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

Optimal. Leaf size=51 \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0535306, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 (a+2 b x)}{3 a^2 b \sqrt{a x+b x^2}}-\frac{2 x}{3 b \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a*x + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.45877, size = 44, normalized size = 0.86 \[ - \frac{2 x}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 4 b x}{3 a^{2} b \sqrt{a x + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0273374, size = 29, normalized size = 0.57 \[ \frac{2 x^2 (3 a+2 b x)}{3 a^2 (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 33, normalized size = 0.7 \[{\frac{2\,{x}^{3} \left ( bx+a \right ) \left ( 2\,bx+3\,a \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a*x)^(5/2),x)

[Out]

2/3*x^3*(b*x+a)*(2*b*x+3*a)/a^2/(b*x^2+a*x)^(5/2)

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Maxima [A]  time = 0.708414, size = 73, normalized size = 1.43 \[ \frac{4 \, x}{3 \, \sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} + \frac{2}{3 \, \sqrt{b x^{2} + a x} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219028, size = 49, normalized size = 0.96 \[ \frac{2 \,{\left (2 \, b x^{2} + 3 \, a x\right )}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225079, size = 82, normalized size = 1.61 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} b + 2 \, a \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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