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

Optimal. Leaf size=159 \[ \frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

[Out]

(d^2*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (2*d*(c*d - b*e)*(2*c*d -
b*e)*(d + e*x)^(2 + m))/(e^5*(2 + m)) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d +
e*x)^(3 + m))/(e^5*(3 + m)) - (2*c*(2*c*d - b*e)*(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 = 0.23841, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac{2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac{2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac{c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (2*d*(c*d - b*e)*(2*c*d -
b*e)*(d + e*x)^(2 + m))/(e^5*(2 + m)) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d +
e*x)^(3 + m))/(e^5*(3 + m)) - (2*c*(2*c*d - b*e)*(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 = 46.2054, size = 144, normalized size = 0.91 \[ \frac{c^{2} \left (d + e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} + \frac{2 c \left (d + e x\right )^{m + 4} \left (b e - 2 c d\right )}{e^{5} \left (m + 4\right )} + \frac{d^{2} \left (d + e x\right )^{m + 1} \left (b e - c d\right )^{2}}{e^{5} \left (m + 1\right )} - \frac{2 d \left (d + e x\right )^{m + 2} \left (b e - 2 c d\right ) \left (b e - c d\right )}{e^{5} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 3} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{5} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.227456, size = 230, normalized size = 1.45 \[ \frac{(d+e x)^{m+1} \left (b^2 e^2 \left (m^2+9 m+20\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+2 b c e (m+5) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+c^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )}{e^5 (m+1) (m+2) (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(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) + 2*b*c*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) + 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 = 0.014, size = 547, normalized size = 3.4 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ({c}^{2}{e}^{4}{m}^{4}{x}^{4}+2\,bc{e}^{4}{m}^{4}{x}^{3}+10\,{c}^{2}{e}^{4}{m}^{3}{x}^{4}+{b}^{2}{e}^{4}{m}^{4}{x}^{2}+22\,bc{e}^{4}{m}^{3}{x}^{3}-4\,{c}^{2}d{e}^{3}{m}^{3}{x}^{3}+35\,{c}^{2}{e}^{4}{m}^{2}{x}^{4}+12\,{b}^{2}{e}^{4}{m}^{3}{x}^{2}-6\,bcd{e}^{3}{m}^{3}{x}^{2}+82\,bc{e}^{4}{m}^{2}{x}^{3}-24\,{c}^{2}d{e}^{3}{m}^{2}{x}^{3}+50\,{c}^{2}{e}^{4}m{x}^{4}-2\,{b}^{2}d{e}^{3}{m}^{3}x+49\,{b}^{2}{e}^{4}{m}^{2}{x}^{2}-48\,bcd{e}^{3}{m}^{2}{x}^{2}+122\,bc{e}^{4}m{x}^{3}+12\,{c}^{2}{d}^{2}{e}^{2}{m}^{2}{x}^{2}-44\,{c}^{2}d{e}^{3}m{x}^{3}+24\,{c}^{2}{x}^{4}{e}^{4}-20\,{b}^{2}d{e}^{3}{m}^{2}x+78\,{b}^{2}{e}^{4}m{x}^{2}+12\,bc{d}^{2}{e}^{2}{m}^{2}x-102\,bcd{e}^{3}m{x}^{2}+60\,bc{e}^{4}{x}^{3}+36\,{c}^{2}{d}^{2}{e}^{2}m{x}^{2}-24\,{c}^{2}d{e}^{3}{x}^{3}+2\,{b}^{2}{d}^{2}{e}^{2}{m}^{2}-58\,{b}^{2}d{e}^{3}mx+40\,{b}^{2}{e}^{4}{x}^{2}+72\,bc{d}^{2}{e}^{2}mx-60\,bcd{e}^{3}{x}^{2}-24\,{c}^{2}{d}^{3}emx+24\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+18\,{b}^{2}{d}^{2}{e}^{2}m-40\,{b}^{2}d{e}^{3}x-12\,bc{d}^{3}em+60\,bc{d}^{2}{e}^{2}x-24\,{c}^{2}{d}^{3}ex+40\,{b}^{2}{d}^{2}{e}^{2}-60\,bc{d}^{3}e+24\,{c}^{2}{d}^{4} \right ) }{{e}^{5} \left ({m}^{5}+15\,{m}^{4}+85\,{m}^{3}+225\,{m}^{2}+274\,m+120 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x)^2,x)

