(e^5*(2 + m)) + (6*b^2*(b*d - a*e)^2*(d + e*x)^(3 + m))/(e^5*(3 + m)) -
(4*b^3*(b*d - a*e)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (b^4*(d + e*x)^(5 + m))/(e
^5*(5 + m))

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Rubi in Sympy [A]  time = 79.7479, size = 124, normalized size = 0.87 \[ \frac{b^{4} \left (d + e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} + \frac{4 b^{3} \left (d + e x\right )^{m + 4} \left (a e - b d\right )}{e^{5} \left (m + 4\right )} + \frac{6 b^{2} \left (d + e x\right )^{m + 3} \left (a e - b d\right )^{2}}{e^{5} \left (m + 3\right )} + \frac{4 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )^{3}}{e^{5} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{4}}{e^{5} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**4*(d + e*x)**(m + 5)/(e**5*(m + 5)) + 4*b**3*(d + e*x)**(m + 4)*(a*e - b*d)/(
e**5*(m + 4)) + 6*b**2*(d + e*x)**(m + 3)*(a*e - b*d)**2/(e**5*(m + 3)) + 4*b*(d
+ e*x)**(m + 2)*(a*e - b*d)**3/(e**5*(m + 2)) + (d + e*x)**(m + 1)*(a*e - b*d)*
*4/(e**5*(m + 1))

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Mathematica [B]  time = 0.30627, size = 292, normalized size = 2.06 \[ \frac{(d+e x)^{m+1} \left (a^4 e^4 \left (m^4+14 m^3+71 m^2+154 m+120\right )-4 a^3 b e^3 \left (m^3+12 m^2+47 m+60\right ) (d-e (m+1) x)+6 a^2 b^2 e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+4 a b^3 e (m+5) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+b^4 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )}{e^5 (m+1) (m+2) (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(a^4*e^4*(120 + 154*m + 71*m^2 + 14*m^3 + m^4) - 4*a^3*b*e^3*
(60 + 47*m + 12*m^2 + m^3)*(d - e*(1 + m)*x) + 6*a^2*b^2*e^2*(20 + 9*m + m^2)*(2
*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 4*a*b^3*e*(5 + m)*(-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) + b^4*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^
3*(6 + 11*m + 6*m^2 + m^3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4)))/
(e^5*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m))

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Maple [B]  time = 0.016, size = 768, normalized size = 5.4 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({b}^{4}{e}^{4}{m}^{4}{x}^{4}+4\,a{b}^{3}{e}^{4}{m}^{4}{x}^{3}+10\,{b}^{4}{e}^{4}{m}^{3}{x}^{4}+6\,{a}^{2}{b}^{2}{e}^{4}{m}^{4}{x}^{2}+44\,a{b}^{3}{e}^{4}{m}^{3}{x}^{3}-4\,{b}^{4}d{e}^{3}{m}^{3}{x}^{3}+35\,{b}^{4}{e}^{4}{m}^{2}{x}^{4}+4\,{a}^{3}b{e}^{4}{m}^{4}x+72\,{a}^{2}{b}^{2}{e}^{4}{m}^{3}{x}^{2}-12\,a{b}^{3}d{e}^{3}{m}^{3}{x}^{2}+164\,a{b}^{3}{e}^{4}{m}^{2}{x}^{3}-24\,{b}^{4}d{e}^{3}{m}^{2}{x}^{3}+50\,{b}^{4}{e}^{4}m{x}^{4}+{a}^{4}{e}^{4}{m}^{4}+52\,{a}^{3}b{e}^{4}{m}^{3}x-12\,{a}^{2}{b}^{2}d{e}^{3}{m}^{3}x+294\,{a}^{2}{b}^{2}{e}^{4}{m}^{2}{x}^{2}-96\,a{b}^{3}d{e}^{3}{m}^{2}{x}^{2}+244\,a{b}^{3}{e}^{4}m{x}^{3}+12\,{b}^{4}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{b}^{4}d{e}^{3}m{x}^{3}+24\,{x}^{4}{b}^{4}{e}^{4}+14\,{a}^{4}{e}^{4}{m}^{3}-4\,{a}^{3}bd{e}^{3}{m}^{3}+236\,{a}^{3}b{e}^{4}{m}^{2}x-120\,{a}^{2}{b}^{2}d{e}^{3}{m}^{2}x+468\,{a}^{2}{b}^{2}{e}^{4}m{x}^{2}+24\,a{b}^{3}{d}^{2}{e}^{2}{m}^{2}x-204\,a{b}^{3}d{e}^{3}m{x}^{2}+120\,{x}^{3}a{b}^{3}{e}^{4}+36\,{b}^{4}{d}^{2}{e}^{2}m{x}^{2}-24\,{x}^{3}{b}^{4}d{e}^{3}+71\,{a}^{4}{e}^{4}{m}^{2}-48\,{a}^{3}bd{e}^{3}{m}^{2}+428\,{a}^{3}b{e}^{4}mx+12\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}{m}^{2}-348\,{a}^{2}{b}^{2}d{e}^{3}mx+240\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+144\,a{b}^{3}{d}^{2}{e}^{2}mx-120\,{x}^{2}a{b}^{3}d{e}^{3}-24\,{b}^{4}{d}^{3}emx+24\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+154\,{a}^{4}{e}^{4}m-188\,{a}^{3}bd{e}^{3}m+240\,x{a}^{3}b{e}^{4}+108\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}m-240\,x{a}^{2}{b}^{2}d{e}^{3}-24\,a{b}^{3}{d}^{3}em+120\,xa{b}^{3}{d}^{2}{e}^{2}-24\,x{b}^{4}{d}^{3}e+120\,{a}^{4}{e}^{4}-240\,{a}^{3}bd{e}^{3}+240\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-120\,a{b}^{3}{d}^{3}e+24\,{b}^{4}{d}^{4} \right ) }{{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*(b^4*e^4*m^4*x^4+4*a*b^3*e^4*m^4*x^3+10*b^4*e^4*m^3*x^4+6*a^2*b^2*
