3.2147 \(\int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)
Optimal. Leaf size=175 \[ -\frac{5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}-\frac{(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac{b^5 (d+e x)^{m+6}}{e^6 (m+6)} \]
[Out]
-(((b*d - a*e)^5*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x
)^(2 + m))/(e^6*(2 + m)) - (10*b^2*(b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)
) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (5*b^4*(b*d - a*e)*
(d + e*x)^(5 + m))/(e^6*(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + m))
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Rubi [A] time = 0.202351, antiderivative size = 175, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065
\[ -\frac{5 b^4 (b d-a e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac{10 b^3 (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4)}-\frac{10 b^2 (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3)}-\frac{(b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{5 b (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2)}+\frac{b^5 (d+e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
-(((b*d - a*e)^5*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (5*b*(b*d - a*e)^4*(d + e*x
)^(2 + m))/(e^6*(2 + m)) - (10*b^2*(b*d - a*e)^3*(d + e*x)^(3 + m))/(e^6*(3 + m)
) + (10*b^3*(b*d - a*e)^2*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (5*b^4*(b*d - a*e)*
(d + e*x)^(5 + m))/(e^6*(5 + m)) + (b^5*(d + e*x)^(6 + m))/(e^6*(6 + m))
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Rubi in Sympy [A] time = 111.258, size = 153, normalized size = 0.87 \[ \frac{b^{5} \left (d + e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} + \frac{5 b^{4} \left (d + e x\right )^{m + 5} \left (a e - b d\right )}{e^{6} \left (m + 5\right )} + \frac{10 b^{3} \left (d + e x\right )^{m + 4} \left (a e - b d\right )^{2}}{e^{6} \left (m + 4\right )} + \frac{10 b^{2} \left (d + e x\right )^{m + 3} \left (a e - b d\right )^{3}}{e^{6} \left (m + 3\right )} + \frac{5 b \left (d + e x\right )^{m + 2} \left (a e - b d\right )^{4}}{e^{6} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right )^{5}}{e^{6} \left (m + 1\right )} \]
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)**2,x)
[Out]
b**5*(d + e*x)**(m + 6)/(e**6*(m + 6)) + 5*b**4*(d + e*x)**(m + 5)*(a*e - b*d)/(
e**6*(m + 5)) + 10*b**3*(d + e*x)**(m + 4)*(a*e - b*d)**2/(e**6*(m + 4)) + 10*b*
*2*(d + e*x)**(m + 3)*(a*e - b*d)**3/(e**6*(m + 3)) + 5*b*(d + e*x)**(m + 2)*(a*
e - b*d)**4/(e**6*(m + 2)) + (d + e*x)**(m + 1)*(a*e - b*d)**5/(e**6*(m + 1))
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Mathematica [B] time = 0.67475, size = 449, normalized size = 2.57 \[ \frac{(d+e x)^{m+1} \left (a^5 e^5 \left (m^5+20 m^4+155 m^3+580 m^2+1044 m+720\right )-5 a^4 b e^4 \left (m^4+18 m^3+119 m^2+342 m+360\right ) (d-e (m+1) x)+10 a^3 b^2 e^3 \left (m^3+15 m^2+74 m+120\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+10 a^2 b^3 e^2 \left (m^2+11 m+30\right ) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+5 a b^4 e (m+6) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+b^5 \left (-\left (120 d^5-120 d^4 e (m+1) x+60 d^3 e^2 \left (m^2+3 m+2\right ) x^2-20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(a^5*e^5*(720 + 1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5) -
