3.353 \(\int \frac{x^2}{(a+b x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0386187, 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 \[ -\frac{2 a^2}{3 b^3 (a+b x)^{3/2}}+\frac{4 a}{b^3 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.96574, size = 46, normalized size = 0.94 \[ - \frac{2 a^{2}}{3 b^{3} \left (a + b x\right )^{\frac{3}{2}}} + \frac{4 a}{b^{3} \sqrt{a + b x}} + \frac{2 \sqrt{a + b x}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0244345, size = 35, normalized size = 0.71 \[ \frac{2 \left (8 a^2+12 a b x+3 b^2 x^2\right )}{3 b^3 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(8*a^2 + 12*a*b*x + 3*b^2*x^2))/(3*b^3*(a + b*x)^(3/2))

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Maple [A]  time = 0.009, size = 32, normalized size = 0.7 \[{\frac{6\,{b}^{2}{x}^{2}+24\,abx+16\,{a}^{2}}{3\,{b}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)^(5/2),x)

[Out]

2/3/(b*x+a)^(3/2)*(3*b^2*x^2+12*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 1.72629, size = 55, normalized size = 1.12 \[ \frac{2 \, \sqrt{b x + a}}{b^{3}} + \frac{4 \, a}{\sqrt{b x + a} b^{3}} - \frac{2 \, a^{2}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225132, size = 55, normalized size = 1.12 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} + 12 \, a b x + 8 \, a^{2}\right )}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(3*b^2*x^2 + 12*a*b*x + 8*a^2)/((b^4*x + a*b^3)*sqrt(b*x + a))

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Sympy [A]  time = 3.58932, size = 121, normalized size = 2.47 \[ \begin{cases} \frac{16 a^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{24 a b x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{6 b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((16*a**2/(3*a*b**3*sqrt(a + b*x) + 3*b**4*x*sqrt(a + b*x)) + 24*a*b*x/
(3*a*b**3*sqrt(a + b*x) + 3*b**4*x*sqrt(a + b*x)) + 6*b**2*x**2/(3*a*b**3*sqrt(a
+ b*x) + 3*b**4*x*sqrt(a + b*x)), Ne(b, 0)), (x**3/(3*a**(5/2)), True))

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GIAC/XCAS [A]  time = 0.204343, size = 53, normalized size = 1.08 \[ \frac{2 \, \sqrt{b x + a}}{b^{3}} + \frac{2 \,{\left (6 \,{\left (b x + a\right )} a - a^{2}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

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