e^4*m^4*x^2+44*a*b^3*e^4*m^3*x^3-4*b^4*d*e^3*m^3*x^3+35*b^4*e^4*m^2*x^4+4*a^3*b*
e^4*m^4*x+72*a^2*b^2*e^4*m^3*x^2-12*a*b^3*d*e^3*m^3*x^2+164*a*b^3*e^4*m^2*x^3-24
*b^4*d*e^3*m^2*x^3+50*b^4*e^4*m*x^4+a^4*e^4*m^4+52*a^3*b*e^4*m^3*x-12*a^2*b^2*d*
e^3*m^3*x+294*a^2*b^2*e^4*m^2*x^2-96*a*b^3*d*e^3*m^2*x^2+244*a*b^3*e^4*m*x^3+12*
b^4*d^2*e^2*m^2*x^2-44*b^4*d*e^3*m*x^3+24*b^4*e^4*x^4+14*a^4*e^4*m^3-4*a^3*b*d*e
^3*m^3+236*a^3*b*e^4*m^2*x-120*a^2*b^2*d*e^3*m^2*x+468*a^2*b^2*e^4*m*x^2+24*a*b^
3*d^2*e^2*m^2*x-204*a*b^3*d*e^3*m*x^2+120*a*b^3*e^4*x^3+36*b^4*d^2*e^2*m*x^2-24*
b^4*d*e^3*x^3+71*a^4*e^4*m^2-48*a^3*b*d*e^3*m^2+428*a^3*b*e^4*m*x+12*a^2*b^2*d^2
*e^2*m^2-348*a^2*b^2*d*e^3*m*x+240*a^2*b^2*e^4*x^2+144*a*b^3*d^2*e^2*m*x-120*a*b
^3*d*e^3*x^2-24*b^4*d^3*e*m*x+24*b^4*d^2*e^2*x^2+154*a^4*e^4*m-188*a^3*b*d*e^3*m
+240*a^3*b*e^4*x+108*a^2*b^2*d^2*e^2*m-240*a^2*b^2*d*e^3*x-24*a*b^3*d^3*e*m+120*
a*b^3*d^2*e^2*x-24*b^4*d^3*e*x+120*a^4*e^4-240*a^3*b*d*e^3+240*a^2*b^2*d^2*e^2-1
20*a*b^3*d^3*e+24*b^4*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)

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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)^2*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225146, size = 1216, normalized size = 8.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^4*d*e^4*m^4 + 24*b^4*d^5 - 120*a*b^3*d^4*e + 240*a^2*b^2*d^3*e^2 - 240*a^3*b*
d^2*e^3 + 120*a^4*d*e^4 + (b^4*e^5*m^4 + 10*b^4*e^5*m^3 + 35*b^4*e^5*m^2 + 50*b^
4*e^5*m + 24*b^4*e^5)*x^5 + (120*a*b^3*e^5 + (b^4*d*e^4 + 4*a*b^3*e^5)*m^4 + 2*(
3*b^4*d*e^4 + 22*a*b^3*e^5)*m^3 + (11*b^4*d*e^4 + 164*a*b^3*e^5)*m^2 + 2*(3*b^4*
d*e^4 + 122*a*b^3*e^5)*m)*x^4 - 2*(2*a^3*b*d^2*e^3 - 7*a^4*d*e^4)*m^3 + 2*(120*a
^2*b^2*e^5 + (2*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*m^4 - 2*(b^4*d^2*e^3 - 8*a*b^3*d*e^
4 - 18*a^2*b^2*e^5)*m^3 - (6*b^4*d^2*e^3 - 34*a*b^3*d*e^4 - 147*a^2*b^2*e^5)*m^2
- 2*(2*b^4*d^2*e^3 - 10*a*b^3*d*e^4 - 117*a^2*b^2*e^5)*m)*x^3 + (12*a^2*b^2*d^3
*e^2 - 48*a^3*b*d^2*e^3 + 71*a^4*d*e^4)*m^2 + 2*(120*a^3*b*e^5 + (3*a^2*b^2*d*e^
4 + 2*a^3*b*e^5)*m^4 - 2*(3*a*b^3*d^2*e^3 - 15*a^2*b^2*d*e^4 - 13*a^3*b*e^5)*m^3
+ (6*b^4*d^3*e^2 - 36*a*b^3*d^2*e^3 + 87*a^2*b^2*d*e^4 + 118*a^3*b*e^5)*m^2 + 2
*(3*b^4*d^3*e^2 - 15*a*b^3*d^2*e^3 + 30*a^2*b^2*d*e^4 + 107*a^3*b*e^5)*m)*x^2 -
2*(12*a*b^3*d^4*e - 54*a^2*b^2*d^3*e^2 + 94*a^3*b*d^2*e^3 - 77*a^4*d*e^4)*m + (1
20*a^4*e^5 + (4*a^3*b*d*e^4 + a^4*e^5)*m^4 - 2*(6*a^2*b^2*d^2*e^3 - 24*a^3*b*d*e
^4 - 7*a^4*e^5)*m^3 + (24*a*b^3*d^3*e^2 - 108*a^2*b^2*d^2*e^3 + 188*a^3*b*d*e^4
+ 71*a^4*e^5)*m^2 - 2*(12*b^4*d^4*e - 60*a*b^3*d^3*e^2 + 120*a^2*b^2*d^2*e^3 - 1
20*a^3*b*d*e^4 - 77*a^4*e^5)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^
3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)

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Sympy [A]  time = 24.8759, size = 8789, normalized size = 61.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*
x**5/5), Eq(e, 0)), (-3*a**4*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*
x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*a**3*b*d*e**3/(12*d**4*e**5 + 48*d**3*
e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 16*a**3*b*e**4*x/(
12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**
4) - 6*a**2*b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4) - 24*a**2*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e
**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 36*a**2*b**2*e**4*x
**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**
9*x**4) - 12*a*b**3*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4) - 48*a*b**3*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e
**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 72*a*b**3*d*e**3*x*
*2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9
*x**4) - 48*a*b**3*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2
