3.1720 \(\int \frac{x^3}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=120 \[ \frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{35 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^4}+\frac{35 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^3}-\frac{7 b x^3 \sqrt{a+\frac{b}{x}}}{24 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x}}}{4 a} \]

[Out]

(-35*b^3*Sqrt[a + b/x]*x)/(64*a^4) + (35*b^2*Sqrt[a + b/x]*x^2)/(96*a^3) - (7*b*
Sqrt[a + b/x]*x^3)/(24*a^2) + (Sqrt[a + b/x]*x^4)/(4*a) + (35*b^4*ArcTanh[Sqrt[a
+ b/x]/Sqrt[a]])/(64*a^(9/2))

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Rubi [A]  time = 0.166242, antiderivative size = 120, 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 \[ \frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{35 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^4}+\frac{35 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^3}-\frac{7 b x^3 \sqrt{a+\frac{b}{x}}}{24 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x}}}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b/x],x]

[Out]

(-35*b^3*Sqrt[a + b/x]*x)/(64*a^4) + (35*b^2*Sqrt[a + b/x]*x^2)/(96*a^3) - (7*b*
Sqrt[a + b/x]*x^3)/(24*a^2) + (Sqrt[a + b/x]*x^4)/(4*a) + (35*b^4*ArcTanh[Sqrt[a
+ b/x]/Sqrt[a]])/(64*a^(9/2))

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Rubi in Sympy [A]  time = 16.352, size = 104, normalized size = 0.87 \[ \frac{x^{4} \sqrt{a + \frac{b}{x}}}{4 a} - \frac{7 b x^{3} \sqrt{a + \frac{b}{x}}}{24 a^{2}} + \frac{35 b^{2} x^{2} \sqrt{a + \frac{b}{x}}}{96 a^{3}} - \frac{35 b^{3} x \sqrt{a + \frac{b}{x}}}{64 a^{4}} + \frac{35 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{64 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b/x)**(1/2),x)

[Out]

x**4*sqrt(a + b/x)/(4*a) - 7*b*x**3*sqrt(a + b/x)/(24*a**2) + 35*b**2*x**2*sqrt(
a + b/x)/(96*a**3) - 35*b**3*x*sqrt(a + b/x)/(64*a**4) + 35*b**4*atanh(sqrt(a +
b/x)/sqrt(a))/(64*a**(9/2))

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Mathematica [A]  time = 0.134642, size = 90, normalized size = 0.75 \[ \frac{2 \sqrt{a} x \sqrt{a+\frac{b}{x}} \left (48 a^3 x^3-56 a^2 b x^2+70 a b^2 x-105 b^3\right )+105 b^4 \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{384 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b/x],x]

[Out]

(2*Sqrt[a]*Sqrt[a + b/x]*x*(-105*b^3 + 70*a*b^2*x - 56*a^2*b*x^2 + 48*a^3*x^3) +
105*b^4*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(384*a^(9/2))

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Maple [A]  time = 0.017, size = 188, normalized size = 1.6 \[ -{\frac{x}{384}\sqrt{{\frac{ax+b}{x}}} \left ( -96\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{13/2}+208\,b \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{11/2}-348\,{b}^{2}\sqrt{a{x}^{2}+bx}x{a}^{11/2}-174\,{b}^{3}\sqrt{a{x}^{2}+bx}{a}^{9/2}+384\,{b}^{3}\sqrt{x \left ( ax+b \right ) }{a}^{9/2}+87\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}-192\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x)^(1/2),x)

[Out]

-1/384*((a*x+b)/x)^(1/2)*x*(-96*x*(a*x^2+b*x)^(3/2)*a^(13/2)+208*b*(a*x^2+b*x)^(
3/2)*a^(11/2)-348*b^2*(a*x^2+b*x)^(1/2)*x*a^(11/2)-174*b^3*(a*x^2+b*x)^(1/2)*a^(
9/2)+384*b^3*(x*(a*x+b))^(1/2)*a^(9/2)+87*b^4*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*a^4-192*b^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(
1/2))*a^4)/(x*(a*x+b))^(1/2)/a^(17/2)

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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^3/sqrt(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242826, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (48 \, a^{3} x^{4} - 56 \, a^{2} b x^{3} + 70 \, a b^{2} x^{2} - 105 \, b^{3} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{384 \, a^{\frac{9}{2}}}, -\frac{105 \, b^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (48 \, a^{3} x^{4} - 56 \, a^{2} b x^{3} + 70 \, a b^{2} x^{2} - 105 \, b^{3} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{192 \, \sqrt{-a} a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(105*b^4*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(48*a^3*x
^4 - 56*a^2*b*x^3 + 70*a*b^2*x^2 - 105*b^3*x)*sqrt(a)*sqrt((a*x + b)/x))/a^(9/2)
, -1/192*(105*b^4*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (48*a^3*x^4 - 56*a^2*
b*x^3 + 70*a*b^2*x^2 - 105*b^3*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a^4)]

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Sympy [A]  time = 31.0611, size = 155, normalized size = 1.29 \[ \frac{x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{7}{2}}}{24 a \sqrt{\frac{a x}{b} + 1}} + \frac{7 b^{\frac{3}{2}} x^{\frac{5}{2}}}{96 a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{\frac{5}{2}} x^{\frac{3}{2}}}{192 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{\frac{7}{2}} \sqrt{x}}{64 a^{4} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x)**(1/2),x)

[Out]

x**(9/2)/(4*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(7/2)/(24*a*sqrt(a*x/b + 1)) +
7*b**(3/2)*x**(5/2)/(96*a**2*sqrt(a*x/b + 1)) - 35*b**(5/2)*x**(3/2)/(192*a**3*
sqrt(a*x/b + 1)) - 35*b**(7/2)*sqrt(x)/(64*a**4*sqrt(a*x/b + 1)) + 35*b**4*asinh
(sqrt(a)*sqrt(x)/sqrt(b))/(64*a**(9/2))

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GIAC/XCAS [A]  time = 0.249167, size = 208, normalized size = 1.73 \[ -\frac{1}{192} \, b{\left (\frac{105 \, b^{3} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{279 \, a^{3} b^{3} \sqrt{\frac{a x + b}{x}} - \frac{511 \,{\left (a x + b\right )} a^{2} b^{3} \sqrt{\frac{a x + b}{x}}}{x} + \frac{385 \,{\left (a x + b\right )}^{2} a b^{3} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{105 \,{\left (a x + b\right )}^{3} b^{3} \sqrt{\frac{a x + b}{x}}}{x^{3}}}{{\left (a - \frac{a x + b}{x}\right )}^{4} a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*b*(105*b^3*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) - (279*a^3*b
^3*sqrt((a*x + b)/x) - 511*(a*x + b)*a^2*b^3*sqrt((a*x + b)/x)/x + 385*(a*x + b)
^2*a*b^3*sqrt((a*x + b)/x)/x^2 - 105*(a*x + b)^3*b^3*sqrt((a*x + b)/x)/x^3)/((a
- (a*x + b)/x)^4*a^4))