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

Optimal. Leaf size=231 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.20252, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{a b^4 x^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^5 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{5 a^3 b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.1133, size = 192, normalized size = 0.83 \[ \frac{a^{5} x^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2772 \left (a + b x\right )} + \frac{a^{4} x^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{462} + \frac{a^{3} x^{6} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{396} + \frac{2 a^{2} x^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99} + \frac{a x^{6} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{110} + \frac{x^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2772*(a + b*x)) + a**4*x**6*sqrt(a**
2 + 2*a*b*x + b**2*x**2)/462 + a**3*x**6*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**
2*x**2)/396 + 2*a**2*x**6*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/99 + a*x**6*(5*a +
5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/110 + x**6*(a**2 + 2*a*b*x + b**2*x*
*2)**(5/2)/11

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Mathematica [A]  time = 0.0392709, size = 77, normalized size = 0.33 \[ \frac{x^6 \sqrt{(a+b x)^2} \left (462 a^5+1980 a^4 b x+3465 a^3 b^2 x^2+3080 a^2 b^3 x^3+1386 a b^4 x^4+252 b^5 x^5\right )}{2772 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*Sqrt[(a + b*x)^2]*(462*a^5 + 1980*a^4*b*x + 3465*a^3*b^2*x^2 + 3080*a^2*b^3
*x^3 + 1386*a*b^4*x^4 + 252*b^5*x^5))/(2772*(a + b*x))

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Maple [A]  time = 0.009, size = 74, normalized size = 0.3 \[{\frac{{x}^{6} \left ( 252\,{b}^{5}{x}^{5}+1386\,a{b}^{4}{x}^{4}+3080\,{a}^{2}{b}^{3}{x}^{3}+3465\,{a}^{3}{b}^{2}{x}^{2}+1980\,{a}^{4}bx+462\,{a}^{5} \right ) }{2772\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2772*x^6*(252*b^5*x^5+1386*a*b^4*x^4+3080*a^2*b^3*x^3+3465*a^3*b^2*x^2+1980*a^
4*b*x+462*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)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216341, size = 77, normalized size = 0.33 \[ \frac{1}{11} \, b^{5} x^{11} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{9} \, a^{2} b^{3} x^{9} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{6} \, a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^5*x^11 + 1/2*a*b^4*x^10 + 10/9*a^2*b^3*x^9 + 5/4*a^3*b^2*x^8 + 5/7*a^4*b*
x^7 + 1/6*a^5*x^6

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210142, size = 144, normalized size = 0.62 \[ \frac{1}{11} \, b^{5} x^{11}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a b^{4} x^{10}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, a^{2} b^{3} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, a^{3} b^{2} x^{8}{\rm sign}\left (b x + a\right ) + \frac{5}{7} \, a^{4} b x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, a^{5} x^{6}{\rm sign}\left (b x + a\right ) - \frac{a^{11}{\rm sign}\left (b x + a\right )}{2772 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^5*x^11*sign(b*x + a) + 1/2*a*b^4*x^10*sign(b*x + a) + 10/9*a^2*b^3*x^9*si
gn(b*x + a) + 5/4*a^3*b^2*x^8*sign(b*x + a) + 5/7*a^4*b*x^7*sign(b*x + a) + 1/6*
a^5*x^6*sign(b*x + a) - 1/2772*a^11*sign(b*x + a)/b^6