+ 48*d*e**8*x**3 + 12*e**9*x**4) + 12*b**4*d**4*log(d/e + x)/(12*d**4*e**5 + 48*
d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 25*b**4*d**4/
(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x*
*4) + 48*b**4*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**
7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*b**4*d**3*e*x/(12*d**4*e**5 + 48*d*
*3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 72*b**4*d**2*e*
*2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e
**8*x**3 + 12*e**9*x**4) + 108*b**4*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*
x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*b**4*d*e**3*x**3*log
(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 +
12*e**9*x**4) + 48*b**4*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**
7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*b**4*e**4*x**4*log(d/e + x)/(12*d**
4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4), Eq
(m, -5)), (-a**4*d**2*e**4/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d
**2*e**8*x**3) + 6*a**3*b*d*e**5*x**2/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7
*x**2 + 3*d**2*e**8*x**3) + 2*a**3*b*e**6*x**3/(3*d**5*e**5 + 9*d**4*e**6*x + 9*
d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 6*a**2*b**2*d*e**5*x**3/(3*d**5*e**5 + 9*d*
*4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 12*a*b**3*d**5*e*log(d/e + x)
/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 4*a*b**3*
d**5*e/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 36*
a*b**3*d**4*e**2*x*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2
+ 3*d**2*e**8*x**3) + 36*a*b**3*d**3*e**3*x**2*log(d/e + x)/(3*d**5*e**5 + 9*d**
4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 18*a*b**3*d**3*e**3*x**2/(3*d*
*5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 12*a*b**3*d**2*
e**4*x**3*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*
e**8*x**3) - 18*a*b**3*d**2*e**4*x**3/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7
*x**2 + 3*d**2*e**8*x**3) - 12*b**4*d**6*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6
*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 4*b**4*d**6/(3*d**5*e**5 + 9*d**4*e*
*6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 36*b**4*d**5*e*x*log(d/e + x)/(3*d
**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) - 36*b**4*d**4*e
**2*x**2*log(d/e + x)/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e
**8*x**3) + 18*b**4*d**4*e**2*x**2/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x*
*2 + 3*d**2*e**8*x**3) - 12*b**4*d**3*e**3*x**3*log(d/e + x)/(3*d**5*e**5 + 9*d*
*4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 18*b**4*d**3*e**3*x**3/(3*d**
5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3) + 3*b**4*d**2*e**4
*x**4/(3*d**5*e**5 + 9*d**4*e**6*x + 9*d**3*e**7*x**2 + 3*d**2*e**8*x**3), Eq(m,
-4)), (-a**4*d*e**4/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 4*a**3*b*e*
*5*x**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 12*a**2*b**2*d**3*e**2*l
og(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 6*a**2*b**2*d**3*e**
2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 24*a**2*b**2*d**2*e**3*x*log(d
/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 12*a**2*b**2*d*e**4*x**2
*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 12*a**2*b**2*d*e**
4*x**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 24*a*b**3*d**4*e*log(d/e
+ x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 12*a*b**3*d**4*e/(2*d**3*e*
*5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 48*a*b**3*d**3*e**2*x*log(d/e + x)/(2*d**3
*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 24*a*b**3*d**2*e**3*x**2*log(d/e + x)/(
2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 24*a*b**3*d**2*e**3*x**2/(2*d**3*
e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + 8*a*b**3*d*e**4*x**3/(2*d**3*e**5 + 4*d*
*2*e**6*x + 2*d*e**7*x**2) + 12*b**4*d**5*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**
