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3.1888 \(\int (A+B x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=471 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) (d+e x)^{m+1}}{e^7 (m+1) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+2} (-a B e-5 A b e+6 b B d)}{e^7 (m+2) (a+b x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+3} (-a B e-2 A b e+3 b B d)}{e^7 (m+3) (a+b x)}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+4} (-a B e-A b e+2 b B d)}{e^7 (m+4) (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+6} (-5 a B e-A b e+6 b B d)}{e^7 (m+6) (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+5} (-2 a B e-A b e+3 b B d)}{e^7 (m+5) (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+7}}{e^7 (m+7) (a+b x)} \]

[Out]

((b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7
*(1 + m)*(a + b*x)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(2 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(2 + m)*(a + b*x)) + (5*b*(b*d - a*e)^3*(
3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7
*(3 + m)*(a + b*x)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^
(4 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(4 + m)*(a + b*x)) + (5*b^3*(b*d - a
*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^7*(5 + m)*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(6 + m)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(6 + m)*(a + b*x)) + (b^5*B*(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 = 0.88077, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e) (d+e x)^{m+1}}{e^7 (m+1) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+2} (-a B e-5 A b e+6 b B d)}{e^7 (m+2) (a+b x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+3} (-a B e-2 A b e+3 b B d)}{e^7 (m+3) (a+b x)}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+4} (-a B e-A b e+2 b B d)}{e^7 (m+4) (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+6} (-5 a B e-A b e+6 b B d)}{e^7 (m+6) (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+5} (-2 a B e-A b e+3 b B d)}{e^7 (m+5) (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+7}}{e^7 (m+7) (a+b x)} \]

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)^5*(B*d - A*e)*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7
*(1 + m)*(a + b*x)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(2 +
m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(2 + m)*(a + b*x)) + (5*b*(b*d - a*e)^3*(
3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7
*(3 + m)*(a + b*x)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^
(4 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(4 + m)*(a + b*x)) + (5*b^3*(b*d - a
