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3.2151 \(\int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=395 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1) (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2) (a+b x)}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3) (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+7}}{e^7 (m+7) (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+6}}{e^7 (m+6) (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5) (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4) (a+b x)} \]

[Out]

((b*d - a*e)^6*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(1 + m)*(a
+ b*x)) - (6*b*(b*d - a*e)^5*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e
^7*(2 + m)*(a + b*x)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^7*(3 + m)*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(4 + m)*(a + b*x)) + (15*b^4*(b*d - a*e)^2*
(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(5 + m)*(a + b*x)) - (6*b^
5*(b*d - a*e)*(d + e*x)^(6 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(6 + m)*(a +
 b*x)) + (b^6*(d + e*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(7 + m)*(a +
 b*x))

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Rubi [A]  time = 0.541218, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6 (d+e x)^{m+1}}{e^7 (m+1) (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+2}}{e^7 (m+2) (a+b x)}+\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+3}}{e^7 (m+3) (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+7}}{e^7 (m+7) (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+6}}{e^7 (m+6) (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+5}}{e^7 (m+5) (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+4}}{e^7 (m+4) (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(1 + m)*(a
+ b*x)) - (6*b*(b*d - a*e)^5*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e
^7*(2 + m)*(a + b*x)) + (15*b^2*(b*d - a*e)^4*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^7*(3 + m)*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^(4 + m)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(4 + m)*(a + b*x)) + (15*b^4*(b*d - a*e)^2*
(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(5 + m)*(a + b*x)) - (6*b^
5*(b*d - a*e)*(d + e*x)^(6 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(6 + m)*(a +
 b*x)) + (b^6*(d + e*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(7 + m)*(a +
 b*x))

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Rubi in Sympy [A]  time = 175.489, size = 405, normalized size = 1.03 \[ \frac{\left (a + b x\right ) \left (d + e x\right )^{m + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (m + 7\right )} + \frac{6 \left (d + e x\right )^{m + 1} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e^{2} \left (m + 6\right ) \left (m + 7\right )} + \frac{6 \left (5 a + 5 b x\right ) \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{3} \left (m + 5\right ) \left (m + 6\right ) \left (m + 7\right )} + \frac{120 \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4} \left (m + 4\right ) \left (m + 5\right ) \left (m + 6\right ) \left (m + 7\right )} + \frac{120 \left (3 a + 3 b x\right ) \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5} \left (m + 3\right ) \left (m + 4\right ) \left (m + 5\right ) \left (m + 6\right ) \left (m + 7\right )} + \frac{720 \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6} \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right ) \left (m + 5\right ) \left (m + 6\right ) \left (m + 7\right )} + \frac{720 \left (d + e x\right )^{m + 1} \left (a e - b d\right )^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{7} \left (a + b x\right ) \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right ) \left (m + 4\right ) \left (m + 5\right ) \left (m + 6\right ) \left (m + 7\right )} \]

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)**(5/2),x)

[Out]

(a + b*x)*(d + e*x)**(m + 1)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e*(m + 7)) + 6
*(d + e*x)**(m + 1)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e**2*(m + 6
)*(m + 7)) + 6*(5*a + 5*b*x)*(d + e*x)**(m + 1)*(a*e - b*d)**2*(a**2 + 2*a*b*x +
 b**2*x**2)**(3/2)/(e**3*(m + 5)*(m + 6)*(m + 7)) + 120*(d + e*x)**(m + 1)*(a*e
- b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(e**4*(m + 4)*(m + 5)*(m + 6)*(m +
 7)) + 120*(3*a + 3*b*x)*(d + e*x)**(m + 1)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x +
 b**2*x**2)/(e**5*(m + 3)*(m + 4)*(m + 5)*(m + 6)*(m + 7)) + 720*(d + e*x)**(m +
 1)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(e**6*(m + 2)*(m + 3)*(m + 4
)*(m + 5)*(m + 6)*(m + 7)) + 720*(d + e*x)**(m + 1)*(a*e - b*d)**6*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(e**7*(a + b*x)*(m + 1)*(m + 2)*(m + 3)*(m + 4)*(m + 5)*(m +
 6)*(m + 7))

