3.2242 \(\int \frac{1}{\sqrt{a+b \sqrt{x}} x^3} \, dx\)

Optimal. Leaf size=136 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*a*x^2) + (7*b*Sqrt[a + b*Sqrt[x]])/(12*a^2*x^(3/2)) - (3
5*b^2*Sqrt[a + b*Sqrt[x]])/(48*a^3*x) + (35*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^4*Sqr
t[x]) - (35*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.174682, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]

[Out]

-Sqrt[a + b*Sqrt[x]]/(2*a*x^2) + (7*b*Sqrt[a + b*Sqrt[x]])/(12*a^2*x^(3/2)) - (3
5*b^2*Sqrt[a + b*Sqrt[x]])/(48*a^3*x) + (35*b^3*Sqrt[a + b*Sqrt[x]])/(32*a^4*Sqr
t[x]) - (35*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.6339, size = 124, normalized size = 0.91 \[ - \frac{\sqrt{a + b \sqrt{x}}}{2 a x^{2}} + \frac{7 b \sqrt{a + b \sqrt{x}}}{12 a^{2} x^{\frac{3}{2}}} - \frac{35 b^{2} \sqrt{a + b \sqrt{x}}}{48 a^{3} x} + \frac{35 b^{3} \sqrt{a + b \sqrt{x}}}{32 a^{4} \sqrt{x}} - \frac{35 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{32 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*sqrt(x))/(2*a*x**2) + 7*b*sqrt(a + b*sqrt(x))/(12*a**2*x**(3/2)) - 3
5*b**2*sqrt(a + b*sqrt(x))/(48*a**3*x) + 35*b**3*sqrt(a + b*sqrt(x))/(32*a**4*sq
rt(x)) - 35*b**4*atanh(sqrt(a + b*sqrt(x))/sqrt(a))/(32*a**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0978351, size = 90, normalized size = 0.66 \[ \frac{\sqrt{a+b \sqrt{x}} \left (-48 a^3+56 a^2 b \sqrt{x}-70 a b^2 x+105 b^3 x^{3/2}\right )}{96 a^4 x^2}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^3),x]

[Out]

(Sqrt[a + b*Sqrt[x]]*(-48*a^3 + 56*a^2*b*Sqrt[x] - 70*a*b^2*x + 105*b^3*x^(3/2))
)/(96*a^4*x^2) - (35*b^4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/(32*a^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 124, normalized size = 0.9 \[ 4\,{b}^{4} \left ( -1/8\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{4}{x}^{2}}}-{\frac{7}{8\,a} \left ( -1/6\,{\frac{\sqrt{a+b\sqrt{x}}}{a{x}^{3/2}{b}^{3}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{2}x}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b^4*(-1/8*(a+b*x^(1/2))^(1/2)/a/x^2/b^4-7/8/a*(-1/6*(a+b*x^(1/2))^(1/2)/a/x^(3
/2)/b^3-5/6/a*(-1/4*(a+b*x^(1/2))^(1/2)/a/x/b^2-3/4/a*(-1/2*(a+b*x^(1/2))^(1/2)/
a/x^(1/2)/b+1/2/a^(3/2)*arctanh((a+b*x^(1/2))^(1/2)/a^(1/2))))))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.255696, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} x^{2} \log \left (\frac{\sqrt{a} b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) + 2 \,{\left (7 \,{\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{a} \sqrt{x} - 2 \,{\left (35 \, a b^{2} x + 24 \, a^{3}\right )} \sqrt{a}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{\frac{9}{2}} x^{2}}, \frac{105 \, b^{4} x^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left (7 \,{\left (15 \, b^{3} x + 8 \, a^{2} b\right )} \sqrt{-a} \sqrt{x} - 2 \,{\left (35 \, a b^{2} x + 24 \, a^{3}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{x} + a}}{96 \, \sqrt{-a} a^{4} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(x) + a)*x^3),x, algorithm="fricas")

[Out]

[1/192*(105*b^4*x^2*log((sqrt(a)*b*sqrt(x) - 2*sqrt(b*sqrt(x) + a)*a + 2*a^(3/2)
)/sqrt(x)) + 2*(7*(15*b^3*x + 8*a^2*b)*sqrt(a)*sqrt(x) - 2*(35*a*b^2*x + 24*a^3)
*sqrt(a))*sqrt(b*sqrt(x) + a))/(a^(9/2)*x^2), 1/96*(105*b^4*x^2*arctan(a/(sqrt(b
*sqrt(x) + a)*sqrt(-a))) + (7*(15*b^3*x + 8*a^2*b)*sqrt(-a)*sqrt(x) - 2*(35*a*b^
2*x + 24*a^3)*sqrt(-a))*sqrt(b*sqrt(x) + a))/(sqrt(-a)*a^4*x^2)]

_______________________________________________________________________________________

Sympy [A]  time = 43.1338, size = 173, normalized size = 1.27 \[ - \frac{1}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 b^{\frac{3}{2}}}{48 a^{2} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{5}{2}}}{96 a^{3} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{7}{2}}}{32 a^{4} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*sqrt(b)*x**(9/4)*sqrt(a/(b*sqrt(x)) + 1)) + sqrt(b)/(12*a*x**(7/4)*sqrt(a/
(b*sqrt(x)) + 1)) - 7*b**(3/2)/(48*a**2*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) + 35*b
**(5/2)/(96*a**3*x**(3/4)*sqrt(a/(b*sqrt(x)) + 1)) + 35*b**(7/2)/(32*a**4*x**(1/
4)*sqrt(a/(b*sqrt(x)) + 1)) - 35*b**4*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(32*a**(
9/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.276574, size = 127, normalized size = 0.93 \[ \frac{1}{96} \, b^{4}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 385 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 511 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 279 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{4} b^{4} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(x) + a)*x^3),x, algorithm="giac")

[Out]

1/96*b^4*(105*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a^4) + (105*(b*sqrt
(x) + a)^(7/2) - 385*(b*sqrt(x) + a)^(5/2)*a + 511*(b*sqrt(x) + a)^(3/2)*a^2 - 2
79*sqrt(b*sqrt(x) + a)*a^3)/(a^4*b^4*x^2))