3.2236 \(\int \frac{\sqrt{a+b \sqrt{x}}}{x^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^3 \sqrt{a+b \sqrt{x}}}{32 a^3 \sqrt{x}}+\frac{5 b^2 \sqrt{a+b \sqrt{x}}}{48 a^2 x}-\frac{b \sqrt{a+b \sqrt{x}}}{12 a x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 x^2} \]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*x^2) - (b*Sqrt[a + b*Sqrt[x]])/(12*a*x^(3/2)) + (5*b^2*S
qrt[a + b*Sqrt[x]])/(48*a^2*x) - (5*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^3*Sqrt[x]) +
(5*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(7/2))

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Rubi [A]  time = 0.172506, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{5 b^3 \sqrt{a+b \sqrt{x}}}{32 a^3 \sqrt{x}}+\frac{5 b^2 \sqrt{a+b \sqrt{x}}}{48 a^2 x}-\frac{b \sqrt{a+b \sqrt{x}}}{12 a x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[x]]/x^3,x]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*x^2) - (b*Sqrt[a + b*Sqrt[x]])/(12*a*x^(3/2)) + (5*b^2*S
qrt[a + b*Sqrt[x]])/(48*a^2*x) - (5*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^3*Sqrt[x]) +
(5*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(7/2))

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Rubi in Sympy [A]  time = 17.9616, size = 119, normalized size = 0.89 \[ - \frac{\sqrt{a + b \sqrt{x}}}{2 x^{2}} - \frac{b \sqrt{a + b \sqrt{x}}}{12 a x^{\frac{3}{2}}} + \frac{5 b^{2} \sqrt{a + b \sqrt{x}}}{48 a^{2} x} - \frac{5 b^{3} \sqrt{a + b \sqrt{x}}}{32 a^{3} \sqrt{x}} + \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*sqrt(x))/(2*x**2) - b*sqrt(a + b*sqrt(x))/(12*a*x**(3/2)) + 5*b**2*s
qrt(a + b*sqrt(x))/(48*a**2*x) - 5*b**3*sqrt(a + b*sqrt(x))/(32*a**3*sqrt(x)) +
5*b**4*atanh(sqrt(a + b*sqrt(x))/sqrt(a))/(32*a**(7/2))

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Mathematica [A]  time = 0.0882718, size = 90, normalized size = 0.68 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{7/2}}-\frac{\sqrt{a+b \sqrt{x}} \left (48 a^3+8 a^2 b \sqrt{x}-10 a b^2 x+15 b^3 x^{3/2}\right )}{96 a^3 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[x]]/x^3,x]

[Out]

-(Sqrt[a + b*Sqrt[x]]*(48*a^3 + 8*a^2*b*Sqrt[x] - 10*a*b^2*x + 15*b^3*x^(3/2)))/
(96*a^3*x^2) + (5*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(7/2))

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Maple [A]  time = 0.015, size = 87, normalized size = 0.7 \[ 4\,{b}^{4} \left ({\frac{1}{{x}^{2}{b}^{4}} \left ( -{\frac{5\, \left ( a+b\sqrt{x} \right ) ^{7/2}}{128\,{a}^{3}}}+{\frac{55\, \left ( a+b\sqrt{x} \right ) ^{5/2}}{384\,{a}^{2}}}-{\frac{73\, \left ( a+b\sqrt{x} \right ) ^{3/2}}{384\,a}}-{\frac{5\,\sqrt{a+b\sqrt{x}}}{128}} \right ) }+{\frac{5}{128\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^(1/2)/x^3,x)

[Out]

4*b^4*((-5/128/a^3*(a+b*x^(1/2))^(7/2)+55/384/a^2*(a+b*x^(1/2))^(5/2)-73/384/a*(
a+b*x^(1/2))^(3/2)-5/128*(a+b*x^(1/2))^(1/2))/x^2/b^4+5/128/a^(7/2)*arctanh((a+b
*x^(1/2))^(1/2)/a^(1/2)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255883, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} x^{2} \log \left (\frac{\sqrt{a} b \sqrt{x} + 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) - 2 \,{\left ({\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{a} \sqrt{x} - 2 \,{\left (5 \, a b^{2} x - 24 \, a^{3}\right )} \sqrt{a}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{\frac{7}{2}} x^{2}}, -\frac{15 \, b^{4} x^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left ({\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{-a} \sqrt{x} - 2 \,{\left (5 \, a b^{2} x - 24 \, a^{3}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{x} + a}}{96 \, \sqrt{-a} a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(15*b^4*x^2*log((sqrt(a)*b*sqrt(x) + 2*sqrt(b*sqrt(x) + a)*a + 2*a^(3/2))
/sqrt(x)) - 2*((15*b^3*x + 8*a^2*b)*sqrt(a)*sqrt(x) - 2*(5*a*b^2*x - 24*a^3)*sqr
t(a))*sqrt(b*sqrt(x) + a))/(a^(7/2)*x^2), -1/96*(15*b^4*x^2*arctan(a/(sqrt(b*sqr
t(x) + a)*sqrt(-a))) + ((15*b^3*x + 8*a^2*b)*sqrt(-a)*sqrt(x) - 2*(5*a*b^2*x - 2
4*a^3)*sqrt(-a))*sqrt(b*sqrt(x) + a))/(sqrt(-a)*a^3*x^2)]

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Sympy [A]  time = 35.6483, size = 170, normalized size = 1.28 \[ - \frac{a}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 \sqrt{b}}{12 x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{b^{\frac{3}{2}}}{48 a x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{5 b^{\frac{5}{2}}}{96 a^{2} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{5 b^{\frac{7}{2}}}{32 a^{3} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*sqrt(b)*x**(9/4)*sqrt(a/(b*sqrt(x)) + 1)) - 7*sqrt(b)/(12*x**(7/4)*sqrt(a/
(b*sqrt(x)) + 1)) + b**(3/2)/(48*a*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) - 5*b**(5/2
)/(96*a**2*x**(3/4)*sqrt(a/(b*sqrt(x)) + 1)) - 5*b**(7/2)/(32*a**3*x**(1/4)*sqrt
(a/(b*sqrt(x)) + 1)) + 5*b**4*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(32*a**(7/2))

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GIAC/XCAS [A]  time = 0.228737, size = 127, normalized size = 0.95 \[ -\frac{1}{96} \, b^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 55 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 73 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} + 15 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{3} b^{4} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*b^4*(15*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*sqrt(
x) + a)^(7/2) - 55*(b*sqrt(x) + a)^(5/2)*a + 73*(b*sqrt(x) + a)^(3/2)*a^2 + 15*s
qrt(b*sqrt(x) + a)*a^3)/(a^3*b^4*x^2))