3.1739 \(\int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=383 \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac{B e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \]

[Out]

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

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Rubi [A]  time = 2.18545, antiderivative size = 383, 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 \[ \frac{e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac{10 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac{5 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac{B e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 77.651, size = 410, normalized size = 1.07 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{24 b e} + \frac{\left (d + e x\right )^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (2 A b e - B a e - B b d\right )}{22 b e^{2}} + \frac{\left (5 a + 5 b x\right ) \left (d + e x\right )^{6} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (2 A b e - B a e - B b d\right )}{220 b e^{3}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (2 A b e - B a e - B b d\right )}{99 b e^{4}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{6} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{792 b e^{5}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{924 b e^{6}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{5544 b e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(2*a + 2*b*x)*(d + e*x)**6*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(24*b*e) + (d +
e*x)**6*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(2*A*b*e - B*a*e - B*b*d)/(22*b*e**
2) + (5*a + 5*b*x)*(d + e*x)**6*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*
(2*A*b*e - B*a*e - B*b*d)/(220*b*e**3) + (d + e*x)**6*(a*e - b*d)**2*(a**2 + 2*a
*b*x + b**2*x**2)**(3/2)*(2*A*b*e - B*a*e - B*b*d)/(99*b*e**4) + (3*a + 3*b*x)*(
d + e*x)**6*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(2*A*b*e - B*a*e - B
*b*d)/(792*b*e**5) + (d + e*x)**6*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2
)*(2*A*b*e - B*a*e - B*b*d)/(924*b*e**6) + (d + e*x)**6*(a*e - b*d)**5*sqrt(a**2
+ 2*a*b*x + b**2*x**2)*(2*A*b*e - B*a*e - B*b*d)/(5544*b*e**7*(a + b*x))

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Mathematica [A]  time = 1.32472, size = 740, normalized size = 1.93 \[ \frac{x \sqrt{(a+b x)^2} \left (132 a^5 \left (7 A \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+B x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )\right )+165 a^4 b x \left (4 A \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+B x \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )\right )+110 a^3 b^2 x^2 \left (3 A \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+B x \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right )+22 a^2 b^3 x^3 \left (5 A \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+2 B x \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )+2 a b^4 x^4 \left (11 A \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+5 B x \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )\right )+b^5 x^5 \left (A \left (924 d^5+3960 d^4 e x+6930 d^3 e^2 x^2+6160 d^2 e^3 x^3+2772 d e^4 x^4+504 e^5 x^5\right )+B x \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )\right )\right )}{5544 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(132*a^5*(7*A*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2
*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + B*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 +
84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)) + 165*a^4*b*x*(4*A*(21*d^5 + 70*d^4
*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + B*x*(56*d^
5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5
)) + 110*a^3*b^2*x^2*(3*A*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*
x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + B*x*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2
+ 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)) + 22*a^2*b^3*x^3*(5*A*(126*d^5
+ 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)
+ 2*B*x*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e
^4*x^4 + 126*e^5*x^5)) + 2*a*b^4*x^4*(11*A*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^
2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 5*B*x*(462*d^5 + 1980*
d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)) +
b^5*x^5*(B*x*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 25
20*d*e^4*x^4 + 462*e^5*x^5) + A*(924*d^5 + 3960*d^4*e*x + 6930*d^3*e^2*x^2 + 616
0*d^2*e^3*x^3 + 2772*d*e^4*x^4 + 504*e^5*x^5))))/(5544*(a + b*x))

