3.2981 \(\int \frac{x}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

Optimal. Leaf size=172 \[ \frac{35 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{9/2}}-\frac{35 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^4 \sqrt{\frac{c}{x}}}+\frac{35 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^3}-\frac{7 b c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a^2 \left (\frac{c}{x}\right )^{3/2}}+\frac{x^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a} \]

[Out]

(-7*b*c^2*Sqrt[a + b*Sqrt[c/x]])/(12*a^2*(c/x)^(3/2)) - (35*b^3*c^2*Sqrt[a + b*S
qrt[c/x]])/(32*a^4*Sqrt[c/x]) + (35*b^2*c*Sqrt[a + b*Sqrt[c/x]]*x)/(48*a^3) + (S
qrt[a + b*Sqrt[c/x]]*x^2)/(2*a) + (35*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sqrt
[a]])/(32*a^(9/2))

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Rubi [A]  time = 0.235741, antiderivative size = 175, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{35 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{32 a^{9/2}}-\frac{35 b^3 c^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{32 a^4 \sqrt{\frac{c}{x}}}+\frac{35 b^2 c x \sqrt{a+b \sqrt{\frac{c}{x}}}}{48 a^3}-\frac{7 b x^3 \left (\frac{c}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{c}{x}}}}{12 a^2 c}+\frac{x^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(-35*b^3*c^2*Sqrt[a + b*Sqrt[c/x]])/(32*a^4*Sqrt[c/x]) + (35*b^2*c*Sqrt[a + b*Sq
rt[c/x]]*x)/(48*a^3) + (Sqrt[a + b*Sqrt[c/x]]*x^2)/(2*a) - (7*b*Sqrt[a + b*Sqrt[
c/x]]*(c/x)^(3/2)*x^3)/(12*a^2*c) + (35*b^4*c^2*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sq
rt[a]])/(32*a^(9/2))

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Rubi in Sympy [A]  time = 25.588, size = 148, normalized size = 0.86 \[ \frac{x^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{2 a} - \frac{7 b c^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{12 a^{2} \left (\frac{c}{x}\right )^{\frac{3}{2}}} + \frac{35 b^{2} c x \sqrt{a + b \sqrt{\frac{c}{x}}}}{48 a^{3}} - \frac{35 b^{3} c^{2} \sqrt{a + b \sqrt{\frac{c}{x}}}}{32 a^{4} \sqrt{\frac{c}{x}}} + \frac{35 b^{4} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{32 a^{\frac{9}{2}}} \]

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

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Mathematica [A]  time = 0.209333, size = 126, normalized size = 0.73 \[ \frac{35 b^4 c^2 \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b \sqrt{\frac{c}{x}}}}\right )}{32 a^{9/2}}+\frac{48 a^4 x^2-8 a^3 b x^2 \sqrt{\frac{c}{x}}+14 a^2 b^2 c x-35 a b^3 c x \sqrt{\frac{c}{x}}-105 b^4 c^2}{96 a^4 \sqrt{a+b \sqrt{\frac{c}{x}}}} \]

Antiderivative was successfully verified.

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

[Out]

(-105*b^4*c^2 + 14*a^2*b^2*c*x - 35*a*b^3*c*Sqrt[c/x]*x + 48*a^4*x^2 - 8*a^3*b*S
qrt[c/x]*x^2)/(96*a^4*Sqrt[a + b*Sqrt[c/x]]) + (35*b^4*c^2*ArcTanh[Sqrt[a]/Sqrt[
a + b*Sqrt[c/x]]])/(32*a^(9/2))

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Maple [B]  time = 0.062, size = 302, normalized size = 1.8 \[{\frac{1}{192}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 174\,{b}^{3} \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{9/2}-384\,{b}^{3} \left ({\frac{c}{x}} \right ) ^{3/2}{x}^{3/2}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{9/2}+348\,{b}^{2}c\sqrt{x}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{11/2}+96\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{13/2}-208\,b\sqrt{{\frac{c}{x}}}\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{11/2}-87\,{b}^{4}{c}^{2}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{4}+192\,{b}^{4}{c}^{2}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{4} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{17}{2}}}} \]

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)*(174*b^3*(c/x)^(3/2)*x^(3/2)*(a*x+b*(c/x)^
(1/2)*x)^(1/2)*a^(9/2)-384*b^3*(c/x)^(3/2)*x^(3/2)*(x*(a+b*(c/x)^(1/2)))^(1/2)*a
^(9/2)+348*b^2*c*x^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(11/2)+96*x^(1/2)*(a*x+b*
(c/x)^(1/2)*x)^(3/2)*a^(13/2)-208*b*(c/x)^(1/2)*x^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3
/2)*a^(11/2)-87*b^4*c^2*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))*a^4+192*b^4*c^2*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+
2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^4)/(x*(a+b*(c/x)^(
1/2)))^(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/sqrt(b*sqrt(c/x) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259712, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} c^{2} \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right ) + 2 \,{\left (70 \, a b^{2} c x + 48 \, a^{3} x^{2} - 7 \,{\left (15 \, b^{3} c x + 8 \, a^{2} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a}}{192 \, a^{\frac{9}{2}}}, -\frac{105 \, b^{4} c^{2} \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right ) +{\left (7 \,{\left (15 \, b^{3} c x + 8 \, a^{2} b x^{2}\right )} \sqrt{-a} \sqrt{\frac{c}{x}} - 2 \,{\left (35 \, a b^{2} c x + 24 \, a^{3} x^{2}\right )} \sqrt{-a}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{96 \, \sqrt{-a} a^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(105*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*(70*a*b^2*c*x + 48*a^3*x^2 - 7*(15*b^3*c*x + 8*a^2*b*x^2)*sqrt(
c/x))*sqrt(b*sqrt(c/x) + a)*sqrt(a))/a^(9/2), -1/96*(105*b^4*c^2*arctan(a/(sqrt(
b*sqrt(c/x) + a)*sqrt(-a))) + (7*(15*b^3*c*x + 8*a^2*b*x^2)*sqrt(-a)*sqrt(c/x) -
2*(35*a*b^2*c*x + 24*a^3*x^2)*sqrt(-a))*sqrt(b*sqrt(c/x) + a))/(sqrt(-a)*a^4)]

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

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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError