3.2190 \(\int \frac{x^2}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{2 a^5 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 a^4 \sqrt{x}}{b^5}-\frac{a^3 x}{b^4}+\frac{2 a^2 x^{3/2}}{3 b^3}-\frac{a x^2}{2 b^2}+\frac{2 x^{5/2}}{5 b} \]

[Out]

(2*a^4*Sqrt[x])/b^5 - (a^3*x)/b^4 + (2*a^2*x^(3/2))/(3*b^3) - (a*x^2)/(2*b^2) +
(2*x^(5/2))/(5*b) - (2*a^5*Log[a + b*Sqrt[x]])/b^6

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Rubi [A]  time = 0.115811, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^5 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 a^4 \sqrt{x}}{b^5}-\frac{a^3 x}{b^4}+\frac{2 a^2 x^{3/2}}{3 b^3}-\frac{a x^2}{2 b^2}+\frac{2 x^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*Sqrt[x]),x]

[Out]

(2*a^4*Sqrt[x])/b^5 - (a^3*x)/b^4 + (2*a^2*x^(3/2))/(3*b^3) - (a*x^2)/(2*b^2) +
(2*x^(5/2))/(5*b) - (2*a^5*Log[a + b*Sqrt[x]])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{5} \log{\left (a + b \sqrt{x} \right )}}{b^{6}} - \frac{2 a^{3} \int ^{\sqrt{x}} x\, dx}{b^{4}} + \frac{2 a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{a x^{2}}{2 b^{2}} + \frac{2 x^{\frac{5}{2}}}{5 b} + \frac{2 \int ^{\sqrt{x}} a^{4}\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b*x**(1/2)),x)

[Out]

-2*a**5*log(a + b*sqrt(x))/b**6 - 2*a**3*Integral(x, (x, sqrt(x)))/b**4 + 2*a**2
*x**(3/2)/(3*b**3) - a*x**2/(2*b**2) + 2*x**(5/2)/(5*b) + 2*Integral(a**4, (x, s
qrt(x)))/b**5

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Mathematica [A]  time = 0.0235146, size = 75, normalized size = 0.95 \[ \frac{-60 a^5 \log \left (a+b \sqrt{x}\right )+60 a^4 b \sqrt{x}-30 a^3 b^2 x+20 a^2 b^3 x^{3/2}-15 a b^4 x^2+12 b^5 x^{5/2}}{30 b^6} \]

Antiderivative was successfully verified.

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

[Out]

(60*a^4*b*Sqrt[x] - 30*a^3*b^2*x + 20*a^2*b^3*x^(3/2) - 15*a*b^4*x^2 + 12*b^5*x^
(5/2) - 60*a^5*Log[a + b*Sqrt[x]])/(30*b^6)

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Maple [A]  time = 0.005, size = 66, normalized size = 0.8 \[ -{\frac{{a}^{3}x}{{b}^{4}}}+{\frac{2\,{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{a{x}^{2}}{2\,{b}^{2}}}+{\frac{2}{5\,b}{x}^{{\frac{5}{2}}}}-2\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+2\,{\frac{{a}^{4}\sqrt{x}}{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/2)),x)

[Out]

-a^3*x/b^4+2/3*a^2*x^(3/2)/b^3-1/2*a*x^2/b^2+2/5*x^(5/2)/b-2*a^5*ln(a+b*x^(1/2))
/b^6+2*a^4*x^(1/2)/b^5

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Maxima [A]  time = 1.44257, size = 128, normalized size = 1.62 \[ -\frac{2 \, a^{5} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{3 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )} a^{4}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a^5*log(b*sqrt(x) + a)/b^6 + 2/5*(b*sqrt(x) + a)^5/b^6 - 5/2*(b*sqrt(x) + a)^
4*a/b^6 + 20/3*(b*sqrt(x) + a)^3*a^2/b^6 - 10*(b*sqrt(x) + a)^2*a^3/b^6 + 10*(b*
sqrt(x) + a)*a^4/b^6

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Fricas [A]  time = 0.240372, size = 89, normalized size = 1.13 \[ -\frac{15 \, a b^{4} x^{2} + 30 \, a^{3} b^{2} x + 60 \, a^{5} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (3 \, b^{5} x^{2} + 5 \, a^{2} b^{3} x + 15 \, a^{4} b\right )} \sqrt{x}}{30 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(15*a*b^4*x^2 + 30*a^3*b^2*x + 60*a^5*log(b*sqrt(x) + a) - 4*(3*b^5*x^2 +
5*a^2*b^3*x + 15*a^4*b)*sqrt(x))/b^6

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Sympy [A]  time = 1.47042, size = 82, normalized size = 1.04 \[ \begin{cases} - \frac{2 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{6}} + \frac{2 a^{4} \sqrt{x}}{b^{5}} - \frac{a^{3} x}{b^{4}} + \frac{2 a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{a x^{2}}{2 b^{2}} + \frac{2 x^{\frac{5}{2}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b*x**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220805, size = 90, normalized size = 1.14 \[ -\frac{2 \, a^{5}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{12 \, b^{4} x^{\frac{5}{2}} - 15 \, a b^{3} x^{2} + 20 \, a^{2} b^{2} x^{\frac{3}{2}} - 30 \, a^{3} b x + 60 \, a^{4} \sqrt{x}}{30 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a^5*ln(abs(b*sqrt(x) + a))/b^6 + 1/30*(12*b^4*x^(5/2) - 15*a*b^3*x^2 + 20*a^2
*b^2*x^(3/2) - 30*a^3*b*x + 60*a^4*sqrt(x))/b^5