*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(e^7*(5 + m)*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(6 + m)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(6 + m)*(a + b*x)) + (b^5*B*(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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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Mathematica [B]  time = 2.84933, size = 969, normalized size = 2.06 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} \left (\left (A e (m+7) \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )+B \left (720 d^6-720 e (m+1) x d^5+360 e^2 \left (m^2+3 m+2\right ) x^2 d^4-120 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^3+30 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^2-6 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d+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 ) b^5+5 a e (m+7) \left (A e (m+6) \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+B \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )\right ) b^4+10 a^2 e^2 \left (m^2+13 m+42\right ) \left (A e (m+5) \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+B \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right ) b^3+10 a^3 e^3 \left (m^3+18 m^2+107 m+210\right ) \left (A e (m+4) \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right )+B \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right ) b^2+5 a^4 e^4 \left (m^4+22 m^3+179 m^2+638 m+840\right ) \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right )\right ) b+a^5 e^5 \left (m^5+25 m^4+245 m^3+1175 m^2+2754 m+2520\right ) (-B d+A e (m+2)+B e (m+1) x)\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^5*e^5*(2520 + 2754*m + 1175*m^2 + 245*m^
3 + 25*m^4 + m^5)*(-(B*d) + A*e*(2 + m) + B*e*(1 + m)*x) + 5*a^4*b*e^4*(840 + 63
8*m + 179*m^2 + 22*m^3 + m^4)*(A*e*(3 + m)*(-d + e*(1 + m)*x) + B*(2*d^2 - 2*d*e
*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) + 10*a^3*b^2*e^3*(210 + 107*m + 18*m^2 +
m^3)*(A*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + B*(-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)) + 10*a^2*b^3*e^2*(42 + 13*m + m^2)*(A*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*(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)) + 5*a*b^4*e*(7 +
 m)*(A*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*(-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^5*(A*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*(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*(120
 + 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 = 0.026, size = 3931, normalized size = 8.4 \[ \text{output too large to display} \]