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Mathematica [A]  time = 1.44366, size = 664, normalized size = 1.68 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} \left (a^6 e^6 \left (m^6+27 m^5+295 m^4+1665 m^3+5104 m^2+8028 m+5040\right )-6 a^5 b e^5 \left (m^5+25 m^4+245 m^3+1175 m^2+2754 m+2520\right ) (d-e (m+1) x)+15 a^4 b^2 e^4 \left (m^4+22 m^3+179 m^2+638 m+840\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+20 a^3 b^3 e^3 \left (m^3+18 m^2+107 m+210\right ) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+15 a^2 b^4 e^2 \left (m^2+13 m+42\right ) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+6 a b^5 e (m+7) \left (-120 d^5+120 d^4 e (m+1) x-60 d^3 e^2 \left (m^2+3 m+2\right ) x^2+20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3-5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )+b^6 \left (720 d^6-720 d^5 e (m+1) x+360 d^4 e^2 \left (m^2+3 m+2\right ) x^2-120 d^3 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+30 d^2 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-6 d e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right )}{e^7 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*(a^6*e^6*(5040 + 8028*m + 5104*m^2 + 1665*m
^3 + 295*m^4 + 27*m^5 + m^6) - 6*a^5*b*e^5*(2520 + 2754*m + 1175*m^2 + 245*m^3 +
 25*m^4 + m^5)*(d - e*(1 + m)*x) + 15*a^4*b^2*e^4*(840 + 638*m + 179*m^2 + 22*m^
3 + m^4)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 20*a^3*b^3*e^3*(2
10 + 107*m + 18*m^2 + m^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) + 15*a^2*b^4*e^2*(42 + 13*m + m^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) + 6*a*b^5*e*(7 +
 m)*(-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 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5) + b^6*(720*d^6 - 720
*d^5*e*(1 + m)*x + 360*d^4*e^2*(2 + 3*m + m^2)*x^2 - 120*d^3*e^3*(6 + 11*m + 6*m
^2 + m^3)*x^3 + 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e^5*(12
0 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2
+ 735*m^3 + 175*m^4 + 21*m^5 + m^6)*x^6)))/(e^7*(1 + m)*(2 + m)*(3 + m)*(4 + m)*
(5 + m)*(6 + m)*(7 + m)*(a + b*x))

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Maple [B]  time = 0.021, size = 2173, normalized size = 5.5 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

