3.164 \(\int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=181 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5} \]

[Out]

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

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Rubi [A]  time = 0.155792, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.6444, size = 180, normalized size = 0.99 \[ \frac{a^{4} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{360 b^{5}} - \frac{a^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{210 b^{5}} + \frac{a^{2} x^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{120 b^{3}} - \frac{a x^{3} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{45 b^{2}} + \frac{x^{4} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{20 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

a**4*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(360*b**5) - a**3*(a**2 +
2*a*b*x + b**2*x**2)**(7/2)/(210*b**5) + a**2*x**2*(2*a + 2*b*x)*(a**2 + 2*a*b*
x + b**2*x**2)**(5/2)/(120*b**3) - a*x**3*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x
**2)**(5/2)/(45*b**2) + x**4*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(
20*b)

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Mathematica [A]  time = 0.0325231, size = 77, normalized size = 0.43 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (252 a^5+1050 a^4 b x+1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+700 a b^4 x^4+126 b^5 x^5\right )}{1260 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(252*a^5 + 1050*a^4*b*x + 1800*a^3*b^2*x^2 + 1575*a^2*b^3
*x^3 + 700*a*b^4*x^4 + 126*b^5*x^5))/(1260*(a + b*x))

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Maple [A]  time = 0.008, size = 74, normalized size = 0.4 \[{\frac{{x}^{5} \left ( 126\,{b}^{5}{x}^{5}+700\,a{b}^{4}{x}^{4}+1575\,{a}^{2}{b}^{3}{x}^{3}+1800\,{a}^{3}{b}^{2}{x}^{2}+1050\,{a}^{4}bx+252\,{a}^{5} \right ) }{1260\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/1260*x^5*(126*b^5*x^5+700*a*b^4*x^4+1575*a^2*b^3*x^3+1800*a^3*b^2*x^2+1050*a^4
*b*x+252*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238955, size = 77, normalized size = 0.43 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} \]

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^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**4*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.213294, size = 144, normalized size = 0.8 \[ \frac{1}{10} \, b^{5} x^{10}{\rm sign}\left (b x + a\right ) + \frac{5}{9} \, a b^{4} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, a^{2} b^{3} x^{8}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, a^{3} b^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{6} \, a^{4} b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a^{5} x^{5}{\rm sign}\left (b x + a\right ) + \frac{a^{10}{\rm sign}\left (b x + a\right )}{1260 \, b^{5}} \]

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^4,x, algorithm="giac")

[Out]

1/10*b^5*x^10*sign(b*x + a) + 5/9*a*b^4*x^9*sign(b*x + a) + 5/4*a^2*b^3*x^8*sign
(b*x + a) + 10/7*a^3*b^2*x^7*sign(b*x + a) + 5/6*a^4*b*x^6*sign(b*x + a) + 1/5*a
^5*x^5*sign(b*x + a) + 1/1260*a^10*sign(b*x + a)/b^5