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*b^5*e^6*m^6*x^6+A*b^5*e^6*m^6*x^5+5*B*a*b^4*e^6*m^6*x^5+21*B*b^
5*e^6*m^5*x^6+5*A*a*b^4*e^6*m^6*x^4+22*A*b^5*e^6*m^5*x^5+10*B*a^2*b^3*e^6*m^6*x^
4+110*B*a*b^4*e^6*m^5*x^5-6*B*b^5*d*e^5*m^5*x^5+175*B*b^5*e^6*m^4*x^6+10*A*a^2*b
^3*e^6*m^6*x^3+115*A*a*b^4*e^6*m^5*x^4-5*A*b^5*d*e^5*m^5*x^4+190*A*b^5*e^6*m^4*x
^5+10*B*a^3*b^2*e^6*m^6*x^3+230*B*a^2*b^3*e^6*m^5*x^4-25*B*a*b^4*d*e^5*m^5*x^4+9
50*B*a*b^4*e^6*m^4*x^5-90*B*b^5*d*e^5*m^4*x^5+735*B*b^5*e^6*m^3*x^6+10*A*a^3*b^2
*e^6*m^6*x^2+240*A*a^2*b^3*e^6*m^5*x^3-20*A*a*b^4*d*e^5*m^5*x^3+1035*A*a*b^4*e^6
*m^4*x^4-85*A*b^5*d*e^5*m^4*x^4+820*A*b^5*e^6*m^3*x^5+5*B*a^4*b*e^6*m^6*x^2+240*
B*a^3*b^2*e^6*m^5*x^3-40*B*a^2*b^3*d*e^5*m^5*x^3+2070*B*a^2*b^3*e^6*m^4*x^4-425*
B*a*b^4*d*e^5*m^4*x^4+4100*B*a*b^4*e^6*m^3*x^5+30*B*b^5*d^2*e^4*m^4*x^4-510*B*b^
5*d*e^5*m^3*x^5+1624*B*b^5*e^6*m^2*x^6+5*A*a^4*b*e^6*m^6*x+250*A*a^3*b^2*e^6*m^5
*x^2-30*A*a^2*b^3*d*e^5*m^5*x^2+2260*A*a^2*b^3*e^6*m^4*x^3-380*A*a*b^4*d*e^5*m^4
*x^3+4625*A*a*b^4*e^6*m^3*x^4+20*A*b^5*d^2*e^4*m^4*x^3-525*A*b^5*d*e^5*m^3*x^4+1
849*A*b^5*e^6*m^2*x^5+B*a^5*e^6*m^6*x+125*B*a^4*b*e^6*m^5*x^2-30*B*a^3*b^2*d*e^5
*m^5*x^2+2260*B*a^3*b^2*e^6*m^4*x^3-760*B*a^2*b^3*d*e^5*m^4*x^3+9250*B*a^2*b^3*e
^6*m^3*x^4+100*B*a*b^4*d^2*e^4*m^4*x^3-2625*B*a*b^4*d*e^5*m^3*x^4+9245*B*a*b^4*e
^6*m^2*x^5+300*B*b^5*d^2*e^4*m^3*x^4-1350*B*b^5*d*e^5*m^2*x^5+1764*B*b^5*e^6*m*x
^6+A*a^5*e^6*m^6+130*A*a^4*b*e^6*m^5*x-20*A*a^3*b^2*d*e^5*m^5*x+2470*A*a^3*b^2*e
^6*m^4*x^2-630*A*a^2*b^3*d*e^5*m^4*x^2+10560*A*a^2*b^3*e^6*m^3*x^3+60*A*a*b^4*d^
2*e^4*m^4*x^2-2620*A*a*b^4*d*e^5*m^3*x^3+10720*A*a*b^4*e^6*m^2*x^4+260*A*b^5*d^2
*e^4*m^3*x^3-1475*A*b^5*d*e^5*m^2*x^4+2038*A*b^5*e^6*m*x^5+26*B*a^5*e^6*m^5*x-10
*B*a^4*b*d*e^5*m^5*x+1235*B*a^4*b*e^6*m^4*x^2-630*B*a^3*b^2*d*e^5*m^4*x^2+10560*
B*a^3*b^2*e^6*m^3*x^3+120*B*a^2*b^3*d^2*e^4*m^4*x^2-5240*B*a^2*b^3*d*e^5*m^3*x^3
+21440*B*a^2*b^3*e^6*m^2*x^4+1300*B*a*b^4*d^2*e^4*m^3*x^3-7375*B*a*b^4*d*e^5*m^2
*x^4+10190*B*a*b^4*e^6*m*x^5-120*B*b^5*d^3*e^3*m^3*x^3+1050*B*b^5*d^2*e^4*m^2*x^
4-1644*B*b^5*d*e^5*m*x^5+720*B*b^5*e^6*x^6+27*A*a^5*e^6*m^5-5*A*a^4*b*d*e^5*m^5+
1350*A*a^4*b*e^6*m^4*x-460*A*a^3*b^2*d*e^5*m^4*x+12190*A*a^3*b^2*e^6*m^3*x^2+60*
A*a^2*b^3*d^2*e^4*m^4*x-4890*A*a^2*b^3*d*e^5*m^3*x^2+25450*A*a^2*b^3*e^6*m^2*x^3
+960*A*a*b^4*d^2*e^4*m^3*x^2-8020*A*a*b^4*d*e^5*m^2*x^3+12060*A*a*b^4*e^6*m*x^4-
60*A*b^5*d^3*e^3*m^3*x^2+1060*A*b^5*d^2*e^4*m^2*x^3-1870*A*b^5*d*e^5*m*x^4+840*A
*b^5*e^6*x^5-B*a^5*d*e^5*m^5+270*B*a^5*e^6*m^4*x-230*B*a^4*b*d*e^5*m^4*x+6095*B*
a^4*b*e^6*m^3*x^2+60*B*a^3*b^2*d^2*e^4*m^4*x-4890*B*a^3*b^2*d*e^5*m^3*x^2+25450*