(e*x+d)^(1+m)*(b^6*e^6*m^6*x^6+6*a*b^5*e^6*m^6*x^5+21*b^6*e^6*m^5*x^6+15*a^2*b^4
*e^6*m^6*x^4+132*a*b^5*e^6*m^5*x^5-6*b^6*d*e^5*m^5*x^5+175*b^6*e^6*m^4*x^6+20*a^
3*b^3*e^6*m^6*x^3+345*a^2*b^4*e^6*m^5*x^4-30*a*b^5*d*e^5*m^5*x^4+1140*a*b^5*e^6*
m^4*x^5-90*b^6*d*e^5*m^4*x^5+735*b^6*e^6*m^3*x^6+15*a^4*b^2*e^6*m^6*x^2+480*a^3*
b^3*e^6*m^5*x^3-60*a^2*b^4*d*e^5*m^5*x^3+3105*a^2*b^4*e^6*m^4*x^4-510*a*b^5*d*e^
5*m^4*x^4+4920*a*b^5*e^6*m^3*x^5+30*b^6*d^2*e^4*m^4*x^4-510*b^6*d*e^5*m^3*x^5+16
24*b^6*e^6*m^2*x^6+6*a^5*b*e^6*m^6*x+375*a^4*b^2*e^6*m^5*x^2-60*a^3*b^3*d*e^5*m^
5*x^2+4520*a^3*b^3*e^6*m^4*x^3-1140*a^2*b^4*d*e^5*m^4*x^3+13875*a^2*b^4*e^6*m^3*
x^4+120*a*b^5*d^2*e^4*m^4*x^3-3150*a*b^5*d*e^5*m^3*x^4+11094*a*b^5*e^6*m^2*x^5+3
00*b^6*d^2*e^4*m^3*x^4-1350*b^6*d*e^5*m^2*x^5+1764*b^6*e^6*m*x^6+a^6*e^6*m^6+156
*a^5*b*e^6*m^5*x-30*a^4*b^2*d*e^5*m^5*x+3705*a^4*b^2*e^6*m^4*x^2-1260*a^3*b^3*d*
e^5*m^4*x^2+21120*a^3*b^3*e^6*m^3*x^3+180*a^2*b^4*d^2*e^4*m^4*x^2-7860*a^2*b^4*d
*e^5*m^3*x^3+32160*a^2*b^4*e^6*m^2*x^4+1560*a*b^5*d^2*e^4*m^3*x^3-8850*a*b^5*d*e
^5*m^2*x^4+12228*a*b^5*e^6*m*x^5-120*b^6*d^3*e^3*m^3*x^3+1050*b^6*d^2*e^4*m^2*x^
4-1644*b^6*d*e^5*m*x^5+720*b^6*e^6*x^6+27*a^6*e^6*m^5-6*a^5*b*d*e^5*m^5+1620*a^5
*b*e^6*m^4*x-690*a^4*b^2*d*e^5*m^4*x+18285*a^4*b^2*e^6*m^3*x^2+120*a^3*b^3*d^2*e
^4*m^4*x-9780*a^3*b^3*d*e^5*m^3*x^2+50900*a^3*b^3*e^6*m^2*x^3+2880*a^2*b^4*d^2*e
^4*m^3*x^2-24060*a^2*b^4*d*e^5*m^2*x^3+36180*a^2*b^4*e^6*m*x^4-360*a*b^5*d^3*e^3
*m^3*x^2+6360*a*b^5*d^2*e^4*m^2*x^3-11220*a*b^5*d*e^5*m*x^4+5040*a*b^5*e^6*x^5-7
20*b^6*d^3*e^3*m^2*x^3+1500*b^6*d^2*e^4*m*x^4-720*b^6*d*e^5*x^5+295*a^6*e^6*m^4-
150*a^5*b*d*e^5*m^4+8520*a^5*b*e^6*m^3*x+30*a^4*b^2*d^2*e^4*m^4-6030*a^4*b^2*d*e