[Out]

(e*x+d)^(1+m)*(c^2*e^4*m^4*x^4+2*b*c*e^4*m^4*x^3+10*c^2*e^4*m^3*x^4+b^2*e^4*m^4*
x^2+22*b*c*e^4*m^3*x^3-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+12*b^2*e^4*m^3*x^2
-6*b*c*d*e^3*m^3*x^2+82*b*c*e^4*m^2*x^3-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4-2*
b^2*d*e^3*m^3*x+49*b^2*e^4*m^2*x^2-48*b*c*d*e^3*m^2*x^2+122*b*c*e^4*m*x^3+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^4-20*b^2*d*e^3*m^2*x+78*b^2*e^4
*m*x^2+12*b*c*d^2*e^2*m^2*x-102*b*c*d*e^3*m*x^2+60*b*c*e^4*x^3+36*c^2*d^2*e^2*m*
x^2-24*c^2*d*e^3*x^3+2*b^2*d^2*e^2*m^2-58*b^2*d*e^3*m*x+40*b^2*e^4*x^2+72*b*c*d^
2*e^2*m*x-60*b*c*d*e^3*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+18*b^2*d^2*e^2*m-
40*b^2*d*e^3*x-12*b*c*d^3*e*m+60*b*c*d^2*e^2*x-24*c^2*d^3*e*x+40*b^2*d^2*e^2-60*
b*c*d^3*e+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 [A]  time = 0.737096, size = 429, normalized size = 2.7 \[ \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac{2 \,{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} +{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \,{\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )}{\left (e x + d\right )}^{m} b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac{{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} +{\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \,{\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \,{\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )}{\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^
m*b^2/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^
3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x
+ d)^m*b*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2
+ 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 +
2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^
m*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)