B*a^3*b^2*e^6*m^2*x^3+1920*B*a^2*b^3*d^2*e^4*m^3*x^2-16040*B*a^2*b^3*d*e^5*m^2*x
^3+24120*B*a^2*b^3*e^6*m*x^4-300*B*a*b^4*d^3*e^3*m^3*x^2+5300*B*a*b^4*d^2*e^4*m^
2*x^3-9350*B*a*b^4*d*e^5*m*x^4+4200*B*a*b^4*e^6*x^5-720*B*b^5*d^3*e^3*m^2*x^3+15
00*B*b^5*d^2*e^4*m*x^4-720*B*b^5*d*e^5*x^5+295*A*a^5*e^6*m^4-125*A*a^4*b*d*e^5*m
^4+7100*A*a^4*b*e^6*m^3*x+20*A*a^3*b^2*d^2*e^4*m^4-4020*A*a^3*b^2*d*e^5*m^3*x+31
120*A*a^3*b^2*e^6*m^2*x^2+1140*A*a^2*b^3*d^2*e^4*m^3*x-17010*A*a^2*b^3*d*e^5*m^2
*x^2+29520*A*a^2*b^3*e^6*m*x^3-120*A*a*b^4*d^3*e^3*m^3*x+4980*A*a*b^4*d^2*e^4*m^
2*x^2-10800*A*a*b^4*d*e^5*m*x^3+5040*A*a*b^4*e^6*x^4-600*A*b^5*d^3*e^3*m^2*x^2+1
660*A*b^5*d^2*e^4*m*x^3-840*A*b^5*d*e^5*x^4-25*B*a^5*d*e^5*m^4+1420*B*a^5*e^6*m^
3*x+10*B*a^4*b*d^2*e^4*m^4-2010*B*a^4*b*d*e^5*m^3*x+15560*B*a^4*b*e^6*m^2*x^2+11
40*B*a^3*b^2*d^2*e^4*m^3*x-17010*B*a^3*b^2*d*e^5*m^2*x^2+29520*B*a^3*b^2*e^6*m*x
^3-240*B*a^2*b^3*d^3*e^3*m^3*x+9960*B*a^2*b^3*d^2*e^4*m^2*x^2-21600*B*a^2*b^3*d*
e^5*m*x^3+10080*B*a^2*b^3*e^6*x^4-3000*B*a*b^4*d^3*e^3*m^2*x^2+8300*B*a*b^4*d^2*
e^4*m*x^3-4200*B*a*b^4*d*e^5*x^4+360*B*b^5*d^4*e^2*m^2*x^2-1320*B*b^5*d^3*e^3*m*
x^3+720*B*b^5*d^2*e^4*x^4+1665*A*a^5*e^6*m^3-1225*A*a^4*b*d*e^5*m^3+19645*A*a^4*
b*e^6*m^2*x+440*A*a^3*b^2*d^2*e^4*m^3-16340*A*a^3*b^2*d*e^5*m^2*x+37960*A*a^3*b^
2*e^6*m*x^2-60*A*a^2*b^3*d^3*e^3*m^3+7500*A*a^2*b^3*d^2*e^4*m^2*x-25320*A*a^2*b^
3*d*e^5*m*x^2+12600*A*a^2*b^3*e^6*x^3-1680*A*a*b^4*d^3*e^3*m^2*x+9120*A*a*b^4*d^
2*e^4*m*x^2-5040*A*a*b^4*d*e^5*x^3+120*A*b^5*d^4*e^2*m^2*x-1380*A*b^5*d^3*e^3*m*
x^2+840*A*b^5*d^2*e^4*x^3-245*B*a^5*d*e^5*m^3+3929*B*a^5*e^6*m^2*x+220*B*a^4*b*d
^2*e^4*m^3-8170*B*a^4*b*d*e^5*m^2*x+18980*B*a^4*b*e^6*m*x^2-60*B*a^3*b^2*d^3*e^3
*m^3+7500*B*a^3*b^2*d^2*e^4*m^2*x-25320*B*a^3*b^2*d*e^5*m*x^2+12600*B*a^3*b^2*e^
6*x^3-3360*B*a^2*b^3*d^3*e^3*m^2*x+18240*B*a^2*b^3*d^2*e^4*m*x^2-10080*B*a^2*b^3
*d*e^5*x^3+600*B*a*b^4*d^4*e^2*m^2*x-6900*B*a*b^4*d^3*e^3*m*x^2+4200*B*a*b^4*d^2
*e^4*x^3+1080*B*b^5*d^4*e^2*m*x^2-720*B*b^5*d^3*e^3*x^3+5104*A*a^5*e^6*m^2-5875*
A*a^4*b*d*e^5*m^2+26370*A*a^4*b*e^6*m*x+3580*A*a^3*b^2*d^2*e^4*m^2-29560*A*a^3*b
^2*d*e^5*m*x+16800*A*a^3*b^2*e^6*x^2-1080*A*a^2*b^3*d^3*e^3*m^2+19020*A*a^2*b^3*
d^2*e^4*m*x-12600*A*a^2*b^3*d*e^5*x^2+120*A*a*b^4*d^4*e^2*m^2-6600*A*a*b^4*d^3*e
^3*m*x+5040*A*a*b^4*d^2*e^4*x^2+960*A*b^5*d^4*e^2*m*x-840*A*b^5*d^3*e^3*x^2-1175
*B*a^5*d*e^5*m^2+5274*B*a^5*e^6*m*x+1790*B*a^4*b*d^2*e^4*m^2-14780*B*a^4*b*d*e^5
*m*x+8400*B*a^4*b*e^6*x^2-1080*B*a^3*b^2*d^3*e^3*m^2+19020*B*a^3*b^2*d^2*e^4*m*x
-12600*B*a^3*b^2*d*e^5*x^2+240*B*a^2*b^3*d^4*e^2*m^2-13200*B*a^2*b^3*d^3*e^3*m*x
+10080*B*a^2*b^3*d^2*e^4*x^2+4800*B*a*b^4*d^4*e^2*m*x-4200*B*a*b^4*d^3*e^3*x^2-7
