3.1738 \(\int (A+B x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=436 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{8 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (-5 a B e-A b e+6 b B d)}{12 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13}}{13 e^7 (a+b x)} \]

[Out]

((b*d - a*e)^5*(B*d - A*e)*(d + e*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a
+ b*x)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^8*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(8*e^7*(a + b*x)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*
e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (b^2*(b*d - a*
e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*
(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^11*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d
+ e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)) + (b^5*B*(d + e*x)^
13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

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Rubi [A]  time = 3.10237, antiderivative size = 436, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{8 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (-5 a B e-A b e+6 b B d)}{12 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13}}{13 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^6*(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)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a
+ b*x)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^8*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(8*e^7*(a + b*x)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*
e)*(d + e*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (b^2*(b*d - a*
e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*
(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^11*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d
+ e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)) + (b^5*B*(d + e*x)^
13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 78.278, size = 430, normalized size = 0.99 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{7} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{26 b e} + \frac{\left (d + e x\right )^{7} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{156 b e^{2}} + \frac{\left (5 a + 5 b x\right ) \left (d + e x\right )^{7} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{1716 b e^{3}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{858 b e^{4}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{7} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{7722 b e^{5}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{10296 b e^{6}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (13 A b e - 7 B a e - 6 B b d\right )}{72072 b e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**7*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(26*b*e) + (d +
e*x)**7*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(13*A*b*e - 7*B*a*e - 6*B*b*d)/(156
*b*e**2) + (5*a + 5*b*x)*(d + e*x)**7*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**
(3/2)*(13*A*b*e - 7*B*a*e - 6*B*b*d)/(1716*b*e**3) + (d + e*x)**7*(a*e - b*d)**2
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(13*A*b*e - 7*B*a*e - 6*B*b*d)/(858*b*e**4)
+ (3*a + 3*b*x)*(d + e*x)**7*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(1
3*A*b*e - 7*B*a*e - 6*B*b*d)/(7722*b*e**5) + (d + e*x)**7*(a*e - b*d)**4*sqrt(a*
*2 + 2*a*b*x + b**2*x**2)*(13*A*b*e - 7*B*a*e - 6*B*b*d)/(10296*b*e**6) + (d + e
*x)**7*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(13*A*b*e - 7*B*a*e - 6*B
*b*d)/(72072*b*e**7*(a + b*x))

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Mathematica [B]  time = 1.55316, size = 876, normalized size = 2.01 \[ \frac{x \sqrt{(a+b x)^2} \left (1287 \left (8 A \left (7 d^6+21 e x d^5+35 e^2 x^2 d^4+35 e^3 x^3 d^3+21 e^4 x^4 d^2+7 e^5 x^5 d+e^6 x^6\right )+B x \left (28 d^6+112 e x d^5+210 e^2 x^2 d^4+224 e^3 x^3 d^3+140 e^4 x^4 d^2+48 e^5 x^5 d+7 e^6 x^6\right )\right ) a^5+715 b x \left (9 A \left (28 d^6+112 e x d^5+210 e^2 x^2 d^4+224 e^3 x^3 d^3+140 e^4 x^4 d^2+48 e^5 x^5 d+7 e^6 x^6\right )+2 B x \left (84 d^6+378 e x d^5+756 e^2 x^2 d^4+840 e^3 x^3 d^3+540 e^4 x^4 d^2+189 e^5 x^5 d+28 e^6 x^6\right )\right ) a^4+286 b^2 x^2 \left (10 A \left (84 d^6+378 e x d^5+756 e^2 x^2 d^4+840 e^3 x^3 d^3+540 e^4 x^4 d^2+189 e^5 x^5 d+28 e^6 x^6\right )+3 B x \left (210 d^6+1008 e x d^5+2100 e^2 x^2 d^4+2400 e^3 x^3 d^3+1575 e^4 x^4 d^2+560 e^5 x^5 d+84 e^6 x^6\right )\right ) a^3+78 b^3 x^3 \left (11 A \left (210 d^6+1008 e x d^5+2100 e^2 x^2 d^4+2400 e^3 x^3 d^3+1575 e^4 x^4 d^2+560 e^5 x^5 d+84 e^6 x^6\right )+4 B x \left (462 d^6+2310 e x d^5+4950 e^2 x^2 d^4+5775 e^3 x^3 d^3+3850 e^4 x^4 d^2+1386 e^5 x^5 d+210 e^6 x^6\right )\right ) a^2+13 b^4 x^4 \left (12 A \left (462 d^6+2310 e x d^5+4950 e^2 x^2 d^4+5775 e^3 x^3 d^3+3850 e^4 x^4 d^2+1386 e^5 x^5 d+210 e^6 x^6\right )+5 B x \left (924 d^6+4752 e x d^5+10395 e^2 x^2 d^4+12320 e^3 x^3 d^3+8316 e^4 x^4 d^2+3024 e^5 x^5 d+462 e^6 x^6\right )\right ) a+b^5 x^5 \left (13 A \left (924 d^6+4752 e x d^5+10395 e^2 x^2 d^4+12320 e^3 x^3 d^3+8316 e^4 x^4 d^2+3024 e^5 x^5 d+462 e^6 x^6\right )+6 B x \left (1716 d^6+9009 e x d^5+20020 e^2 x^2 d^4+24024 e^3 x^3 d^3+16380 e^4 x^4 d^2+6006 e^5 x^5 d+924 e^6 x^6\right )\right )\right )}{72072 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(1287*a^5*(8*A*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^
3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5*x^5 + e^6*x^6) + B*x*(28*d^6 + 112*d^5*e*x
+ 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 + 48*d*e^5*x^5 + 7*e^6*x^6
)) + 715*a^4*b*x*(9*A*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3
+ 140*d^2*e^4*x^4 + 48*d*e^5*x^5 + 7*e^6*x^6) + 2*B*x*(84*d^6 + 378*d^5*e*x + 75
6*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6))
+ 286*a^3*b^2*x^2*(10*A*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x
^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 3*B*x*(210*d^6 + 1008*d^5*e
*x + 2100*d^4*e^2*x^2 + 2400*d^3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84
*e^6*x^6)) + 78*a^2*b^3*x^3*(11*A*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^2 + 2
400*d^3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + 4*B*x*(462*d^
6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386
*d*e^5*x^5 + 210*e^6*x^6)) + 13*a*b^4*x^4*(12*A*(462*d^6 + 2310*d^5*e*x + 4950*d
^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6)
+ 5*B*x*(924*d^6 + 4752*d^5*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3*x^3 + 8316*
d^2*e^4*x^4 + 3024*d*e^5*x^5 + 462*e^6*x^6)) + b^5*x^5*(13*A*(924*d^6 + 4752*d^5
*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3*x^3 + 8316*d^2*e^4*x^4 + 3024*d*e^5*x^5
+ 462*e^6*x^6) + 6*B*x*(1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^2*x^2 + 24024*d^3
*e^3*x^3 + 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6*x^6))))/(72072*(a + b*x)
)

