3.1730 \(\int \frac{x^2}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{35 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^4}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^3}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{3 a^2}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

(35*b^2*Sqrt[a + b/x]*x)/(8*a^4) - (35*b*Sqrt[a + b/x]*x^2)/(12*a^3) - (2*x^3)/(
a*Sqrt[a + b/x]) + (7*Sqrt[a + b/x]*x^3)/(3*a^2) - (35*b^3*ArcTanh[Sqrt[a + b/x]
/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.162194, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{35 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^4}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^3}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{3 a^2}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x)^(3/2),x]

[Out]

(35*b^2*Sqrt[a + b/x]*x)/(8*a^4) - (35*b*Sqrt[a + b/x]*x^2)/(12*a^3) - (2*x^3)/(
a*Sqrt[a + b/x]) + (7*Sqrt[a + b/x]*x^3)/(3*a^2) - (35*b^3*ArcTanh[Sqrt[a + b/x]
/Sqrt[a]])/(8*a^(9/2))

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Rubi in Sympy [A]  time = 16.8451, size = 100, normalized size = 0.87 \[ - \frac{2 x^{3}}{a \sqrt{a + \frac{b}{x}}} + \frac{7 x^{3} \sqrt{a + \frac{b}{x}}}{3 a^{2}} - \frac{35 b x^{2} \sqrt{a + \frac{b}{x}}}{12 a^{3}} + \frac{35 b^{2} x \sqrt{a + \frac{b}{x}}}{8 a^{4}} - \frac{35 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3/(a*sqrt(a + b/x)) + 7*x**3*sqrt(a + b/x)/(3*a**2) - 35*b*x**2*sqrt(a + b
/x)/(12*a**3) + 35*b**2*x*sqrt(a + b/x)/(8*a**4) - 35*b**3*atanh(sqrt(a + b/x)/s
qrt(a))/(8*a**(9/2))

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Mathematica [A]  time = 0.185158, size = 95, normalized size = 0.83 \[ \frac{x \sqrt{a+\frac{b}{x}} \left (8 a^3 x^3-14 a^2 b x^2+35 a b^2 x+105 b^3\right )}{24 a^4 (a x+b)}-\frac{35 b^3 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{16 a^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(105*b^3 + 35*a*b^2*x - 14*a^2*b*x^2 + 8*a^3*x^3))/(24*a^4*(b +
a*x)) - (35*b^3*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(16*a^(9/2))

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Maple [B]  time = 0.023, size = 462, normalized size = 4. \[{\frac{x}{48\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 16\,{a}^{15/2} \left ( a{x}^{2}+bx \right ) ^{3/2}{x}^{2}-60\,{a}^{15/2}\sqrt{a{x}^{2}+bx}{x}^{3}b+32\,{a}^{13/2} \left ( a{x}^{2}+bx \right ) ^{3/2}xb-150\,{a}^{13/2}\sqrt{a{x}^{2}+bx}{x}^{2}{b}^{2}+240\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{2}+16\,{a}^{11/2} \left ( a{x}^{2}+bx \right ) ^{3/2}{b}^{2}-120\,{a}^{11/2}\sqrt{a{x}^{2}+bx}x{b}^{3}-96\,{b}^{2}{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}+480\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }x{b}^{3}-30\,{a}^{9/2}\sqrt{a{x}^{2}+bx}{b}^{4}+240\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{b}^{4}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{6}{b}^{3}-120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{6}{b}^{3}+30\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{5}{b}^{4}-240\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{5}{b}^{4}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5}-120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5} \right ){a}^{-{\frac{17}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x)^(3/2),x)

[Out]

1/48*((a*x+b)/x)^(1/2)*x/a^(17/2)*(16*a^(15/2)*(a*x^2+b*x)^(3/2)*x^2-60*a^(15/2)
*(a*x^2+b*x)^(1/2)*x^3*b+32*a^(13/2)*(a*x^2+b*x)^(3/2)*x*b-150*a^(13/2)*(a*x^2+b
*x)^(1/2)*x^2*b^2+240*a^(13/2)*(x*(a*x+b))^(1/2)*x^2*b^2+16*a^(11/2)*(a*x^2+b*x)
^(3/2)*b^2-120*a^(11/2)*(a*x^2+b*x)^(1/2)*x*b^3-96*b^2*a^(11/2)*(x*(a*x+b))^(3/2
)+480*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^3-30*a^(9/2)*(a*x^2+b*x)^(1/2)*b^4+240*a^(9
/2)*(x*(a*x+b))^(1/2)*b^4+15*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*x^2*a^6*b^3-120*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^6
*b^3+30*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^5*b^4-240*ln(1
/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^5*b^4+15*ln(1/2*(2*(a*x^2+
b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*b^5-120*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(
1/2)+2*a*x+b)/a^(1/2))*a^4*b^5)/(x*(a*x+b))^(1/2)/(a*x+b)^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/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243995, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{3} \sqrt{\frac{a x + b}{x}} \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (8 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 35 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{a}}{48 \, a^{\frac{9}{2}} \sqrt{\frac{a x + b}{x}}}, \frac{105 \, b^{3} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (8 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 35 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{-a}}{24 \, \sqrt{-a} a^{4} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(105*b^3*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt
(a)) + 2*(8*a^3*x^3 - 14*a^2*b*x^2 + 35*a*b^2*x + 105*b^3)*sqrt(a))/(a^(9/2)*sqr
t((a*x + b)/x)), 1/24*(105*b^3*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x +
b)/x))) + (8*a^3*x^3 - 14*a^2*b*x^2 + 35*a*b^2*x + 105*b^3)*sqrt(-a))/(sqrt(-a)*
a^4*sqrt((a*x + b)/x))]

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Sympy [A]  time = 27.0753, size = 133, normalized size = 1.16 \[ \frac{x^{\frac{7}{2}}}{3 a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{7 \sqrt{b} x^{\frac{5}{2}}}{12 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a^{3} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{\frac{5}{2}} \sqrt{x}}{8 a^{4} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(7/2)/(3*a*sqrt(b)*sqrt(a*x/b + 1)) - 7*sqrt(b)*x**(5/2)/(12*a**2*sqrt(a*x/b
+ 1)) + 35*b**(3/2)*x**(3/2)/(24*a**3*sqrt(a*x/b + 1)) + 35*b**(5/2)*sqrt(x)/(8*
a**4*sqrt(a*x/b + 1)) - 35*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(8*a**(9/2))

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GIAC/XCAS [A]  time = 0.260497, size = 194, normalized size = 1.69 \[ \frac{1}{24} \, b{\left (\frac{105 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{48 \, b^{2}}{a^{4} \sqrt{\frac{a x + b}{x}}} - \frac{87 \, a^{2} b^{2} \sqrt{\frac{a x + b}{x}} - \frac{136 \,{\left (a x + b\right )} a b^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{57 \,{\left (a x + b\right )}^{2} b^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{{\left (a - \frac{a x + b}{x}\right )}^{3} a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b*(105*b^2*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) + 48*b^2/(a^4*
sqrt((a*x + b)/x)) - (87*a^2*b^2*sqrt((a*x + b)/x) - 136*(a*x + b)*a*b^2*sqrt((a
*x + b)/x)/x + 57*(a*x + b)^2*b^2*sqrt((a*x + b)/x)/x^2)/((a - (a*x + b)/x)^3*a^
4))