20*B*b^5*d^5*e*m*x+720*B*b^5*d^4*e^2*x^2+8028*A*a^5*e^6*m-13770*A*a^4*b*d*e^5*m+
12600*A*a^4*b*e^6*x+12760*A*a^3*b^2*d^2*e^4*m-16800*A*a^3*b^2*d*e^5*x-6420*A*a^2
*b^3*d^3*e^3*m+12600*A*a^2*b^3*d^2*e^4*x+1560*A*a*b^4*d^4*e^2*m-5040*A*a*b^4*d^3
*e^3*x-120*A*b^5*d^5*e*m+840*A*b^5*d^4*e^2*x-2754*B*a^5*d*e^5*m+2520*B*a^5*e^6*x
+6380*B*a^4*b*d^2*e^4*m-8400*B*a^4*b*d*e^5*x-6420*B*a^3*b^2*d^3*e^3*m+12600*B*a^
3*b^2*d^2*e^4*x+3120*B*a^2*b^3*d^4*e^2*m-10080*B*a^2*b^3*d^3*e^3*x-600*B*a*b^4*d
^5*e*m+4200*B*a*b^4*d^4*e^2*x-720*B*b^5*d^5*e*x+5040*A*a^5*e^6-12600*A*a^4*b*d*e
^5+16800*A*a^3*b^2*d^2*e^4-12600*A*a^2*b^3*d^3*e^3+5040*A*a*b^4*d^4*e^2-840*A*b^
5*d^5*e-2520*B*a^5*d*e^5+8400*B*a^4*b*d^2*e^4-12600*B*a^3*b^2*d^3*e^3+10080*B*a^
2*b^3*d^4*e^2-4200*B*a*b^4*d^5*e+720*B*b^5*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+13132*m^2+13068*m+5040)

_______________________________________________________________________________________

Maxima [A]  time = 0.773395, size = 2516, normalized size = 5.34 \[ \text{result too large to display} \]

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 = 0.346156, size = 4705, normalized size = 9.99 \[ \text{result too large to display} \]

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*a^5*d*e^6*m^6 + 720*B*b^5*d^7 + 5040*A*a^5*d*e^6 - 840*(5*B*a*b^4 + A*b^5)*d^
6*e + 5040*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^2 - 12600*(B*a^3*b^2 + A*a^2*b^3)*d^4*e
^3 + 8400*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 2520*(B*a^5 + 5*A*a^4*b)*d^2*e^5 + (
B*b^5*e^7*m^6 + 21*B*b^5*e^7*m^5 + 175*B*b^5*e^7*m^4 + 735*B*b^5*e^7*m^3 + 1624*
B*b^5*e^7*m^2 + 1764*B*b^5*e^7*m + 720*B*b^5*e^7)*x^7 + (840*(5*B*a*b^4 + A*b^5)
*e^7 + (B*b^5*d*e^6 + (5*B*a*b^4 + A*b^5)*e^7)*m^6 + (15*B*b^5*d*e^6 + 22*(5*B*a
*b^4 + A*b^5)*e^7)*m^5 + 5*(17*B*b^5*d*e^6 + 38*(5*B*a*b^4 + A*b^5)*e^7)*m^4 + 5
*(45*B*b^5*d*e^6 + 164*(5*B*a*b^4 + A*b^5)*e^7)*m^3 + (274*B*b^5*d*e^6 + 1849*(5
*B*a*b^4 + A*b^5)*e^7)*m^2 + 2*(60*B*b^5*d*e^6 + 1019*(5*B*a*b^4 + A*b^5)*e^7)*m
)*x^6 + (27*A*a^5*d*e^6 - (B*a^5 + 5*A*a^4*b)*d^2*e^5)*m^5 + (5040*(2*B*a^2*b^3
+ A*a*b^4)*e^7 + ((5*B*a*b^4 + A*b^5)*d*e^6 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m^6
 - (6*B*b^5*d^2*e^5 - 17*(5*B*a*b^4 + A*b^5)*d*e^6 - 115*(2*B*a^2*b^3 + A*a*b^4)
*e^7)*m^5 - 15*(4*B*b^5*d^2*e^5 - 7*(5*B*a*b^4 + A*b^5)*d*e^6 - 69*(2*B*a^2*b^3
+ A*a*b^4)*e^7)*m^4 - 5*(42*B*b^5*d^2*e^5 - 59*(5*B*a*b^4 + A*b^5)*d*e^6 - 925*(
2*B*a^2*b^3 + A*a*b^4)*e^7)*m^3 - 2*(150*B*b^5*d^2*e^5 - 187*(5*B*a*b^4 + A*b^5)
*d*e^6 - 5360*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m^2 - 12*(12*B*b^5*d^2*e^5 - 14*(5*B*
