3.711 \(\int (d+e x)^m \left (a+c x^2\right )^2 \, dx\)
Optimal. Leaf size=140 \[ \frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac{2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
[Out]
((c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*c*d*(c*d^2 + a*e^2)*(d
+ e*x)^(2 + m))/(e^5*(2 + m)) + (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^5*(
3 + m)) - (4*c^2*d*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/(e
^5*(5 + m))
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Rubi [A] time = 0.17888, antiderivative size = 140, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059
\[ \frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac{2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac{4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
((c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*c*d*(c*d^2 + a*e^2)*(d
+ e*x)^(2 + m))/(e^5*(2 + m)) + (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^5*(
3 + m)) - (4*c^2*d*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/(e
^5*(5 + m))
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Rubi in Sympy [A] time = 37.968, size = 128, normalized size = 0.91 \[ - \frac{4 c^{2} d \left (d + e x\right )^{m + 4}}{e^{5} \left (m + 4\right )} + \frac{c^{2} \left (d + e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} - \frac{4 c d \left (d + e x\right )^{m + 2} \left (a e^{2} + c d^{2}\right )}{e^{5} \left (m + 2\right )} + \frac{2 c \left (d + e x\right )^{m + 3} \left (a e^{2} + 3 c d^{2}\right )}{e^{5} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} + c d^{2}\right )^{2}}{e^{5} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+a)**2,x)
[Out]
-4*c**2*d*(d + e*x)**(m + 4)/(e**5*(m + 4)) + c**2*(d + e*x)**(m + 5)/(e**5*(m +
5)) - 4*c*d*(d + e*x)**(m + 2)*(a*e**2 + c*d**2)/(e**5*(m + 2)) + 2*c*(d + e*x)
**(m + 3)*(a*e**2 + 3*c*d**2)/(e**5*(m + 3)) + (d + e*x)**(m + 1)*(a*e**2 + c*d*
*2)**2/(e**5*(m + 1))
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Mathematica [A] time = 0.186281, size = 194, normalized size = 1.39 \[ \frac{(d+e x)^{m+1} \left (a^2 e^4 \left (m^4+14 m^3+71 m^2+154 m+120\right )+2 a c e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+c^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )}{e^5 (m+1) (m+2) (m+3) (m+4) (m+5)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m*(a + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(a^2*e^4*(120 + 154*m + 71*m^2 + 14*m^3 + m^4) + 2*a*c*e^2*(2
0 + 9*m + m^2)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + c^2*(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 = 0.013, size = 420, normalized size = 3. \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({c}^{2}{e}^{4}{m}^{4}{x}^{4}+10\,{c}^{2}{e}^{4}{m}^{3}{x}^{4}+2\,ac{e}^{4}{m}^{4}{x}^{2}-4\,{c}^{2}d{e}^{3}{m}^{3}{x}^{3}+35\,{c}^{2}{e}^{4}{m}^{2}{x}^{4}+24\,ac{e}^{4}{m}^{3}{x}^{2}-24\,{c}^{2}d{e}^{3}{m}^{2}{x}^{3}+50\,{c}^{2}{e}^{4}m{x}^{4}+{a}^{2}{e}^{4}{m}^{4}-4\,acd{e}^{3}{m}^{3}x+98\,ac{e}^{4}{m}^{2}{x}^{2}+12\,{c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{c}^{2}d{e}^{3}m{x}^{3}+24\,{x}^{4}{c}^{2}{e}^{4}+14\,{a}^{2}{e}^{4}{m}^{3}-40\,acd{e}^{3}{m}^{2}x+156\,ac{e}^{4}m{x}^{2}+36\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}-24\,{x}^{3}{c}^{2}d{e}^{3}+71\,{a}^{2}{e}^{4}{m}^{2}+4\,ac{d}^{2}{e}^{2}{m}^{2}-116\,acd{e}^{3}mx+80\,{x}^{2}ac{e}^{4}-24\,{c}^{2}{d}^{3}emx+24\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+154\,{a}^{2}{e}^{4}m+36\,ac{d}^{2}{e}^{2}m-80\,xacd{e}^{3}-24\,x{c}^{2}{d}^{3}e+120\,{a}^{2}{e}^{4}+80\,ac{d}^{2}{e}^{2}+24\,{c}^{2}{d}^{4} \right ) }{{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+a)^2,x)
[Out]
(e*x+d)^(1+m)*(c^2*e^4*m^4*x^4+10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^4*x^2-4*c^2*d*e^3*
m^3*x^3+35*c^2*e^4*m^2*x^4+24*a*c*e^4*m^3*x^2-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*
x^4+a^2*e^4*m^4-4*a*c*d*e^3*m^3*x+98*a*c*e^4*m^2*x^2+12*c^2*d^2*e^2*m^2*x^2-44*c
^2*d*e^3*m*x^3+24*c^2*e^4*x^4+14*a^2*e^4*m^3-40*a*c*d*e^3*m^2*x+156*a*c*e^4*m*x^
2+36*c^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2+4*a*c*d^2*e^2*m^2-116*a*c
