3.149 \(\int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=151 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^2 b x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{a^3 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.133821, 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 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^2 b x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{a^3 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.6559, size = 124, normalized size = 0.82 \[ \frac{a^{3} x^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 \left (a + b x\right )} + \frac{a^{2} x^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56} + \frac{a x^{5} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56} + \frac{x^{5} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8} \]

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)**(3/2),x)

[Out]

a**3*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*(a + b*x)) + a**2*x**5*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/56 + a*x**5*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/56 + x**5*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/8

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Mathematica [A]  time = 0.0234432, size = 55, normalized size = 0.36 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (56 a^3+140 a^2 b x+120 a b^2 x^2+35 b^3 x^3\right )}{280 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(56*a^3 + 140*a^2*b*x + 120*a*b^2*x^2 + 35*b^3*x^3))/(280
*(a + b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.3 \[{\frac{{x}^{5} \left ( 35\,{b}^{3}{x}^{3}+120\,a{b}^{2}{x}^{2}+140\,{a}^{2}bx+56\,{a}^{3} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/280*x^5*(35*b^3*x^3+120*a*b^2*x^2+140*a^2*b*x+56*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224403, size = 47, normalized size = 0.31 \[ \frac{1}{8} \, b^{3} x^{8} + \frac{3}{7} \, a b^{2} x^{7} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{5} \, a^{3} x^{5} \]

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^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral(x**4*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.210503, size = 99, normalized size = 0.66 \[ \frac{1}{8} \, b^{3} x^{8}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, a b^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{2} b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{a^{8}{\rm sign}\left (b x + a\right )}{280 \, b^{5}} \]

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^4,x, algorithm="giac")

[Out]

1/8*b^3*x^8*sign(b*x + a) + 3/7*a*b^2*x^7*sign(b*x + a) + 1/2*a^2*b*x^6*sign(b*x
+ a) + 1/5*a^3*x^5*sign(b*x + a) + 1/280*a^8*sign(b*x + a)/b^5