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

Optimal. Leaf size=306 \[ \frac{10 a^2 b^2 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{b^4 x^{11} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{a b^3 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac{b^5 B x^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 (a+b x)}+\frac{a^5 A x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{a^4 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{5 a^3 b x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.460913, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{10 a^2 b^2 x^9 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{b^4 x^{11} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{a b^3 x^{10} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac{b^5 B x^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{12 (a+b x)}+\frac{a^5 A x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{a^4 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{5 a^3 b x^8 \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.4004, size = 287, normalized size = 0.94 \[ \frac{B x^{6} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{24 b} + \frac{a^{5} x^{6} \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5544 b \left (a + b x\right )} + \frac{a^{4} x^{6} \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{924 b} + \frac{a^{3} x^{6} \left (3 a + 3 b x\right ) \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{792 b} + \frac{a^{2} x^{6} \left (2 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 b} + \frac{a x^{6} \left (5 a + 5 b x\right ) \left (2 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{220 b} + \frac{x^{6} \left (2 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{22 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.083985, size = 125, normalized size = 0.41 \[ \frac{x^6 \sqrt{(a+b x)^2} \left (132 a^5 (7 A+6 B x)+495 a^4 b x (8 A+7 B x)+770 a^3 b^2 x^2 (9 A+8 B x)+616 a^2 b^3 x^3 (10 A+9 B x)+252 a b^4 x^4 (11 A+10 B x)+42 b^5 x^5 (12 A+11 B x)\right )}{5544 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*Sqrt[(a + b*x)^2]*(132*a^5*(7*A + 6*B*x) + 495*a^4*b*x*(8*A + 7*B*x) + 770*
a^3*b^2*x^2*(9*A + 8*B*x) + 616*a^2*b^3*x^3*(10*A + 9*B*x) + 252*a*b^4*x^4*(11*A
+ 10*B*x) + 42*b^5*x^5*(12*A + 11*B*x)))/(5544*(a + b*x))

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Maple [A]  time = 0.01, size = 140, normalized size = 0.5 \[{\frac{{x}^{6} \left ( 462\,B{b}^{5}{x}^{6}+504\,{x}^{5}A{b}^{5}+2520\,{x}^{5}Ba{b}^{4}+2772\,{x}^{4}Aa{b}^{4}+5544\,{x}^{4}B{a}^{2}{b}^{3}+6160\,{x}^{3}A{a}^{2}{b}^{3}+6160\,{x}^{3}B{a}^{3}{b}^{2}+6930\,{x}^{2}A{a}^{3}{b}^{2}+3465\,{x}^{2}B{a}^{4}b+3960\,xA{a}^{4}b+792\,xB{a}^{5}+924\,A{a}^{5} \right ) }{5544\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5544*x^6*(462*B*b^5*x^6+504*A*b^5*x^5+2520*B*a*b^4*x^5+2772*A*a*b^4*x^4+5544*B
*a^2*b^3*x^4+6160*A*a^2*b^3*x^3+6160*B*a^3*b^2*x^3+6930*A*a^3*b^2*x^2+3465*B*a^4
*b*x^2+3960*A*a^4*b*x+792*B*a^5*x+924*A*a^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)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273507, size = 161, normalized size = 0.53 \[ \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{11} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{5}{8} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]

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)*x^5,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272333, size = 297, normalized size = 0.97 \[ \frac{1}{12} \, B b^{5} x^{12}{\rm sign}\left (b x + a\right ) + \frac{5}{11} \, B a b^{4} x^{11}{\rm sign}\left (b x + a\right ) + \frac{1}{11} \, A b^{5} x^{11}{\rm sign}\left (b x + a\right ) + B a^{2} b^{3} x^{10}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A a b^{4} x^{10}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, B a^{3} b^{2} x^{9}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, A a^{2} b^{3} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{8} \, B a^{4} b x^{8}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, A a^{3} b^{2} x^{8}{\rm sign}\left (b x + a\right ) + \frac{1}{7} \, B a^{5} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{7} \, A a^{4} b x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, A a^{5} x^{6}{\rm sign}\left (b x + a\right ) + \frac{{\left (B a^{12} - 2 \, A a^{11} b\right )}{\rm sign}\left (b x + a\right )}{5544 \, b^{7}} \]

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)*x^5,x, algorithm="giac")

[Out]

1/12*B*b^5*x^12*sign(b*x + a) + 5/11*B*a*b^4*x^11*sign(b*x + a) + 1/11*A*b^5*x^1
1*sign(b*x + a) + B*a^2*b^3*x^10*sign(b*x + a) + 1/2*A*a*b^4*x^10*sign(b*x + a)
+ 10/9*B*a^3*b^2*x^9*sign(b*x + a) + 10/9*A*a^2*b^3*x^9*sign(b*x + a) + 5/8*B*a^
4*b*x^8*sign(b*x + a) + 5/4*A*a^3*b^2*x^8*sign(b*x + a) + 1/7*B*a^5*x^7*sign(b*x
+ a) + 5/7*A*a^4*b*x^7*sign(b*x + a) + 1/6*A*a^5*x^6*sign(b*x + a) + 1/5544*(B*
a^12 - 2*A*a^11*b)*sign(b*x + a)/b^7