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

Optimal. Leaf size=107 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]

[Out]

(a^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^3) - (2*a*(a + b*x)^6*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(7*b^3) + ((a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(8*b^3)

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Rubi [A]  time = 0.12555, antiderivative size = 107, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^3) - (2*a*(a + b*x)^6*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(7*b^3) + ((a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(8*b^3)

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Rubi in Sympy [A]  time = 14.1659, size = 99, normalized size = 0.93 \[ \frac{a^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 b^{3}} - \frac{a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{28 b^{3}} + \frac{x^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(48*b**3) - a*(a**2 + 2*a
*b*x + b**2*x**2)**(7/2)/(28*b**3) + x**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x
**2)**(5/2)/(16*b)

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Mathematica [A]  time = 0.0370793, size = 77, normalized size = 0.72 \[ \frac{x^3 \sqrt{(a+b x)^2} \left (56 a^5+210 a^4 b x+336 a^3 b^2 x^2+280 a^2 b^3 x^3+120 a b^4 x^4+21 b^5 x^5\right )}{168 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[(a + b*x)^2]*(56*a^5 + 210*a^4*b*x + 336*a^3*b^2*x^2 + 280*a^2*b^3*x^3
+ 120*a*b^4*x^4 + 21*b^5*x^5))/(168*(a + b*x))

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Maple [A]  time = 0.008, size = 74, normalized size = 0.7 \[{\frac{{x}^{3} \left ( 21\,{b}^{5}{x}^{5}+120\,a{b}^{4}{x}^{4}+280\,{a}^{2}{b}^{3}{x}^{3}+336\,{a}^{3}{b}^{2}{x}^{2}+210\,{a}^{4}bx+56\,{a}^{5} \right ) }{168\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*x^3*(21*b^5*x^5+120*a*b^4*x^4+280*a^2*b^3*x^3+336*a^3*b^2*x^2+210*a^4*b*x+
56*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216974, size = 77, normalized size = 0.72 \[ \frac{1}{8} \, b^{5} x^{8} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{4} \, a^{4} b x^{4} + \frac{1}{3} \, a^{5} x^{3} \]

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^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212302, size = 144, normalized size = 1.35 \[ \frac{1}{8} \, b^{5} x^{8}{\rm sign}\left (b x + a\right ) + \frac{5}{7} \, a b^{4} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, a^{2} b^{3} x^{6}{\rm sign}\left (b x + a\right ) + 2 \, a^{3} b^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, a^{4} b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{5} x^{3}{\rm sign}\left (b x + a\right ) + \frac{a^{8}{\rm sign}\left (b x + a\right )}{168 \, b^{3}} \]

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^2,x, algorithm="giac")

[Out]

1/8*b^5*x^8*sign(b*x + a) + 5/7*a*b^4*x^7*sign(b*x + a) + 5/3*a^2*b^3*x^6*sign(b
*x + a) + 2*a^3*b^2*x^5*sign(b*x + a) + 5/4*a^4*b*x^4*sign(b*x + a) + 1/3*a^5*x^
3*sign(b*x + a) + 1/168*a^8*sign(b*x + a)/b^3