a*b^4 + A*b^5)*d*e^6 - 1005*(2*B*a^2*b^3 + A*a*b^4)*e^7)*m)*x^5 + 5*(59*A*a^5*d*
e^6 + 2*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 5*(B*a^5 + 5*A*a^4*b)*d^2*e^5)*m^4 + 5
*(2520*(B*a^3*b^2 + A*a^2*b^3)*e^7 + ((2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 2*(B*a^3*b
^2 + A*a^2*b^3)*e^7)*m^6 - ((5*B*a*b^4 + A*b^5)*d^2*e^5 - 19*(2*B*a^2*b^3 + A*a*
b^4)*d*e^6 - 48*(B*a^3*b^2 + A*a^2*b^3)*e^7)*m^5 + (6*B*b^5*d^3*e^4 - 13*(5*B*a*
b^4 + A*b^5)*d^2*e^5 + 131*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 452*(B*a^3*b^2 + A*a^
2*b^3)*e^7)*m^4 + (36*B*b^5*d^3*e^4 - 53*(5*B*a*b^4 + A*b^5)*d^2*e^5 + 401*(2*B*
a^2*b^3 + A*a*b^4)*d*e^6 + 2112*(B*a^3*b^2 + A*a^2*b^3)*e^7)*m^3 + (66*B*b^5*d^3
*e^4 - 83*(5*B*a*b^4 + A*b^5)*d^2*e^5 + 540*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 5090
*(B*a^3*b^2 + A*a^2*b^3)*e^7)*m^2 + 6*(6*B*b^5*d^3*e^4 - 7*(5*B*a*b^4 + A*b^5)*d
^2*e^5 + 42*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 984*(B*a^3*b^2 + A*a^2*b^3)*e^7)*m)*
x^4 + 5*(333*A*a^5*d*e^6 - 12*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^3 + 44*(B*a^4*b + 2*
A*a^3*b^2)*d^3*e^4 - 49*(B*a^5 + 5*A*a^4*b)*d^2*e^5)*m^3 + 5*(1680*(B*a^4*b + 2*
A*a^3*b^2)*e^7 + (2*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 + (B*a^4*b + 2*A*a^3*b^2)*e^7)
*m^6 - (4*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 42*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 2
5*(B*a^4*b + 2*A*a^3*b^2)*e^7)*m^5 + (4*(5*B*a*b^4 + A*b^5)*d^3*e^4 - 64*(2*B*a^
2*b^3 + A*a*b^4)*d^2*e^5 + 326*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 + 247*(B*a^4*b + 2*
A*a^3*b^2)*e^7)*m^4 - (24*B*b^5*d^4*e^3 - 40*(5*B*a*b^4 + A*b^5)*d^3*e^4 + 332*(
2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 1134*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 1219*(B*a^
4*b + 2*A*a^3*b^2)*e^7)*m^3 - 4*(18*B*b^5*d^4*e^3 - 23*(5*B*a*b^4 + A*b^5)*d^3*e
^4 + 152*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 422*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 7
78*(B*a^4*b + 2*A*a^3*b^2)*e^7)*m^2 - 4*(12*B*b^5*d^4*e^3 - 14*(5*B*a*b^4 + A*b^
5)*d^3*e^4 + 84*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 210*(B*a^3*b^2 + A*a^2*b^3)*d*
e^6 - 949*(B*a^4*b + 2*A*a^3*b^2)*e^7)*m)*x^3 + (5104*A*a^5*d*e^6 + 120*(2*B*a^2
*b^3 + A*a*b^4)*d^5*e^2 - 1080*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^3 + 1790*(B*a^4*b +
 2*A*a^3*b^2)*d^3*e^4 - 1175*(B*a^5 + 5*A*a^4*b)*d^2*e^5)*m^2 + (2520*(B*a^5 + 5
*A*a^4*b)*e^7 + (5*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + (B*a^5 + 5*A*a^4*b)*e^7)*m^6
- (30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 - 115*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 - 26*(