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Maple [B]  time = 0.017, size = 1264, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/72072*x*(5544*B*b^5*e^6*x^12+6006*A*b^5*e^6*x^11+30030*B*a*b^4*e^6*x^11+36036*
B*b^5*d*e^5*x^11+32760*A*a*b^4*e^6*x^10+39312*A*b^5*d*e^5*x^10+65520*B*a^2*b^3*e
^6*x^10+196560*B*a*b^4*d*e^5*x^10+98280*B*b^5*d^2*e^4*x^10+72072*A*a^2*b^3*e^6*x
^9+216216*A*a*b^4*d*e^5*x^9+108108*A*b^5*d^2*e^4*x^9+72072*B*a^3*b^2*e^6*x^9+432
432*B*a^2*b^3*d*e^5*x^9+540540*B*a*b^4*d^2*e^4*x^9+144144*B*b^5*d^3*e^3*x^9+8008
0*A*a^3*b^2*e^6*x^8+480480*A*a^2*b^3*d*e^5*x^8+600600*A*a*b^4*d^2*e^4*x^8+160160
*A*b^5*d^3*e^3*x^8+40040*B*a^4*b*e^6*x^8+480480*B*a^3*b^2*d*e^5*x^8+1201200*B*a^
2*b^3*d^2*e^4*x^8+800800*B*a*b^4*d^3*e^3*x^8+120120*B*b^5*d^4*e^2*x^8+45045*A*a^
4*b*e^6*x^7+540540*A*a^3*b^2*d*e^5*x^7+1351350*A*a^2*b^3*d^2*e^4*x^7+900900*A*a*
b^4*d^3*e^3*x^7+135135*A*b^5*d^4*e^2*x^7+9009*B*a^5*e^6*x^7+270270*B*a^4*b*d*e^5
*x^7+1351350*B*a^3*b^2*d^2*e^4*x^7+1801800*B*a^2*b^3*d^3*e^3*x^7+675675*B*a*b^4*
d^4*e^2*x^7+54054*B*b^5*d^5*e*x^7+10296*A*a^5*e^6*x^6+308880*A*a^4*b*d*e^5*x^6+1
544400*A*a^3*b^2*d^2*e^4*x^6+2059200*A*a^2*b^3*d^3*e^3*x^6+772200*A*a*b^4*d^4*e^
2*x^6+61776*A*b^5*d^5*e*x^6+61776*B*a^5*d*e^5*x^6+772200*B*a^4*b*d^2*e^4*x^6+205
9200*B*a^3*b^2*d^3*e^3*x^6+1544400*B*a^2*b^3*d^4*e^2*x^6+308880*B*a*b^4*d^5*e*x^
6+10296*B*b^5*d^6*x^6+72072*A*a^5*d*e^5*x^5+900900*A*a^4*b*d^2*e^4*x^5+2402400*A
*a^3*b^2*d^3*e^3*x^5+1801800*A*a^2*b^3*d^4*e^2*x^5+360360*A*a*b^4*d^5*e*x^5+1201
2*A*b^5*d^6*x^5+180180*B*a^5*d^2*e^4*x^5+1201200*B*a^4*b*d^3*e^3*x^5+1801800*B*a
^3*b^2*d^4*e^2*x^5+720720*B*a^2*b^3*d^5*e*x^5+60060*B*a*b^4*d^6*x^5+216216*A*a^5
*d^2*e^4*x^4+1441440*A*a^4*b*d^3*e^3*x^4+2162160*A*a^3*b^2*d^4*e^2*x^4+864864*A*
a^2*b^3*d^5*e*x^4+72072*A*a*b^4*d^6*x^4+288288*B*a^5*d^3*e^3*x^4+1081080*B*a^4*b
*d^4*e^2*x^4+864864*B*a^3*b^2*d^5*e*x^4+144144*B*a^2*b^3*d^6*x^4+360360*A*a^5*d^
3*e^3*x^3+1351350*A*a^4*b*d^4*e^2*x^3+1081080*A*a^3*b^2*d^5*e*x^3+180180*A*a^2*b
^3*d^6*x^3+270270*B*a^5*d^4*e^2*x^3+540540*B*a^4*b*d^5*e*x^3+180180*B*a^3*b^2*d^
6*x^3+360360*A*a^5*d^4*e^2*x^2+720720*A*a^4*b*d^5*e*x^2+240240*A*a^3*b^2*d^6*x^2
+144144*B*a^5*d^5*e*x^2+120120*B*a^4*b*d^6*x^2+216216*A*a^5*d^5*e*x+180180*A*a^4
*b*d^6*x+36036*B*a^5*d^6*x+72072*A*a^5*d^6)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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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)^(5/2)*(B*x + A)*(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273756, size = 1301, normalized size = 2.98 \[ \text{result too large to display} \]