5*a^4*b*e^4*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*(d - e*(1 + m)*x) + 10*a^3*b^
2*e^3*(120 + 74*m + 15*m^2 + m^3)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)
*x^2) + 10*a^2*b^3*e^2*(30 + 11*m + m^2)*(-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) + 5*a*b^4*e*(6 + m)*(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) - b^5*(120*d^5 -
120*d^4*e*(1 + m)*x + 60*d^3*e^2*(2 + 3*m + m^2)*x^2 - 20*d^2*e^3*(6 + 11*m + 6*
m^2 + m^3)*x^3 + 5*d*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - e^5*(120 + 27
4*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5)))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 +
m)*(5 + m)*(6 + m))
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Maple [B] time = 0.016, size = 1345, normalized size = 7.7 \[ \text{result too large to display} \]
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)^2,x)
[Out]
(e*x+d)^(1+m)*(b^5*e^5*m^5*x^5+5*a*b^4*e^5*m^5*x^4+15*b^5*e^5*m^4*x^5+10*a^2*b^3
*e^5*m^5*x^3+80*a*b^4*e^5*m^4*x^4-5*b^5*d*e^4*m^4*x^4+85*b^5*e^5*m^3*x^5+10*a^3*
b^2*e^5*m^5*x^2+170*a^2*b^3*e^5*m^4*x^3-20*a*b^4*d*e^4*m^4*x^3+475*a*b^4*e^5*m^3
*x^4-50*b^5*d*e^4*m^3*x^4+225*b^5*e^5*m^2*x^5+5*a^4*b*e^5*m^5*x+180*a^3*b^2*e^5*
m^4*x^2-30*a^2*b^3*d*e^4*m^4*x^2+1070*a^2*b^3*e^5*m^3*x^3-240*a*b^4*d*e^4*m^3*x^
3+1300*a*b^4*e^5*m^2*x^4+20*b^5*d^2*e^3*m^3*x^3-175*b^5*d*e^4*m^2*x^4+274*b^5*e^
5*m*x^5+a^5*e^5*m^5+95*a^4*b*e^5*m^4*x-20*a^3*b^2*d*e^4*m^4*x+1210*a^3*b^2*e^5*m
^3*x^2-420*a^2*b^3*d*e^4*m^3*x^2+3070*a^2*b^3*e^5*m^2*x^3+60*a*b^4*d^2*e^3*m^3*x
^2-940*a*b^4*d*e^4*m^2*x^3+1620*a*b^4*e^5*m*x^4+120*b^5*d^2*e^3*m^2*x^3-250*b^5*
d*e^4*m*x^4+120*b^5*e^5*x^5+20*a^5*e^5*m^4-5*a^4*b*d*e^4*m^4+685*a^4*b*e^5*m^3*x
-320*a^3*b^2*d*e^4*m^3*x+3720*a^3*b^2*e^5*m^2*x^2+60*a^2*b^3*d^2*e^3*m^3*x-1950*
a^2*b^3*d*e^4*m^2*x^2+3960*a^2*b^3*e^5*m*x^3+540*a*b^4*d^2*e^3*m^2*x^2-1440*a*b^
4*d*e^4*m*x^3+720*a*b^4*e^5*x^4-60*b^5*d^3*e^2*m^2*x^2+220*b^5*d^2*e^3*m*x^3-120
*b^5*d*e^4*x^4+155*a^5*e^5*m^3-90*a^4*b*d*e^4*m^3+2305*a^4*b*e^5*m^2*x+20*a^3*b^
2*d^2*e^3*m^3-1780*a^3*b^2*d*e^4*m^2*x+5080*a^3*b^2*e^5*m*x^2+720*a^2*b^3*d^2*e^
3*m^2*x-3360*a^2*b^3*d*e^4*m*x^2+1800*a^2*b^3*e^5*x^3-120*a*b^4*d^3*e^2*m^2*x+12
00*a*b^4*d^2*e^3*m*x^2-720*a*b^4*d*e^4*x^3-180*b^5*d^3*e^2*m*x^2+120*b^5*d^2*e^3
*x^3+580*a^5*e^5*m^2-595*a^4*b*d*e^4*m^2+3510*a^4*b*e^5*m*x+300*a^3*b^2*d^2*e^3*
m^2-3880*a^3*b^2*d*e^4*m*x+2400*a^3*b^2*e^5*x^2-60*a^2*b^3*d^3*e^2*m^2+2460*a^2*
b^3*d^2*e^3*m*x-1800*a^2*b^3*d*e^4*x^2-840*a*b^4*d^3*e^2*m*x+720*a*b^4*d^2*e^3*x
^2+120*b^5*d^4*e*m*x-120*b^5*d^3*e^2*x^2+1044*a^5*e^5*m-1710*a^4*b*d*e^4*m+1800*
a^4*b*e^5*x+1480*a^3*b^2*d^2*e^3*m-2400*a^3*b^2*d*e^4*x-660*a^2*b^3*d^3*e^2*m+18