^5*m^3*x+46680*a^4*b^2*e^6*m^2*x^2+2280*a^3*b^3*d^2*e^4*m^3*x-34020*a^3*b^3*d*e^
5*m^2*x^2+59040*a^3*b^3*e^6*m*x^3-360*a^2*b^4*d^3*e^3*m^3*x+14940*a^2*b^4*d^2*e^
4*m^2*x^2-32400*a^2*b^4*d*e^5*m*x^3+15120*a^2*b^4*e^6*x^4-3600*a*b^5*d^3*e^3*m^2
*x^2+9960*a*b^5*d^2*e^4*m*x^3-5040*a*b^5*d*e^5*x^4+360*b^6*d^4*e^2*m^2*x^2-1320*
b^6*d^3*e^3*m*x^3+720*b^6*d^2*e^4*x^4+1665*a^6*e^6*m^3-1470*a^5*b*d*e^5*m^3+2357
4*a^5*b*e^6*m^2*x+660*a^4*b^2*d^2*e^4*m^3-24510*a^4*b^2*d*e^5*m^2*x+56940*a^4*b^
2*e^6*m*x^2-120*a^3*b^3*d^3*e^3*m^3+15000*a^3*b^3*d^2*e^4*m^2*x-50640*a^3*b^3*d*
e^5*m*x^2+25200*a^3*b^3*e^6*x^3-5040*a^2*b^4*d^3*e^3*m^2*x+27360*a^2*b^4*d^2*e^4
*m*x^2-15120*a^2*b^4*d*e^5*x^3+720*a*b^5*d^4*e^2*m^2*x-8280*a*b^5*d^3*e^3*m*x^2+
5040*a*b^5*d^2*e^4*x^3+1080*b^6*d^4*e^2*m*x^2-720*b^6*d^3*e^3*x^3+5104*a^6*e^6*m
^2-7050*a^5*b*d*e^5*m^2+31644*a^5*b*e^6*m*x+5370*a^4*b^2*d^2*e^4*m^2-44340*a^4*b
^2*d*e^5*m*x+25200*a^4*b^2*e^6*x^2-2160*a^3*b^3*d^3*e^3*m^2+38040*a^3*b^3*d^2*e^
4*m*x-25200*a^3*b^3*d*e^5*x^2+360*a^2*b^4*d^4*e^2*m^2-19800*a^2*b^4*d^3*e^3*m*x+
15120*a^2*b^4*d^2*e^4*x^2+5760*a*b^5*d^4*e^2*m*x-5040*a*b^5*d^3*e^3*x^2-720*b^6*
d^5*e*m*x+720*b^6*d^4*e^2*x^2+8028*a^6*e^6*m-16524*a^5*b*d*e^5*m+15120*a^5*b*e^6
*x+19140*a^4*b^2*d^2*e^4*m-25200*a^4*b^2*d*e^5*x-12840*a^3*b^3*d^3*e^3*m+25200*a
^3*b^3*d^2*e^4*x+4680*a^2*b^4*d^4*e^2*m-15120*a^2*b^4*d^3*e^3*x-720*a*b^5*d^5*e*
m+5040*a*b^5*d^4*e^2*x-720*b^6*d^5*e*x+5040*a^6*e^6-15120*a^5*b*d*e^5+25200*a^4*
b^2*d^2*e^4-25200*a^3*b^3*d^3*e^3+15120*a^2*b^4*d^4*e^2-5040*a*b^5*d^5*e+720*b^6
*d^6)*((b*x+a)^2)^(5/2)/(b*x+a)^5/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+1313
2*m^2+13068*m+5040)

