3.2975 \(\int \sqrt{a+b \sqrt{\frac{c}{x}}} x \, dx\)

Optimal. Leaf size=169 \[ -\frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{7/2}}+\frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^2}+\frac{b c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a \left (\frac{c}{x}\right )^{3/2}}+\frac{1}{2} x^2 \sqrt{a+b \sqrt{\frac{c}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.242404, antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{5 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{7/2}}+\frac{5 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^3 \sqrt{\frac{c}{x}}}-\frac{5 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^2}+\frac{b x^3 \left (\frac{c}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a c}+\frac{1}{2} x^2 \sqrt{a+b \sqrt{\frac{c}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c/x]]*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 24.0723, size = 143, normalized size = 0.85 \[ \frac{x^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{2} + \frac{b c^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{12 a \left (\frac{c}{x}\right )^{\frac{3}{2}}} - \frac{5 b^{2} c x \sqrt{a + b \sqrt{\frac{c}{x}}}}{48 a^{2}} + \frac{5 b^{3} c^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{32 a^{3} \sqrt{\frac{c}{x}}} - \frac{5 b^{4} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.185862, size = 111, normalized size = 0.66 \[ \frac{\sqrt{a} x \sqrt{a+b \sqrt{\frac{c}{x}}} \left (48 a^3 x+8 a^2 b x \sqrt{\frac{c}{x}}-10 a b^2 c+15 b^3 c \sqrt{\frac{c}{x}}\right )-15 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{96 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.039, size = 211, normalized size = 1.3 \[ -{\frac{1}{192}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( -30\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x} \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}-96\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2}+80\,{a}^{5/2} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}\sqrt{x}b-60\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}c\sqrt{x}{b}^{2}+15\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){c}^{2}a{b}^{4} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(a+b*(c/x)^(1/2))^(1/2)*x^(1/2)*(-30*a^(3/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*
(c/x)^(3/2)*x^(3/2)*b^3-96*x^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a^(7/2)+80*a^(5/2
)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(c/x)^(1/2)*x^(1/2)*b-60*a^(5/2)*(a*x+b*(c/x)^(1/2
)*x)^(1/2)*c*x^(1/2)*b^2+15*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(a*x+b*(c/x)^(1/2)*x
)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*c^2*a*b^4)/(x*(a+b*(c/x)^(1/2)))^(1/2)/a^(
9/2)

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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(c/x) + a)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304663, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} c^{2} \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right ) - 2 \,{\left (10 \, a b^{2} c x - 48 \, a^{3} x^{2} -{\left (15 \, b^{3} c x + 8 \, a^{2} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a}}{192 \, a^{\frac{7}{2}}}, \frac{15 \, b^{4} c^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right ) +{\left ({\left (15 \, b^{3} c x + 8 \, a^{2} b x^{2}\right )} \sqrt{-a} \sqrt{\frac{c}{x}} - 2 \,{\left (5 \, a b^{2} c x - 24 \, a^{3} x^{2}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{96 \, \sqrt{-a} a^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(15*b^4*c^2*log(((b*sqrt(c/x) + 2*a)*sqrt(a) - 2*sqrt(b*sqrt(c/x) + a)*a)
/sqrt(c/x)) - 2*(10*a*b^2*c*x - 48*a^3*x^2 - (15*b^3*c*x + 8*a^2*b*x^2)*sqrt(c/x
))*sqrt(b*sqrt(c/x) + a)*sqrt(a))/a^(7/2), 1/96*(15*b^4*c^2*arctan(a/(sqrt(b*sqr
t(c/x) + a)*sqrt(-a))) + ((15*b^3*c*x + 8*a^2*b*x^2)*sqrt(-a)*sqrt(c/x) - 2*(5*a
*b^2*c*x - 24*a^3*x^2)*sqrt(-a))*sqrt(b*sqrt(c/x) + a))/(sqrt(-a)*a^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b \sqrt{\frac{c}{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(a + b*sqrt(c/x)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out