3.1721 \(\int \frac{x^2}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]

[Out]

(5*b^2*Sqrt[a + b/x]*x)/(8*a^3) - (5*b*Sqrt[a + b/x]*x^2)/(12*a^2) + (Sqrt[a + b
/x]*x^3)/(3*a) - (5*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.129853, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b/x],x]

[Out]

(5*b^2*Sqrt[a + b/x]*x)/(8*a^3) - (5*b*Sqrt[a + b/x]*x^2)/(12*a^2) + (Sqrt[a + b
/x]*x^3)/(3*a) - (5*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 12.4221, size = 82, normalized size = 0.85 \[ \frac{x^{3} \sqrt{a + \frac{b}{x}}}{3 a} - \frac{5 b x^{2} \sqrt{a + \frac{b}{x}}}{12 a^{2}} + \frac{5 b^{2} x \sqrt{a + \frac{b}{x}}}{8 a^{3}} - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(a + b/x)/(3*a) - 5*b*x**2*sqrt(a + b/x)/(12*a**2) + 5*b**2*x*sqrt(a +
b/x)/(8*a**3) - 5*b**3*atanh(sqrt(a + b/x)/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.11871, size = 77, normalized size = 0.8 \[ \frac{x \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2-10 a b x+15 b^2\right )}{24 a^3}-\frac{5 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{16 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x*(15*b^2 - 10*a*b*x + 8*a^2*x^2))/(24*a^3) - (5*b^3*Log[b + 2*a*
x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(16*a^(7/2))

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Maple [B]  time = 0.014, size = 168, normalized size = 1.8 \[{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}-36\,b\sqrt{a{x}^{2}+bx}x{a}^{9/2}-18\,\sqrt{a{x}^{2}+bx}{b}^{2}{a}^{7/2}+48\,{b}^{2}\sqrt{x \left ( ax+b \right ) }{a}^{7/2}+9\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}-24\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x)^(1/2),x)

[Out]

1/48*((a*x+b)/x)^(1/2)*x*(16*(a*x^2+b*x)^(3/2)*a^(9/2)-36*b*(a*x^2+b*x)^(1/2)*x*
a^(9/2)-18*(a*x^2+b*x)^(1/2)*b^2*a^(7/2)+48*b^2*(x*(a*x+b))^(1/2)*a^(7/2)+9*b^3*
ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3-24*b^3*ln(1/2*(2*(x*(a
*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3)/(x*(a*x+b))^(1/2)/a^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236446, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{2} x^{3} - 10 \, a b x^{2} + 15 \, b^{2} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{48 \, a^{\frac{7}{2}}}, \frac{15 \, b^{3} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (8 \, a^{2} x^{3} - 10 \, a b x^{2} + 15 \, b^{2} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-a} a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*b^3*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(8*a^2*x^3
- 10*a*b*x^2 + 15*b^2*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(7/2), 1/24*(15*b^3*arcta
n(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (8*a^2*x^3 - 10*a*b*x^2 + 15*b^2*x)*sqrt(-a)
*sqrt((a*x + b)/x))/(sqrt(-a)*a^3)]

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Sympy [A]  time = 20.7485, size = 128, normalized size = 1.33 \[ \frac{x^{\frac{7}{2}}}{3 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{5}{2}}}{12 a \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{5}{2}} \sqrt{x}}{8 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(7/2)/(3*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(5/2)/(12*a*sqrt(a*x/b + 1)) +
5*b**(3/2)*x**(3/2)/(24*a**2*sqrt(a*x/b + 1)) + 5*b**(5/2)*sqrt(x)/(8*a**3*sqrt
(a*x/b + 1)) - 5*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(8*a**(7/2))

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GIAC/XCAS [A]  time = 0.253, size = 169, normalized size = 1.76 \[ \frac{1}{24} \, b{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{33 \, a^{2} b^{2} \sqrt{\frac{a x + b}{x}} - \frac{40 \,{\left (a x + b\right )} a b^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{15 \,{\left (a x + b\right )}^{2} b^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{{\left (a - \frac{a x + b}{x}\right )}^{3} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b*(15*b^2*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3) - (33*a^2*b^2*s
qrt((a*x + b)/x) - 40*(a*x + b)*a*b^2*sqrt((a*x + b)/x)/x + 15*(a*x + b)^2*b^2*s
qrt((a*x + b)/x)/x^2)/((a - (a*x + b)/x)^3*a^3))