3.63 \(\int \frac{x^3}{\left (a x+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}} \]

[Out]

(2*x^3)/(3*a*(a*x + b*x^2)^(3/2))

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Rubi [A]  time = 0.0337038, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a*x + b*x^2)^(5/2),x]

[Out]

(2*x^3)/(3*a*(a*x + b*x^2)^(3/2))

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Rubi in Sympy [A]  time = 3.74278, size = 19, normalized size = 0.83 \[ \frac{2 x^{3}}{3 a \left (a x + b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**2+a*x)**(5/2),x)

[Out]

2*x**3/(3*a*(a*x + b*x**2)**(3/2))

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Mathematica [A]  time = 0.026197, size = 21, normalized size = 0.91 \[ \frac{2 x^3}{3 a (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a*x + b*x^2)^(5/2),x]

[Out]

(2*x^3)/(3*a*(x*(a + b*x))^(3/2))

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Maple [A]  time = 0.006, size = 25, normalized size = 1.1 \[{\frac{2\,{x}^{4} \left ( bx+a \right ) }{3\,a} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a*x)^(5/2),x)

[Out]

2/3*x^4*(b*x+a)/a/(b*x^2+a*x)^(5/2)

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Maxima [A]  time = 0.701799, size = 100, normalized size = 4.35 \[ -\frac{x^{2}}{{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} - \frac{a x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b^{2}} + \frac{2 \, x}{3 \, \sqrt{b x^{2} + a x} a b} + \frac{1}{3 \, \sqrt{b x^{2} + a x} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224508, size = 35, normalized size = 1.52 \[ \frac{2 \, x^{2}}{3 \,{\left (a b x + a^{2}\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x**3/(x*(a + b*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.222985, size = 120, normalized size = 5.22 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} b^{\frac{3}{2}} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a b + a^{2} \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a*x)^(5/2),x, algorithm="giac")

[Out]

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