3.1741 \(\int \frac{x}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=114 \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{35 b x \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{35 x^2 \sqrt{a+\frac{b}{x}}}{6 a^3}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-35*b*Sqrt[a + b/x]*x)/(4*a^4) - (2*x^2)/(3*a*(a + b/x)^(3/2)) - (14*x^2)/(3*a^
2*Sqrt[a + b/x]) + (35*Sqrt[a + b/x]*x^2)/(6*a^3) + (35*b^2*ArcTanh[Sqrt[a + b/x
]/Sqrt[a]])/(4*a^(9/2))

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Rubi [A]  time = 0.152204, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{35 b x \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{35 x^2 \sqrt{a+\frac{b}{x}}}{6 a^3}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x)^(5/2),x]

[Out]

(-35*b*Sqrt[a + b/x]*x)/(4*a^4) - (2*x^2)/(3*a*(a + b/x)^(3/2)) - (14*x^2)/(3*a^
2*Sqrt[a + b/x]) + (35*Sqrt[a + b/x]*x^2)/(6*a^3) + (35*b^2*ArcTanh[Sqrt[a + b/x
]/Sqrt[a]])/(4*a^(9/2))

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Rubi in Sympy [A]  time = 16.3529, size = 99, normalized size = 0.87 \[ - \frac{2 x^{2}}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{14 x^{2}}{3 a^{2} \sqrt{a + \frac{b}{x}}} + \frac{35 x^{2} \sqrt{a + \frac{b}{x}}}{6 a^{3}} - \frac{35 b x \sqrt{a + \frac{b}{x}}}{4 a^{4}} + \frac{35 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**2/(3*a*(a + b/x)**(3/2)) - 14*x**2/(3*a**2*sqrt(a + b/x)) + 35*x**2*sqrt(a
+ b/x)/(6*a**3) - 35*b*x*sqrt(a + b/x)/(4*a**4) + 35*b**2*atanh(sqrt(a + b/x)/s
qrt(a))/(4*a**(9/2))

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Mathematica [A]  time = 0.156256, size = 95, normalized size = 0.83 \[ \frac{35 b^2 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 a^{9/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (6 a^3 x^3-21 a^2 b x^2-140 a b^2 x-105 b^3\right )}{12 a^4 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x)^(5/2),x]

[Out]

(Sqrt[a + b/x]*x*(-105*b^3 - 140*a*b^2*x - 21*a^2*b*x^2 + 6*a^3*x^3))/(12*a^4*(b
+ a*x)^2) + (35*b^2*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(8*a^(9/2))

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Maple [B]  time = 0.017, size = 535, normalized size = 4.7 \[ -{\frac{x}{24\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -12\,{a}^{17/2}\sqrt{a{x}^{2}+bx}{x}^{4}-42\,{a}^{15/2}\sqrt{a{x}^{2}+bx}{x}^{3}b+216\,{a}^{15/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}b-54\,{a}^{13/2}\sqrt{a{x}^{2}+bx}{x}^{2}{b}^{2}-144\,{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xb+648\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{2}-30\,{a}^{11/2}\sqrt{a{x}^{2}+bx}x{b}^{3}-128\,{b}^{2}{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}+648\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }x{b}^{3}-6\,{a}^{9/2}\sqrt{a{x}^{2}+bx}{b}^{4}+216\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{b}^{4}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{7}{b}^{2}-108\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{7}{b}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{6}{b}^{3}-324\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{6}{b}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{5}{b}^{4}-324\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{5}{b}^{4}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5}-108\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5} \right ){a}^{-{\frac{17}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x)^(5/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)*x/a^(17/2)*(-12*a^(17/2)*(a*x^2+b*x)^(1/2)*x^4-42*a^(15/
2)*(a*x^2+b*x)^(1/2)*x^3*b+216*a^(15/2)*(x*(a*x+b))^(1/2)*x^3*b-54*a^(13/2)*(a*x
^2+b*x)^(1/2)*x^2*b^2-144*a^(13/2)*(x*(a*x+b))^(3/2)*x*b+648*a^(13/2)*(x*(a*x+b)
)^(1/2)*x^2*b^2-30*a^(11/2)*(a*x^2+b*x)^(1/2)*x*b^3-128*b^2*a^(11/2)*(x*(a*x+b))
^(3/2)+648*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^3-6*a^(9/2)*(a*x^2+b*x)^(1/2)*b^4+216*
a^(9/2)*(x*(a*x+b))^(1/2)*b^4+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(
1/2))*x^3*a^7*b^2-108*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*
a^7*b^2+9*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^3-324*
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+9*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^4-324*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+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*a^4*b^5-108*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)^3

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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/(a + b/x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259129, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (6 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} - 140 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{a}}{24 \,{\left (a^{5} x + a^{4} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, -\frac{105 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (6 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} - 140 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{-a}}{12 \,{\left (a^{5} x + a^{4} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(105*(a*b^2*x + b^3)*sqrt((a*x + b)/x)*log(2*a*x*sqrt((a*x + b)/x) + (2*a*
x + b)*sqrt(a)) + 2*(6*a^3*x^3 - 21*a^2*b*x^2 - 140*a*b^2*x - 105*b^3)*sqrt(a))/
((a^5*x + a^4*b)*sqrt(a)*sqrt((a*x + b)/x)), -1/12*(105*(a*b^2*x + b^3)*sqrt((a*
x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (6*a^3*x^3 - 21*a^2*b*x^2 - 1
40*a*b^2*x - 105*b^3)*sqrt(-a))/((a^5*x + a^4*b)*sqrt(-a)*sqrt((a*x + b)/x))]

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Sympy [A]  time = 26.341, size = 464, normalized size = 4.07 \[ \frac{6 a^{\frac{89}{2}} b^{75} x^{49}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{21 a^{\frac{87}{2}} b^{76} x^{48}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{140 a^{\frac{85}{2}} b^{77} x^{47}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{105 a^{\frac{83}{2}} b^{78} x^{46}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{42} b^{\frac{155}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{41} b^{\frac{157}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x)**(5/2),x)

[Out]

6*a**(89/2)*b**75*x**49/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*
a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 21*a**(87/2)*b**76*x**48/(12*a
**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2
)*sqrt(a*x/b + 1)) - 140*a**(85/2)*b**77*x**47/(12*a**(93/2)*b**(151/2)*x**(93/2
)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 105*a**
(83/2)*b**78*x**46/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(9
1/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**42*b**(155/2)*x**(93/2)*sqrt
(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(12*a**(93/2)*b**(151/2)*x**(93/2)*sq
rt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**41*b
**(157/2)*x**(91/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(12*a**(93/2)
*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a
*x/b + 1))

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GIAC/XCAS [A]  time = 0.262774, size = 169, normalized size = 1.48 \[ -\frac{1}{12} \, b^{2}{\left (\frac{8 \,{\left (a + \frac{9 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{4} \sqrt{\frac{a x + b}{x}}} + \frac{105 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{3 \,{\left (13 \, a \sqrt{\frac{a x + b}{x}} - \frac{11 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*b^2*(8*(a + 9*(a*x + b)/x)*x/((a*x + b)*a^4*sqrt((a*x + b)/x)) + 105*arcta
n(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) - 3*(13*a*sqrt((a*x + b)/x) - 11*(a
*x + b)*sqrt((a*x + b)/x)/x)/((a - (a*x + b)/x)^2*a^4))