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

Optimal. Leaf size=32 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.018848, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 1.93196, size = 31, normalized size = 0.97 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0217726, size = 23, normalized size = 0.72 \[ \frac{(a+b x) \left ((a+b x)^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((a + b*x)^2)^(5/2))/(6*b)

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Maple [B]  time = 0.005, size = 71, normalized size = 2.2 \[{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x*(b^5*x^5+6*a*b^4*x^4+15*a^2*b^3*x^3+20*a^3*b^2*x^2+15*a^4*b*x+6*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218834, size = 72, normalized size = 2.25 \[ \frac{1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac{5}{2} \, a^{2} b^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} x^{3} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} x \]

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, algorithm="fricas")

[Out]

1/6*b^5*x^6 + a*b^4*x^5 + 5/2*a^2*b^3*x^4 + 10/3*a^3*b^2*x^3 + 5/2*a^4*b*x^2 + a
^5*x

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

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.211301, size = 139, normalized size = 4.34 \[ \frac{1}{6} \, b^{5} x^{6}{\rm sign}\left (b x + a\right ) + a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + a^{5} x{\rm sign}\left (b x + a\right ) + \frac{a^{6}{\rm sign}\left (b x + a\right )}{6 \, b} \]

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, algorithm="giac")

[Out]

1/6*b^5*x^6*sign(b*x + a) + a*b^4*x^5*sign(b*x + a) + 5/2*a^2*b^3*x^4*sign(b*x +
a) + 10/3*a^3*b^2*x^3*sign(b*x + a) + 5/2*a^4*b*x^2*sign(b*x + a) + a^5*x*sign(
b*x + a) + 1/6*a^6*sign(b*x + a)/b