6*x + 2*d*e**7*x**2) + 6*b**4*d**5/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2)
+ 24*b**4*d**4*e*x*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) +
12*b**4*d**3*e**2*x**2*log(d/e + x)/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**
2) - 12*b**4*d**3*e**2*x**2/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) - 4*b*
*4*d**2*e**3*x**3/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2) + b**4*d*e**4*x*
*4/(2*d**3*e**5 + 4*d**2*e**6*x + 2*d*e**7*x**2), Eq(m, -3)), (-3*a**4*e**4/(3*d
*e**5 + 3*e**6*x) + 12*a**3*b*d*e**3*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 12*a**
3*b*d*e**3/(3*d*e**5 + 3*e**6*x) + 12*a**3*b*e**4*x*log(d/e + x)/(3*d*e**5 + 3*e
**6*x) - 36*a**2*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 36*a**2*b**
2*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 36*a**2*b**2*d*e**3*x*log(d/e + x)/(3*d*e**5
+ 3*e**6*x) + 18*a**2*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 36*a*b**3*d**3*e*l
og(d/e + x)/(3*d*e**5 + 3*e**6*x) + 36*a*b**3*d**3*e/(3*d*e**5 + 3*e**6*x) + 36*
a*b**3*d**2*e**2*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 18*a*b**3*d*e**3*x**2/(3
*d*e**5 + 3*e**6*x) + 6*a*b**3*e**4*x**3/(3*d*e**5 + 3*e**6*x) - 12*b**4*d**4*lo
g(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*b**4*d**4/(3*d*e**5 + 3*e**6*x) - 12*b**4*
d**3*e*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*b**4*d**2*e**2*x**2/(3*d*e**5 +
3*e**6*x) - 2*b**4*d*e**3*x**3/(3*d*e**5 + 3*e**6*x) + b**4*e**4*x**4/(3*d*e**5
+ 3*e**6*x), Eq(m, -2)), (a**4*log(d/e + x)/e - 4*a**3*b*d*log(d/e + x)/e**2 + 4
*a**3*b*x/e + 6*a**2*b**2*d**2*log(d/e + x)/e**3 - 6*a**2*b**2*d*x/e**2 + 3*a**2
*b**2*x**2/e - 4*a*b**3*d**3*log(d/e + x)/e**4 + 4*a*b**3*d**2*x/e**3 - 2*a*b**3
*d*x**2/e**2 + 4*a*b**3*x**3/(3*e) + b**4*d**4*log(d/e + x)/e**5 - b**4*d**3*x/e
**4 + b**4*d**2*x**2/(2*e**3) - b**4*d*x**3/(3*e**2) + b**4*x**4/(4*e), Eq(m, -1
)), (a**4*d*e**4*m**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) + 14*a**4*d*e**4*m**3*(d + e*x)**m/(e**5*m*
*5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 71*a
**4*d*e**4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5
*m**2 + 274*e**5*m + 120*e**5) + 154*a**4*d*e**4*m*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a**4*d*e
**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*
e**5*m + 120*e**5) + a**4*e**5*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 14*a**4*e**5*m**3*x*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 71*a**4*e**5*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*a**4*e**5*m*x*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5
) + 120*a**4*e**5*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) - 4*a**3*b*d**2*e**3*m**3*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 48
*a**3*b*d**2*e**3*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) - 188*a**3*b*d**2*e**3*m*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) -
240*a**3*b*d**2*e**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225
*e**5*m**2 + 274*e**5*m + 120*e**5) + 4*a**3*b*d*e**4*m**4*x*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 48
*a**3*b*d*e**4*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) + 188*a**3*b*d*e**4*m**2*x*(d + e*x)**m/(e*
*5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
240*a**3*b*d*e**4*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 4*a**3*b*e**5*m**4*x**2*(d + e*x)**m/(e*
*5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
52*a**3*b*e**5*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 236*a**3*b*e**5*m**2*x**2*(d + e*x)**
m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e*
*5) + 428*a**3*b*e**5*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m*
*3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 240*a**3*b*e**5*x**2*(d + e*x)**m/
(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5
) + 12*a**2*b**2*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 108*a**2*b**2*d**3*e**2*m*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 240*a**2*b**2*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*a**2*b**2*d**2*e**3*m**3