*d*e^3*m*x+80*a*c*e^4*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4*m+36*a
*c*d^2*e^2*m-80*a*c*d*e^3*x-24*c^2*d^3*e*x+120*a^2*e^4+80*a*c*d^2*e^2+24*c^2*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((c*x^2 + a)^2*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.240504, size = 702, normalized size = 5.01 \[ \frac{{\left (a^{2} d e^{4} m^{4} + 14 \, a^{2} d e^{4} m^{3} + 24 \, c^{2} d^{5} + 80 \, a c d^{3} e^{2} + 120 \, a^{2} d e^{4} +{\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} +{\left (c^{2} d e^{4} m^{4} + 6 \, c^{2} d e^{4} m^{3} + 11 \, c^{2} d e^{4} m^{2} + 6 \, c^{2} d e^{4} m\right )} x^{4} + 2 \,{\left (a c e^{5} m^{4} + 40 \, a c e^{5} - 2 \,{\left (c^{2} d^{2} e^{3} - 6 \, a c e^{5}\right )} m^{3} -{\left (6 \, c^{2} d^{2} e^{3} - 49 \, a c e^{5}\right )} m^{2} - 2 \,{\left (2 \, c^{2} d^{2} e^{3} - 39 \, a c e^{5}\right )} m\right )} x^{3} +{\left (4 \, a c d^{3} e^{2} + 71 \, a^{2} d e^{4}\right )} m^{2} + 2 \,{\left (a c d e^{4} m^{4} + 10 \, a c d e^{4} m^{3} +{\left (6 \, c^{2} d^{3} e^{2} + 29 \, a c d e^{4}\right )} m^{2} + 2 \,{\left (3 \, c^{2} d^{3} e^{2} + 10 \, a c d e^{4}\right )} m\right )} x^{2} + 2 \,{\left (18 \, a c d^{3} e^{2} + 77 \, a^{2} d e^{4}\right )} m +{\left (a^{2} e^{5} m^{4} + 120 \, a^{2} e^{5} - 2 \,{\left (2 \, a c d^{2} e^{3} - 7 \, a^{2} e^{5}\right )} m^{3} -{\left (36 \, a c d^{2} e^{3} - 71 \, a^{2} e^{5}\right )} m^{2} - 2 \,{\left (12 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 77 \, a^{2} e^{5}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a^2*d*e^4*m^4 + 14*a^2*d*e^4*m^3 + 24*c^2*d^5 + 80*a*c*d^3*e^2 + 120*a^2*d*e^4
+ (c^2*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^
5 + (c^2*d*e^4*m^4 + 6*c^2*d*e^4*m^3 + 11*c^2*d*e^4*m^2 + 6*c^2*d*e^4*m)*x^4 + 2
*(a*c*e^5*m^4 + 40*a*c*e^5 - 2*(c^2*d^2*e^3 - 6*a*c*e^5)*m^3 - (6*c^2*d^2*e^3 -
49*a*c*e^5)*m^2 - 2*(2*c^2*d^2*e^3 - 39*a*c*e^5)*m)*x^3 + (4*a*c*d^3*e^2 + 71*a^
2*d*e^4)*m^2 + 2*(a*c*d*e^4*m^4 + 10*a*c*d*e^4*m^3 + (6*c^2*d^3*e^2 + 29*a*c*d*e
^4)*m^2 + 2*(3*c^2*d^3*e^2 + 10*a*c*d*e^4)*m)*x^2 + 2*(18*a*c*d^3*e^2 + 77*a^2*d
*e^4)*m + (a^2*e^5*m^4 + 120*a^2*e^5 - 2*(2*a*c*d^2*e^3 - 7*a^2*e^5)*m^3 - (36*a
*c*d^2*e^3 - 71*a^2*e^5)*m^2 - 2*(12*c^2*d^4*e + 40*a*c*d^2*e^3 - 77*a^2*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 + 1
20*e^5)
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Sympy [A] time = 16.3539, size = 5044, normalized size = 36.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m*(c*x**2+a)**2,x)
[Out]
Piecewise((d**m*(a**2*x + 2*a*c*x**3/3 + c**2*x**5/5), Eq(e, 0)), (-3*a**2*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) - 2*a*c*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) - 8*a*c*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) - 12*a*c*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*c**2*
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*c**2*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*c**2*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) + 8
8*c**2*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*c**2*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*c**2*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*c**2*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*c**2*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) + 1
2*c**2*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**2*d*e**4/(3*d**4*e**5 + 9*d*
*3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 2*a*c*e**5*x**3/(3*d**4*e**5 + 9
*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 12*c**2*d**5*log(d/e + x)/(3*
d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 4*c**2*d**5/(3*d
**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) - 36*c**2*d**4*e*x*
log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) -
36*c**2*d**3*e**2*x**2*log(d/e + x)/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x
**2 + 3*d*e**8*x**3) + 18*c**2*d**3*e**2*x**2/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d
**2*e**7*x**2 + 3*d*e**8*x**3) - 12*c**2*d**2*e**3*x**3*log(d/e + x)/(3*d**4*e**
5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 18*c**2*d**2*e**3*x**3/(
3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3) + 3*c**2*d*e**4*