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Maple [B]  time = 0.015, size = 1068, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5544*x*(462*B*b^5*e^5*x^11+504*A*b^5*e^5*x^10+2520*B*a*b^4*e^5*x^10+2520*B*b^5
*d*e^4*x^10+2772*A*a*b^4*e^5*x^9+2772*A*b^5*d*e^4*x^9+5544*B*a^2*b^3*e^5*x^9+138
60*B*a*b^4*d*e^4*x^9+5544*B*b^5*d^2*e^3*x^9+6160*A*a^2*b^3*e^5*x^8+15400*A*a*b^4
*d*e^4*x^8+6160*A*b^5*d^2*e^3*x^8+6160*B*a^3*b^2*e^5*x^8+30800*B*a^2*b^3*d*e^4*x
^8+30800*B*a*b^4*d^2*e^3*x^8+6160*B*b^5*d^3*e^2*x^8+6930*A*a^3*b^2*e^5*x^7+34650
*A*a^2*b^3*d*e^4*x^7+34650*A*a*b^4*d^2*e^3*x^7+6930*A*b^5*d^3*e^2*x^7+3465*B*a^4
*b*e^5*x^7+34650*B*a^3*b^2*d*e^4*x^7+69300*B*a^2*b^3*d^2*e^3*x^7+34650*B*a*b^4*d
^3*e^2*x^7+3465*B*b^5*d^4*e*x^7+3960*A*a^4*b*e^5*x^6+39600*A*a^3*b^2*d*e^4*x^6+7
9200*A*a^2*b^3*d^2*e^3*x^6+39600*A*a*b^4*d^3*e^2*x^6+3960*A*b^5*d^4*e*x^6+792*B*
a^5*e^5*x^6+19800*B*a^4*b*d*e^4*x^6+79200*B*a^3*b^2*d^2*e^3*x^6+79200*B*a^2*b^3*
d^3*e^2*x^6+19800*B*a*b^4*d^4*e*x^6+792*B*b^5*d^5*x^6+924*A*a^5*e^5*x^5+23100*A*
a^4*b*d*e^4*x^5+92400*A*a^3*b^2*d^2*e^3*x^5+92400*A*a^2*b^3*d^3*e^2*x^5+23100*A*
a*b^4*d^4*e*x^5+924*A*b^5*d^5*x^5+4620*B*a^5*d*e^4*x^5+46200*B*a^4*b*d^2*e^3*x^5
+92400*B*a^3*b^2*d^3*e^2*x^5+46200*B*a^2*b^3*d^4*e*x^5+4620*B*a*b^4*d^5*x^5+5544
*A*a^5*d*e^4*x^4+55440*A*a^4*b*d^2*e^3*x^4+110880*A*a^3*b^2*d^3*e^2*x^4+55440*A*
a^2*b^3*d^4*e*x^4+5544*A*a*b^4*d^5*x^4+11088*B*a^5*d^2*e^3*x^4+55440*B*a^4*b*d^3
*e^2*x^4+55440*B*a^3*b^2*d^4*e*x^4+11088*B*a^2*b^3*d^5*x^4+13860*A*a^5*d^2*e^3*x
^3+69300*A*a^4*b*d^3*e^2*x^3+69300*A*a^3*b^2*d^4*e*x^3+13860*A*a^2*b^3*d^5*x^3+1
3860*B*a^5*d^3*e^2*x^3+34650*B*a^4*b*d^4*e*x^3+13860*B*a^3*b^2*d^5*x^3+18480*A*a
^5*d^3*e^2*x^2+46200*A*a^4*b*d^4*e*x^2+18480*A*a^3*b^2*d^5*x^2+9240*B*a^5*d^4*e*
x^2+9240*B*a^4*b*d^5*x^2+13860*A*a^5*d^4*e*x+13860*A*a^4*b*d^5*x+2772*B*a^5*d^5*
x+5544*A*a^5*d^5)*((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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272501, size = 1098, normalized size = 2.87 \[ \text{result too large to display} \]

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)^5,x, algorithm="fricas")

[Out]

1/12*B*b^5*e^5*x^12 + A*a^5*d^5*x + 1/11*(5*B*b^5*d*e^4 + (5*B*a*b^4 + A*b^5)*e^
5)*x^11 + 1/2*(2*B*b^5*d^2*e^3 + (5*B*a*b^4 + A*b^5)*d*e^4 + (2*B*a^2*b^3 + A*a*
b^4)*e^5)*x^10 + 5/9*(2*B*b^5*d^3*e^2 + 2*(5*B*a*b^4 + A*b^5)*d^2*e^3 + 5*(2*B*a
^2*b^3 + A*a*b^4)*d*e^4 + 2*(B*a^3*b^2 + A*a^2*b^3)*e^5)*x^9 + 5/8*(B*b^5*d^4*e
+ 2*(5*B*a*b^4 + A*b^5)*d^3*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^3 + 10*(B*a^3
*b^2 + A*a^2*b^3)*d*e^4 + (B*a^4*b + 2*A*a^3*b^2)*e^5)*x^8 + 1/7*(B*b^5*d^5 + 5*
(5*B*a*b^4 + A*b^5)*d^4*e + 50*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^2 + 100*(B*a^3*b^2
+ A*a^2*b^3)*d^2*e^3 + 25*(B*a^4*b + 2*A*a^3*b^2)*d*e^4 + (B*a^5 + 5*A*a^4*b)*e^
5)*x^7 + 1/6*(A*a^5*e^5 + (5*B*a*b^4 + A*b^5)*d^5 + 25*(2*B*a^2*b^3 + A*a*b^4)*d
^4*e + 100*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^2 + 50*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^3
+ 5*(B*a^5 + 5*A*a^4*b)*d*e^4)*x^6 + (A*a^5*d*e^4 + (2*B*a^2*b^3 + A*a*b^4)*d^5
+ 10*(B*a^3*b^2 + A*a^2*b^3)*d^4*e + 10*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^2 + 2*(B*a
^5 + 5*A*a^4*b)*d^2*e^3)*x^5 + 5/4*(2*A*a^5*d^2*e^3 + 2*(B*a^3*b^2 + A*a^2*b^3)*
d^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^4*e + 2*(B*a^5 + 5*A*a^4*b)*d^3*e^2)*x^4 + 5/3
*(2*A*a^5*d^3*e^2 + (B*a^4*b + 2*A*a^3*b^2)*d^5 + (B*a^5 + 5*A*a^4*b)*d^4*e)*x^3
+ 1/2*(5*A*a^5*d^4*e + (B*a^5 + 5*A*a^4*b)*d^5)*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.300821, 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)^5,x, algorithm="giac")

[Out]

Done