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

Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2} \]

[Out]

-(a*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (a^2 + 2*a*b*x + b^2*x^
2)^(7/2)/(7*b^2)

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Rubi [A]  time = 0.053369, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (a^2 + 2*a*b*x + b^2*x^
2)^(7/2)/(7*b^2)

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Rubi in Sympy [A]  time = 7.53478, size = 60, normalized size = 0.98 \[ - \frac{a \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 b^{2}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*b**2) + (a**2 + 2*a*b*x
+ b**2*x**2)**(7/2)/(7*b**2)

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Mathematica [A]  time = 0.0313385, size = 77, normalized size = 1.26 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (21 a^5+70 a^4 b x+105 a^3 b^2 x^2+84 a^2 b^3 x^3+35 a b^4 x^4+6 b^5 x^5\right )}{42 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x)^2]*(21*a^5 + 70*a^4*b*x + 105*a^3*b^2*x^2 + 84*a^2*b^3*x^3 +
35*a*b^4*x^4 + 6*b^5*x^5))/(42*(a + b*x))

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Maple [A]  time = 0.007, size = 74, normalized size = 1.2 \[{\frac{{x}^{2} \left ( 6\,{b}^{5}{x}^{5}+35\,a{b}^{4}{x}^{4}+84\,{a}^{2}{b}^{3}{x}^{3}+105\,{a}^{3}{b}^{2}{x}^{2}+70\,{a}^{4}bx+21\,{a}^{5} \right ) }{42\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*x^2*(6*b^5*x^5+35*a*b^4*x^4+84*a^2*b^3*x^3+105*a^3*b^2*x^2+70*a^4*b*x+21*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225255, size = 77, normalized size = 1.26 \[ \frac{1}{7} \, b^{5} x^{7} + \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{2} \, a^{5} x^{2} \]

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,x, algorithm="fricas")

[Out]

1/7*b^5*x^7 + 5/6*a*b^4*x^6 + 2*a^2*b^3*x^5 + 5/2*a^3*b^2*x^4 + 5/3*a^4*b*x^3 +
1/2*a^5*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.211692, size = 144, normalized size = 2.36 \[ \frac{1}{7} \, b^{5} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{6} \, a b^{4} x^{6}{\rm sign}\left (b x + a\right ) + 2 \, a^{2} b^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{3} b^{2} x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, a^{4} b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{5} x^{2}{\rm sign}\left (b x + a\right ) - \frac{a^{7}{\rm sign}\left (b x + a\right )}{42 \, b^{2}} \]

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,x, algorithm="giac")

[Out]

1/7*b^5*x^7*sign(b*x + a) + 5/6*a*b^4*x^6*sign(b*x + a) + 2*a^2*b^3*x^5*sign(b*x
+ a) + 5/2*a^3*b^2*x^4*sign(b*x + a) + 5/3*a^4*b*x^3*sign(b*x + a) + 1/2*a^5*x^
2*sign(b*x + a) - 1/42*a^7*sign(b*x + a)/b^2