x**4/(3*d**4*e**5 + 9*d**3*e**6*x + 9*d**2*e**7*x**2 + 3*d*e**8*x**3), Eq(m, -4)
), (-a**2*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*d**2*e**2*log(d/
e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*a*c*d**2*e**2/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e
**7*x**2) - 4*a*c*e**4*x**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d
**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*c**2*d**4/(2*d**2*
e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4
*d*e**6*x + 2*e**7*x**2) - 12*c**2*d**2*e**2*x**2/(2*d**2*e**5 + 4*d*e**6*x + 2*
e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c**2*
e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-3*a**2*e**4/(3
*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a
*c*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 12*a*c*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*
e**6*x) + 6*a*c*e**4*x**2/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/(3*d
*e**5 + 3*e**6*x) - 12*c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**3*e*x*log(d/
e + x)/(3*d*e**5 + 3*e**6*x) + 6*c**2*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*c
**2*d*e**3*x**3/(3*d*e**5 + 3*e**6*x) + c**2*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq
(m, -2)), (a**2*log(d/e + x)/e + 2*a*c*d**2*log(d/e + x)/e**3 - 2*a*c*d*x/e**2 +
a*c*x**2/e + c**2*d**4*log(d/e + x)/e**5 - c**2*d**3*x/e**4 + c**2*d**2*x**2/(2
*e**3) - c**2*d*x**3/(3*e**2) + c**2*x**4/(4*e), Eq(m, -1)), (a**2*d*e**4*m**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) + 14*a**2*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**2*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 + 12
0*e**5) + 154*a**2*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**2*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*
*2*e**5*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) + 14*a**2*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**2*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**2*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**2*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*c*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) + 36*a*c*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) + 80*a*c*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) - 4*a*c*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)
- 36*a*c*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) - 80*a*c*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) +
2*a*c*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) + 20*a*c*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)
+ 58*a*c*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) + 40*a*c*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) +
2*a*c*e**5*m**4*x**3*(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) + 24*a*c*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) + 98
*a*c*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) + 156*a*c*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) + 80*a*c
*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*c**2*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*c**2*d**4*e*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*c**2*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*c**2*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*c**2*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*c**2*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*c**2*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) + c**2*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*c**2*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*c**2*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*c**2*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) + c**2*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*c**2*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*c**2*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*c**2*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*c**2*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 = 0.216341, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^m,x, algorithm="giac")
[Out]
Done