_______________________________________________________________________________________

Maxima [A]  time = 0.742955, size = 2516, normalized size = 6.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^m,x, algorithm="maxima")

[Out]

((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5*e^6*x^6 - 60*(m^2 + 11*m +
30)*a^2*b^3*d^4*e^2 + 20*(m^3 + 15*m^2 + 74*m + 120)*a^3*b^2*d^3*e^3 - 5*(m^4 +
18*m^3 + 119*m^2 + 342*m + 360)*a^4*b*d^2*e^4 + (m^5 + 20*m^4 + 155*m^3 + 580*m^
2 + 1044*m + 720)*a^5*d*e^5 + 120*a*b^4*d^5*e*(m + 6) - 120*b^5*d^6 + ((m^5 + 10
*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^5*d*e^5 + 5*(m^5 + 16*m^4 + 95*m^3 + 260*m^2 +
324*m + 144)*a*b^4*e^6)*x^5 - 5*((m^4 + 6*m^3 + 11*m^2 + 6*m)*b^5*d^2*e^4 - (m^5
 + 12*m^4 + 47*m^3 + 72*m^2 + 36*m)*a*b^4*d*e^5 - 2*(m^5 + 17*m^4 + 107*m^3 + 30
7*m^2 + 396*m + 180)*a^2*b^3*e^6)*x^4 + 10*(2*(m^3 + 3*m^2 + 2*m)*b^5*d^3*e^3 -
2*(m^4 + 9*m^3 + 20*m^2 + 12*m)*a*b^4*d^2*e^4 + (m^5 + 14*m^4 + 65*m^3 + 112*m^2
 + 60*m)*a^2*b^3*d*e^5 + (m^5 + 18*m^4 + 121*m^3 + 372*m^2 + 508*m + 240)*a^3*b^
2*e^6)*x^3 - 5*(12*(m^2 + m)*b^5*d^4*e^2 - 12*(m^3 + 7*m^2 + 6*m)*a*b^4*d^3*e^3
+ 6*(m^4 + 12*m^3 + 41*m^2 + 30*m)*a^2*b^3*d^2*e^4 - 2*(m^5 + 16*m^4 + 89*m^3 +
194*m^2 + 120*m)*a^3*b^2*d*e^5 - (m^5 + 19*m^4 + 137*m^3 + 461*m^2 + 702*m + 360
)*a^4*b*e^6)*x^2 - (120*(m^2 + 6*m)*a*b^4*d^4*e^2 - 60*(m^3 + 11*m^2 + 30*m)*a^2
*b^3*d^3*e^3 + 20*(m^4 + 15*m^3 + 74*m^2 + 120*m)*a^3*b^2*d^2*e^4 - 5*(m^5 + 18*
m^4 + 119*m^3 + 342*m^2 + 360*m)*a^4*b*d*e^5 - (m^5 + 20*m^4 + 155*m^3 + 580*m^2
 + 1044*m + 720)*a^5*e^6 - 120*b^5*d^5*e*m)*x)*(e*x + d)^m*a/((m^6 + 21*m^5 + 17
5*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + ((m^6 + 21*m^5 + 175*m^4 + 735
*m^3 + 1624*m^2 + 1764*m + 720)*b^5*e^7*x^7 + 240*(m^2 + 13*m + 42)*a^2*b^3*d^5*
e^2 - 60*(m^3 + 18*m^2 + 107*m + 210)*a^3*b^2*d^4*e^3 + 10*(m^4 + 22*m^3 + 179*m
^2 + 638*m + 840)*a^4*b*d^3*e^4 - (m^5 + 25*m^4 + 245*m^3 + 1175*m^2 + 2754*m +
2520)*a^5*d^2*e^5 - 600*a*b^4*d^6*e*(m + 7) + 720*b^5*d^7 + ((m^6 + 15*m^5 + 85*
m^4 + 225*m^3 + 274*m^2 + 120*m)*b^5*d*e^6 + 5*(m^6 + 22*m^5 + 190*m^4 + 820*m^3
 + 1849*m^2 + 2038*m + 840)*a*b^4*e^7)*x^6 - (6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2
+ 24*m)*b^5*d^2*e^5 - 5*(m^6 + 17*m^5 + 105*m^4 + 295*m^3 + 374*m^2 + 168*m)*a*b
^4*d*e^6 - 10*(m^6 + 23*m^5 + 207*m^4 + 925*m^3 + 2144*m^2 + 2412*m + 1008)*a^2*
b^3*e^7)*x^5 + 5*(6*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b^5*d^3*e^4 - 5*(m^5 + 13*m^4 +
 53*m^3 + 83*m^2 + 42*m)*a*b^4*d^2*e^5 + 2*(m^6 + 19*m^5 + 131*m^4 + 401*m^3 + 5
40*m^2 + 252*m)*a^2*b^3*d*e^6 + 2*(m^6 + 24*m^5 + 226*m^4 + 1056*m^3 + 2545*m^2
+ 2952*m + 1260)*a^3*b^2*e^7)*x^4 - 5*(24*(m^3 + 3*m^2 + 2*m)*b^5*d^4*e^3 - 20*(
m^4 + 10*m^3 + 23*m^2 + 14*m)*a*b^4*d^3*e^4 + 8*(m^5 + 16*m^4 + 83*m^3 + 152*m^2
 + 84*m)*a^2*b^3*d^2*e^5 - 2*(m^6 + 21*m^5 + 163*m^4 + 567*m^3 + 844*m^2 + 420*m
)*a^3*b^2*d*e^6 - (m^6 + 25*m^5 + 247*m^4 + 1219*m^3 + 3112*m^2 + 3796*m + 1680)
*a^4*b*e^7)*x^3 + (360*(m^2 + m)*b^5*d^5*e^2 - 300*(m^3 + 8*m^2 + 7*m)*a*b^4*d^4
*e^3 + 120*(m^4 + 14*m^3 + 55*m^2 + 42*m)*a^2*b^3*d^3*e^4 - 30*(m^5 + 19*m^4 + 1
25*m^3 + 317*m^2 + 210*m)*a^3*b^2*d^2*e^5 + 5*(m^6 + 23*m^5 + 201*m^4 + 817*m^3
+ 1478*m^2 + 840*m)*a^4*b*d*e^6 + (m^6 + 26*m^5 + 270*m^4 + 1420*m^3 + 3929*m^2
+ 5274*m + 2520)*a^5*e^7)*x^2 + (600*(m^2 + 7*m)*a*b^4*d^5*e^2 - 240*(m^3 + 13*m
^2 + 42*m)*a^2*b^3*d^4*e^3 + 60*(m^4 + 18*m^3 + 107*m^2 + 210*m)*a^3*b^2*d^3*e^4
 - 10*(m^5 + 22*m^4 + 179*m^3 + 638*m^2 + 840*m)*a^4*b*d^2*e^5 + (m^6 + 25*m^5 +
 245*m^4 + 1175*m^3 + 2754*m^2 + 2520*m)*a^5*d*e^6 - 720*b^5*d^6*e*m)*x)*(e*x +
d)^m*b/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 50
40)*e^7)