B*a^5 + 5*A*a^4*b)*e^7)*m^5 + 15*(4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^4 - 38*(B*a^3*
b^2 + A*a^2*b^3)*d^2*e^5 + 67*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 18*(B*a^5 + 5*A*a^
4*b)*e^7)*m^4 - 5*(12*(5*B*a*b^4 + A*b^5)*d^4*e^3 - 168*(2*B*a^2*b^3 + A*a*b^4)*
d^3*e^4 + 750*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 - 817*(B*a^4*b + 2*A*a^3*b^2)*d*e^
6 - 284*(B*a^5 + 5*A*a^4*b)*e^7)*m^3 + (360*B*b^5*d^5*e^2 - 480*(5*B*a*b^4 + A*b
^5)*d^4*e^3 + 3300*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^4 - 9510*(B*a^3*b^2 + A*a^2*b^3
)*d^2*e^5 + 7390*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 3929*(B*a^5 + 5*A*a^4*b)*e^7)*m
^2 + 6*(60*B*b^5*d^5*e^2 - 70*(5*B*a*b^4 + A*b^5)*d^4*e^3 + 420*(2*B*a^2*b^3 + A
*a*b^4)*d^3*e^4 - 1050*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 + 700*(B*a^4*b + 2*A*a^3*
b^2)*d*e^6 + 879*(B*a^5 + 5*A*a^4*b)*e^7)*m)*x^2 + 2*(4014*A*a^5*d*e^6 - 60*(5*B
*a*b^4 + A*b^5)*d^6*e + 780*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^2 - 3210*(B*a^3*b^2 +
A*a^2*b^3)*d^4*e^3 + 3190*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 1377*(B*a^5 + 5*A*a^
4*b)*d^2*e^5)*m + (5040*A*a^5*e^7 + (A*a^5*e^7 + (B*a^5 + 5*A*a^4*b)*d*e^6)*m^6
+ (27*A*a^5*e^7 - 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 + 25*(B*a^5 + 5*A*a^4*b)*d*
e^6)*m^5 + 5*(59*A*a^5*e^7 + 12*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^4 - 44*(B*a^4*b +
2*A*a^3*b^2)*d^2*e^5 + 49*(B*a^5 + 5*A*a^4*b)*d*e^6)*m^4 + 5*(333*A*a^5*e^7 - 24
*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^3 + 216*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^4 - 358*(B*
a^4*b + 2*A*a^3*b^2)*d^2*e^5 + 235*(B*a^5 + 5*A*a^4*b)*d*e^6)*m^3 + 2*(2552*A*a^
5*e^7 + 60*(5*B*a*b^4 + A*b^5)*d^5*e^2 - 780*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^3 + 3
210*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^4 - 3190*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 + 137
7*(B*a^5 + 5*A*a^4*b)*d*e^6)*m^2 - 12*(60*B*b^5*d^6*e - 669*A*a^5*e^7 - 70*(5*B*
a*b^4 + A*b^5)*d^5*e^2 + 420*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^3 - 1050*(B*a^3*b^2 +
 A*a^2*b^3)*d^3*e^4 + 700*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 - 210*(B*a^5 + 5*A*a^4
*b)*d*e^6)*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)

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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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.41516, 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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