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)^6,x, algorithm="fricas")

[Out]

1/13*B*b^5*e^6*x^13 + A*a^5*d^6*x + 1/12*(6*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^
6)*x^12 + 1/11*(15*B*b^5*d^2*e^4 + 6*(5*B*a*b^4 + A*b^5)*d*e^5 + 5*(2*B*a^2*b^3
+ A*a*b^4)*e^6)*x^11 + 1/2*(4*B*b^5*d^3*e^3 + 3*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 6*
(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 2*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^10 + 5/9*(3*B*b
^5*d^4*e^2 + 4*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 15*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4
+ 12*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*a^4*b + 2*A*a^3*b^2)*e^6)*x^9 + 1/8*(6*B
*b^5*d^5*e + 15*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 100*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^
3 + 150*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 30*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + (B*
a^5 + 5*A*a^4*b)*e^6)*x^8 + 1/7*(B*b^5*d^6 + A*a^5*e^6 + 6*(5*B*a*b^4 + A*b^5)*d
^5*e + 75*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 200*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3
+ 75*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 6*(B*a^5 + 5*A*a^4*b)*d*e^5)*x^7 + 1/6*(6
*A*a^5*d*e^5 + (5*B*a*b^4 + A*b^5)*d^6 + 30*(2*B*a^2*b^3 + A*a*b^4)*d^5*e + 150*
(B*a^3*b^2 + A*a^2*b^3)*d^4*e^2 + 100*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^3 + 15*(B*a^
5 + 5*A*a^4*b)*d^2*e^4)*x^6 + (3*A*a^5*d^2*e^4 + (2*B*a^2*b^3 + A*a*b^4)*d^6 + 1
2*(B*a^3*b^2 + A*a^2*b^3)*d^5*e + 15*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^2 + 4*(B*a^5
+ 5*A*a^4*b)*d^3*e^3)*x^5 + 5/4*(4*A*a^5*d^3*e^3 + 2*(B*a^3*b^2 + A*a^2*b^3)*d^6
+ 6*(B*a^4*b + 2*A*a^3*b^2)*d^5*e + 3*(B*a^5 + 5*A*a^4*b)*d^4*e^2)*x^4 + 1/3*(1
5*A*a^5*d^4*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^6 + 6*(B*a^5 + 5*A*a^4*b)*d^5*e)*x
^3 + 1/2*(6*A*a^5*d^5*e + (B*a^5 + 5*A*a^4*b)*d^6)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \left (d + e x\right )^{6} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**6*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.314369, 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)^6,x, algorithm="giac")

[Out]

Done