3.345 \(\int \frac{x^2}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]

[Out]

(-2*a^2)/(b^3*Sqrt[a + b*x]) - (4*a*Sqrt[a + b*x])/b^3 + (2*(a + b*x)^(3/2))/(3*
b^3)

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Rubi [A]  time = 0.0382472, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 a^2}{b^3 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^3}+\frac{2 (a+b x)^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^(3/2),x]

[Out]

(-2*a^2)/(b^3*Sqrt[a + b*x]) - (4*a*Sqrt[a + b*x])/b^3 + (2*(a + b*x)^(3/2))/(3*
b^3)

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Rubi in Sympy [A]  time = 7.88459, size = 46, normalized size = 0.94 \[ - \frac{2 a^{2}}{b^{3} \sqrt{a + b x}} - \frac{4 a \sqrt{a + b x}}{b^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(b**3*sqrt(a + b*x)) - 4*a*sqrt(a + b*x)/b**3 + 2*(a + b*x)**(3/2)/(3*b*
*3)

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Mathematica [A]  time = 0.0207877, size = 34, normalized size = 0.69 \[ \frac{2 \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-8*a^2 - 4*a*b*x + b^2*x^2))/(3*b^3*Sqrt[a + b*x])

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Maple [A]  time = 0.009, size = 32, normalized size = 0.7 \[ -{\frac{-2\,{b}^{2}{x}^{2}+8\,abx+16\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(3/2),x)

[Out]

-2/3/(b*x+a)^(1/2)*(-b^2*x^2+4*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 1.34362, size = 55, normalized size = 1.12 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{3}} - \frac{4 \, \sqrt{b x + a} a}{b^{3}} - \frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

2/3*(b*x + a)^(3/2)/b^3 - 4*sqrt(b*x + a)*a/b^3 - 2*a^2/(sqrt(b*x + a)*b^3)

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Fricas [A]  time = 0.224363, size = 41, normalized size = 0.84 \[ \frac{2 \,{\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )}}{3 \, \sqrt{b x + a} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

2/3*(b^2*x^2 - 4*a*b*x - 8*a^2)/(sqrt(b*x + a)*b^3)

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Sympy [A]  time = 5.53629, size = 534, normalized size = 10.9 \[ - \frac{16 a^{\frac{19}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{40 a^{\frac{17}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{30 a^{\frac{15}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{48 a^{\frac{15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac{4 a^{\frac{13}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{16 a^{\frac{13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac{2 a^{\frac{11}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(19/2)*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 +
3*a**5*b**6*x**3) + 16*a**(19/2)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2
+ 3*a**5*b**6*x**3) - 40*a**(17/2)*b*x*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b*
*4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(17/2)*b*x/(3*a**8*b**3 + 9*
a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 30*a**(15/2)*b**2*x**2*sqrt
(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3)
+ 48*a**(15/2)*b**2*x**2/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**
5*b**6*x**3) - 4*a**(13/2)*b**3*x**3*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*
x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 16*a**(13/2)*b**3*x**3/(3*a**8*b**3 +
9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 2*a**(11/2)*b**4*x**4*sq
rt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3
)

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GIAC/XCAS [A]  time = 0.204529, size = 62, normalized size = 1.27 \[ -\frac{2 \, a^{2}}{\sqrt{b x + a} b^{3}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} b^{6} - 6 \, \sqrt{b x + a} a b^{6}\right )}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

-2*a^2/(sqrt(b*x + a)*b^3) + 2/3*((b*x + a)^(3/2)*b^6 - 6*sqrt(b*x + a)*a*b^6)/b
^9