_______________________________________________________________________________________

Fricas [A]  time = 0.33436, size = 3011, normalized size = 7.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(a^6*d*e^6*m^6 + 720*b^6*d^7 - 5040*a*b^5*d^6*e + 15120*a^2*b^4*d^5*e^2 - 25200*
a^3*b^3*d^4*e^3 + 25200*a^4*b^2*d^3*e^4 - 15120*a^5*b*d^2*e^5 + 5040*a^6*d*e^6 +
 (b^6*e^7*m^6 + 21*b^6*e^7*m^5 + 175*b^6*e^7*m^4 + 735*b^6*e^7*m^3 + 1624*b^6*e^
7*m^2 + 1764*b^6*e^7*m + 720*b^6*e^7)*x^7 + (5040*a*b^5*e^7 + (b^6*d*e^6 + 6*a*b
^5*e^7)*m^6 + 3*(5*b^6*d*e^6 + 44*a*b^5*e^7)*m^5 + 5*(17*b^6*d*e^6 + 228*a*b^5*e
^7)*m^4 + 15*(15*b^6*d*e^6 + 328*a*b^5*e^7)*m^3 + 2*(137*b^6*d*e^6 + 5547*a*b^5*
e^7)*m^2 + 12*(10*b^6*d*e^6 + 1019*a*b^5*e^7)*m)*x^6 - 3*(2*a^5*b*d^2*e^5 - 9*a^
6*d*e^6)*m^5 + 3*(5040*a^2*b^4*e^7 + (2*a*b^5*d*e^6 + 5*a^2*b^4*e^7)*m^6 - (2*b^
6*d^2*e^5 - 34*a*b^5*d*e^6 - 115*a^2*b^4*e^7)*m^5 - 5*(4*b^6*d^2*e^5 - 42*a*b^5*
d*e^6 - 207*a^2*b^4*e^7)*m^4 - 5*(14*b^6*d^2*e^5 - 118*a*b^5*d*e^6 - 925*a^2*b^4
*e^7)*m^3 - 4*(25*b^6*d^2*e^5 - 187*a*b^5*d*e^6 - 2680*a^2*b^4*e^7)*m^2 - 12*(4*
b^6*d^2*e^5 - 28*a*b^5*d*e^6 - 1005*a^2*b^4*e^7)*m)*x^5 + 5*(6*a^4*b^2*d^3*e^4 -
 30*a^5*b*d^2*e^5 + 59*a^6*d*e^6)*m^4 + 5*(5040*a^3*b^3*e^7 + (3*a^2*b^4*d*e^6 +
 4*a^3*b^3*e^7)*m^6 - 3*(2*a*b^5*d^2*e^5 - 19*a^2*b^4*d*e^6 - 32*a^3*b^3*e^7)*m^
5 + (6*b^6*d^3*e^4 - 78*a*b^5*d^2*e^5 + 393*a^2*b^4*d*e^6 + 904*a^3*b^3*e^7)*m^4
 + 3*(12*b^6*d^3*e^4 - 106*a*b^5*d^2*e^5 + 401*a^2*b^4*d*e^6 + 1408*a^3*b^3*e^7)
*m^3 + 2*(33*b^6*d^3*e^4 - 249*a*b^5*d^2*e^5 + 810*a^2*b^4*d*e^6 + 5090*a^3*b^3*
e^7)*m^2 + 36*(b^6*d^3*e^4 - 7*a*b^5*d^2*e^5 + 21*a^2*b^4*d*e^6 + 328*a^3*b^3*e^
7)*m)*x^4 - 15*(8*a^3*b^3*d^4*e^3 - 44*a^4*b^2*d^3*e^4 + 98*a^5*b*d^2*e^5 - 111*
a^6*d*e^6)*m^3 + 5*(5040*a^4*b^2*e^7 + (4*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*m^6 - 3
*(4*a^2*b^4*d^2*e^5 - 28*a^3*b^3*d*e^6 - 25*a^4*b^2*e^7)*m^5 + (24*a*b^5*d^3*e^4
 - 192*a^2*b^4*d^2*e^5 + 652*a^3*b^3*d*e^6 + 741*a^4*b^2*e^7)*m^4 - 3*(8*b^6*d^4
