3.148 \(\int x^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=151 \[ \frac{3 a b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^3 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.140513, antiderivative size = 151, 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 \[ \frac{3 a b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{3 a^2 b x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{b^3 x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^3 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.9126, size = 124, normalized size = 0.82 \[ \frac{a^{3} x^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{504 \left (a + b x\right )} + \frac{a^{2} x^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{84} + \frac{a x^{6} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{72} + \frac{x^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9} \]

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)**(3/2),x)

[Out]

a**3*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(504*(a + b*x)) + a**2*x**6*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/84 + a*x**6*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/72 + x**6*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/9

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Mathematica [A]  time = 0.0302662, size = 55, normalized size = 0.36 \[ \frac{x^6 \sqrt{(a+b x)^2} \left (84 a^3+216 a^2 b x+189 a b^2 x^2+56 b^3 x^3\right )}{504 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x^6*Sqrt[(a + b*x)^2]*(84*a^3 + 216*a^2*b*x + 189*a*b^2*x^2 + 56*b^3*x^3))/(504
*(a + b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.3 \[{\frac{{x}^{6} \left ( 56\,{b}^{3}{x}^{3}+189\,a{b}^{2}{x}^{2}+216\,{a}^{2}bx+84\,{a}^{3} \right ) }{504\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/504*x^6*(56*b^3*x^3+189*a*b^2*x^2+216*a^2*b*x+84*a^3)*((b*x+a)^2)^(3/2)/(b*x+a
)^3

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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)^(3/2)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220699, size = 47, normalized size = 0.31 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{7} \, a^{2} b x^{7} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^5,x, algorithm="fricas")

[Out]

1/9*b^3*x^9 + 3/8*a*b^2*x^8 + 3/7*a^2*b*x^7 + 1/6*a^3*x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral(x**5*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.211841, size = 99, normalized size = 0.66 \[ \frac{1}{9} \, b^{3} x^{9}{\rm sign}\left (b x + a\right ) + \frac{3}{8} \, a b^{2} x^{8}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, a^{2} b x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, a^{3} x^{6}{\rm sign}\left (b x + a\right ) - \frac{a^{9}{\rm sign}\left (b x + a\right )}{504 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^5,x, algorithm="giac")

[Out]

1/9*b^3*x^9*sign(b*x + a) + 3/8*a*b^2*x^8*sign(b*x + a) + 3/7*a^2*b*x^7*sign(b*x
+ a) + 1/6*a^3*x^6*sign(b*x + a) - 1/504*a^9*sign(b*x + a)/b^6