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Fricas [A]  time = 0.22747, size = 788, normalized size = 4.96 \[ \frac{{\left (2 \, b^{2} d^{3} e^{2} m^{2} + 24 \, c^{2} d^{5} - 60 \, b c d^{4} e + 40 \, b^{2} d^{3} e^{2} +{\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} +{\left (60 \, b c e^{5} +{\left (c^{2} d e^{4} + 2 \, b c e^{5}\right )} m^{4} + 2 \,{\left (3 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} m^{3} +{\left (11 \, c^{2} d e^{4} + 82 \, b c e^{5}\right )} m^{2} + 2 \,{\left (3 \, c^{2} d e^{4} + 61 \, b c e^{5}\right )} m\right )} x^{4} +{\left (40 \, b^{2} e^{5} +{\left (2 \, b c d e^{4} + b^{2} e^{5}\right )} m^{4} - 4 \,{\left (c^{2} d^{2} e^{3} - 4 \, b c d e^{4} - 3 \, b^{2} e^{5}\right )} m^{3} -{\left (12 \, c^{2} d^{2} e^{3} - 34 \, b c d e^{4} - 49 \, b^{2} e^{5}\right )} m^{2} - 2 \,{\left (4 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} - 39 \, b^{2} e^{5}\right )} m\right )} x^{3} +{\left (b^{2} d e^{4} m^{4} - 2 \,{\left (3 \, b c d^{2} e^{3} - 5 \, b^{2} d e^{4}\right )} m^{3} +{\left (12 \, c^{2} d^{3} e^{2} - 36 \, b c d^{2} e^{3} + 29 \, b^{2} d e^{4}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{3} e^{2} - 15 \, b c d^{2} e^{3} + 10 \, b^{2} d e^{4}\right )} m\right )} x^{2} - 6 \,{\left (2 \, b c d^{4} e - 3 \, b^{2} d^{3} e^{2}\right )} m - 2 \,{\left (b^{2} d^{2} e^{3} m^{3} - 3 \,{\left (2 \, b c d^{3} e^{2} - 3 \, b^{2} d^{2} e^{3}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{4} e - 15 \, b c d^{3} e^{2} + 10 \, b^{2} d^{2} e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*b^2*d^3*e^2*m^2 + 24*c^2*d^5 - 60*b*c*d^4*e + 40*b^2*d^3*e^2 + (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 + (60*b*c*e^5
+ (c^2*d*e^4 + 2*b*c*e^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)*m^3 + (11*c^2*d*e^4
+ 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*m)*x^4 + (40*b^2*e^5 + (2*b*c*
d*e^4 + b^2*e^5)*m^4 - 4*(c^2*d^2*e^3 - 4*b*c*d*e^4 - 3*b^2*e^5)*m^3 - (12*c^2*d
^2*e^3 - 34*b*c*d*e^4 - 49*b^2*e^5)*m^2 - 2*(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*b
^2*e^5)*m)*x^3 + (b^2*d*e^4*m^4 - 2*(3*b*c*d^2*e^3 - 5*b^2*d*e^4)*m^3 + (12*c^2*
d^3*e^2 - 36*b*c*d^2*e^3 + 29*b^2*d*e^4)*m^2 + 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3
+ 10*b^2*d*e^4)*m)*x^2 - 6*(2*b*c*d^4*e - 3*b^2*d^3*e^2)*m - 2*(b^2*d^2*e^3*m^3
- 3*(2*b*c*d^3*e^2 - 3*b^2*d^2*e^3)*m^2 + 2*(6*c^2*d^4*e - 15*b*c*d^3*e^2 + 10*
b^2*d^2*e^3)*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 = 17.6433, size = 6387, normalized size = 40.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((d**m*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-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) - 4*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) - 6*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) - 6*b*c*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) - 24*b
*c*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) - 36*b*c*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) - 24*b*c*e**4*x**3/(12*d**4*e**5 + 4
8*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) + 88*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) + 12*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)), (b**2*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) + 6*b*c*d**4*e*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) + 2*b*c*d**4*e/(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*b*c*d**3*e**2*x*l
og(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) + 1
8*b*c*d**2*e**3*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) - 9*b*c*d**2*e**3*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) + 6*b*c*d*e**4*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) - 9*b*c*d*e**4*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
)), (2*b**2*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + b*
*2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x*log(d/e
+ x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*log(d/e + x)/(2
*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2*b**2*e**4*x**2/(2*d**2*e**5 + 4*d*e**
6*x + 2*e**7*x**2) - 12*b*c*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) - 6*b*c*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e
**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e**3*x**2
*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*b*c*d*e**3*x**2/(2*d
**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b*c*e**4*x**3/(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*l
og(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*lo
g(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), E
q(m, -3)), (-6*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**2*e
**2/(3*d*e**5 + 3*e**6*x) - 6*b**2*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) +
3*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e*log(d/e + x)/(3*d*e**5 +
3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**2*e**2*x*log(d/e +
x)/(3*d*e**5 + 3*e**6*x) - 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e**4*
x**3/(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) - 1
2*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)), (b**2*d**2*
log(d/e + x)/e**3 - b**2*d*x/e**2 + b**2*x**2/(2*e) - 2*b*c*d**3*log(d/e + x)/e*
*4 + 2*b*c*d**2*x/e**3 - b*c*d*x**2/e**2 + 2*b*c*x**3/(3*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)), (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) + 18*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) + 40*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) - 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) - 18*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) - 40*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 + 27
4*e**5*m + 120*e**5) + 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) + 10*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 + 2
74*e**5*m + 120*e**5) + 29*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) + 20*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) + 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 + 274*e**5*m + 120*e**5) + 12*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) + 49*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) + 78*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) + 40*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) - 12*b*c*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) - 60*b*c*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) + 12*b*c*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) + 6
0*b*c*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) - 6*b*c*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) -
36*b*c*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) - 30*b*c*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) + 2*b*c*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) + 16*b*c*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) + 34*b*c*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) + 20*b*c*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) + 2*b*c*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) + 22*b*c*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)
+ 82*b*c*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) + 122*b*c*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) + 60
*b*c*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*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 + 2
74*e**5*m + 120*e**5), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207129, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done