*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) - 108*a**2*b**2*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 240*a**2*b**
2*d**2*e**3*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5
*m**2 + 274*e**5*m + 120*e**5) + 6*a**2*b**2*d*e**4*m**4*x**2*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6
0*a**2*b**2*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m*
*3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 174*a**2*b**2*d*e**4*m**2*x**2*(d
+ e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) + 120*a**2*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*a**2*b**2*e**5*m**4
*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 27
4*e**5*m + 120*e**5) + 72*a**2*b**2*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 294*a**2*b**
2*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**
5*m**2 + 274*e**5*m + 120*e**5) + 468*a**2*b**2*e**5*m*x**3*(d + e*x)**m/(e**5*m
**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 240
*a**2*b**2*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225
*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*a*b**3*d**4*e*m*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 120*a*
b**3*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**
2 + 274*e**5*m + 120*e**5) + 24*a*b**3*d**3*e**2*m**2*x*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a*b
**3*d**3*e**2*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e*
*5*m**2 + 274*e**5*m + 120*e**5) - 12*a*b**3*d**2*e**3*m**3*x**2*(d + e*x)**m/(e
**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
- 72*a*b**3*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 60*a*b**3*d**2*e**3*m*x**2*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 4*a*b**3*d*e**4*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 32*a*b**3*d*e**4*m**3*x*
*3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) + 68*a*b**3*d*e**4*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*a*b**3*d*e**4
*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 4*a*b**3*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e*
*5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 44*a*b**3*e**5
*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 164*a*b**3*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 244*a*b**
3*e**5*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m
**2 + 274*e**5*m + 120*e**5) + 120*a*b**3*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15
*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*b**4*d**
5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e*
*5*m + 120*e**5) - 24*b**4*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 8
5*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b**4*d**3*e**2*m**2*x*
*2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) + 12*b**4*d**3*e**2*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*b**4*d**2*e**3*m
**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) - 12*b**4*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 8*b**4*d*
*2*e**3*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + b**4*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*b**4*d*
e**4*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 11*b**4*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*b**4
*d*e**4*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + b**4*e**5*m**4*x**5*(d + e*x)**m/(e**5*m**5 + 15
*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*b**4*e**
5*m**3*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**
2 + 274*e**5*m + 120*e**5) + 35*b**4*e**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15
*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 50*b**4*e**
5*m*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 24*b**4*e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211673, 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)^2*(e*x + d)^m,x, algorithm="giac")

[Out]

Done