*e^3 - 80*a*b^5*d^3*e^4 + 332*a^2*b^4*d^2*e^5 - 756*a^3*b^3*d*e^6 - 1219*a^4*b^2
*e^7)*m^3 - 8*(9*b^6*d^4*e^3 - 69*a*b^5*d^3*e^4 + 228*a^2*b^4*d^2*e^5 - 422*a^3*
b^3*d*e^6 - 1167*a^4*b^2*e^7)*m^2 - 12*(4*b^6*d^4*e^3 - 28*a*b^5*d^3*e^4 + 84*a^
2*b^4*d^2*e^5 - 140*a^3*b^3*d*e^6 - 949*a^4*b^2*e^7)*m)*x^3 + 2*(180*a^2*b^4*d^5
*e^2 - 1080*a^3*b^3*d^4*e^3 + 2685*a^4*b^2*d^3*e^4 - 3525*a^5*b*d^2*e^5 + 2552*a
^6*d*e^6)*m^2 + 3*(5040*a^5*b*e^7 + (5*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*m^6 - (20*a^
3*b^3*d^2*e^5 - 115*a^4*b^2*d*e^6 - 52*a^5*b*e^7)*m^5 + 5*(12*a^2*b^4*d^3*e^4 -
76*a^3*b^3*d^2*e^5 + 201*a^4*b^2*d*e^6 + 108*a^5*b*e^7)*m^4 - 5*(24*a*b^5*d^4*e^
3 - 168*a^2*b^4*d^3*e^4 + 500*a^3*b^3*d^2*e^5 - 817*a^4*b^2*d*e^6 - 568*a^5*b*e^
7)*m^3 + 2*(60*b^6*d^5*e^2 - 480*a*b^5*d^4*e^3 + 1650*a^2*b^4*d^3*e^4 - 3170*a^3
*b^3*d^2*e^5 + 3695*a^4*b^2*d*e^6 + 3929*a^5*b*e^7)*m^2 + 12*(10*b^6*d^5*e^2 - 7
0*a*b^5*d^4*e^3 + 210*a^2*b^4*d^3*e^4 - 350*a^3*b^3*d^2*e^5 + 350*a^4*b^2*d*e^6
+ 879*a^5*b*e^7)*m)*x^2 - 12*(60*a*b^5*d^6*e - 390*a^2*b^4*d^5*e^2 + 1070*a^3*b^
3*d^4*e^3 - 1595*a^4*b^2*d^3*e^4 + 1377*a^5*b*d^2*e^5 - 669*a^6*d*e^6)*m + (5040
*a^6*e^7 + (6*a^5*b*d*e^6 + a^6*e^7)*m^6 - 3*(10*a^4*b^2*d^2*e^5 - 50*a^5*b*d*e^
6 - 9*a^6*e^7)*m^5 + 5*(24*a^3*b^3*d^3*e^4 - 132*a^4*b^2*d^2*e^5 + 294*a^5*b*d*e
^6 + 59*a^6*e^7)*m^4 - 15*(24*a^2*b^4*d^4*e^3 - 144*a^3*b^3*d^3*e^4 + 358*a^4*b^
2*d^2*e^5 - 470*a^5*b*d*e^6 - 111*a^6*e^7)*m^3 + 4*(180*a*b^5*d^5*e^2 - 1170*a^2
*b^4*d^4*e^3 + 3210*a^3*b^3*d^3*e^4 - 4785*a^4*b^2*d^2*e^5 + 4131*a^5*b*d*e^6 +
1276*a^6*e^7)*m^2 - 36*(20*b^6*d^6*e - 140*a*b^5*d^5*e^2 + 420*a^2*b^4*d^4*e^3 -
 700*a^3*b^3*d^3*e^4 + 700*a^4*b^2*d^2*e^5 - 420*a^5*b*d*e^6 - 223*a^6*e^7)*m)*x
)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3
+ 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.363202, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*(e*x + d)^m,x, algorithm="giac")

[Out]

Done

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