00*a^2*b^3*d^2*e^3*x+120*a*b^4*d^4*e*m-720*a*b^4*d^3*e^2*x+120*b^5*d^4*e*x+720*a
^5*e^5-1800*a^4*b*d*e^4+2400*a^3*b^2*d^2*e^3-1800*a^2*b^3*d^3*e^2+720*a*b^4*d^4*
e-120*b^5*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)
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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*(b*x + a)*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.321021, size = 1971, normalized size = 11.26 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(b*x + a)*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a^5*d*e^5*m^5 - 120*b^5*d^6 + 720*a*b^4*d^5*e - 1800*a^2*b^3*d^4*e^2 + 2400*a^3
*b^2*d^3*e^3 - 1800*a^4*b*d^2*e^4 + 720*a^5*d*e^5 + (b^5*e^6*m^5 + 15*b^5*e^6*m^
4 + 85*b^5*e^6*m^3 + 225*b^5*e^6*m^2 + 274*b^5*e^6*m + 120*b^5*e^6)*x^6 + (720*a
*b^4*e^6 + (b^5*d*e^5 + 5*a*b^4*e^6)*m^5 + 10*(b^5*d*e^5 + 8*a*b^4*e^6)*m^4 + 5*
(7*b^5*d*e^5 + 95*a*b^4*e^6)*m^3 + 50*(b^5*d*e^5 + 26*a*b^4*e^6)*m^2 + 12*(2*b^5
*d*e^5 + 135*a*b^4*e^6)*m)*x^5 - 5*(a^4*b*d^2*e^4 - 4*a^5*d*e^5)*m^4 + 5*(360*a^
2*b^3*e^6 + (a*b^4*d*e^5 + 2*a^2*b^3*e^6)*m^5 - (b^5*d^2*e^4 - 12*a*b^4*d*e^5 -
34*a^2*b^3*e^6)*m^4 - (6*b^5*d^2*e^4 - 47*a*b^4*d*e^5 - 214*a^2*b^3*e^6)*m^3 - (
11*b^5*d^2*e^4 - 72*a*b^4*d*e^5 - 614*a^2*b^3*e^6)*m^2 - 6*(b^5*d^2*e^4 - 6*a*b^
4*d*e^5 - 132*a^2*b^3*e^6)*m)*x^4 + 5*(4*a^3*b^2*d^3*e^3 - 18*a^4*b*d^2*e^4 + 31
*a^5*d*e^5)*m^3 + 10*(240*a^3*b^2*e^6 + (a^2*b^3*d*e^5 + a^3*b^2*e^6)*m^5 - 2*(a
*b^4*d^2*e^4 - 7*a^2*b^3*d*e^5 - 9*a^3*b^2*e^6)*m^4 + (2*b^5*d^3*e^3 - 18*a*b^4*
d^2*e^4 + 65*a^2*b^3*d*e^5 + 121*a^3*b^2*e^6)*m^3 + 2*(3*b^5*d^3*e^3 - 20*a*b^4*
d^2*e^4 + 56*a^2*b^3*d*e^5 + 186*a^3*b^2*e^6)*m^2 + 4*(b^5*d^3*e^3 - 6*a*b^4*d^2
*e^4 + 15*a^2*b^3*d*e^5 + 127*a^3*b^2*e^6)*m)*x^3 - 5*(12*a^2*b^3*d^4*e^2 - 60*a
^3*b^2*d^3*e^3 + 119*a^4*b*d^2*e^4 - 116*a^5*d*e^5)*m^2 + 5*(360*a^4*b*e^6 + (2*
a^3*b^2*d*e^5 + a^4*b*e^6)*m^5 - (6*a^2*b^3*d^2*e^4 - 32*a^3*b^2*d*e^5 - 19*a^4*
b*e^6)*m^4 + (12*a*b^4*d^3*e^3 - 72*a^2*b^3*d^2*e^4 + 178*a^3*b^2*d*e^5 + 137*a^
4*b*e^6)*m^3 - (12*b^5*d^4*e^2 - 84*a*b^4*d^3*e^3 + 246*a^2*b^3*d^2*e^4 - 388*a^
3*b^2*d*e^5 - 461*a^4*b*e^6)*m^2 - 6*(2*b^5*d^4*e^2 - 12*a*b^4*d^3*e^3 + 30*a^2*
b^3*d^2*e^4 - 40*a^3*b^2*d*e^5 - 117*a^4*b*e^6)*m)*x^2 + 2*(60*a*b^4*d^5*e - 330
*a^2*b^3*d^4*e^2 + 740*a^3*b^2*d^3*e^3 - 855*a^4*b*d^2*e^4 + 522*a^5*d*e^5)*m +
(720*a^5*e^6 + (5*a^4*b*d*e^5 + a^5*e^6)*m^5 - 10*(2*a^3*b^2*d^2*e^4 - 9*a^4*b*d
*e^5 - 2*a^5*e^6)*m^4 + 5*(12*a^2*b^3*d^3*e^3 - 60*a^3*b^2*d^2*e^4 + 119*a^4*b*d
*e^5 + 31*a^5*e^6)*m^3 - 10*(12*a*b^4*d^4*e^2 - 66*a^2*b^3*d^3*e^3 + 148*a^3*b^2
*d^2*e^4 - 171*a^4*b*d*e^5 - 58*a^5*e^6)*m^2 + 12*(10*b^5*d^5*e - 60*a*b^4*d^4*e
^2 + 150*a^2*b^3*d^3*e^3 - 200*a^3*b^2*d^2*e^4 + 150*a^4*b*d*e^5 + 87*a^5*e^6)*m
)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^
2 + 1764*e^6*m + 720*e^6)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.301985, 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*(b*x + a)*(e*x + d)^m,x, algorithm="giac")
[Out]
Done