3.3055 \(\int \frac{x}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=248 \[ \frac{5 b d \left (44 a c-21 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{96 a^4 \sqrt{\frac{d}{x}}}-\frac{x \left (36 a c-35 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{48 a^3}-\frac{7 b d^2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{9/2}}+\frac{x^2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

[Out]

(-7*b*d^2*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*a^2*(d/x)^(3/2)) + (5*b*d*(44*a*c - 2
1*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(96*a^4*Sqrt[d/x]) - ((36*a*c - 35*b^2*d)*
Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(48*a^3) + (Sqrt[a + b*Sqrt[d/x] + c/x]*x^2)/(2*a
) + ((48*a^2*c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sq
rt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(9/2))

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Rubi [A]  time = 1.13854, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{5 b d \left (44 a c-21 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{96 a^4 \sqrt{\frac{d}{x}}}-\frac{x \left (36 a c-35 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{48 a^3}-\frac{7 b d^2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{\left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{9/2}}+\frac{x^2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

(-7*b*d^2*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*a^2*(d/x)^(3/2)) + (5*b*d*(44*a*c - 2
1*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(96*a^4*Sqrt[d/x]) - ((36*a*c - 35*b^2*d)*
Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(48*a^3) + (Sqrt[a + b*Sqrt[d/x] + c/x]*x^2)/(2*a
) + ((48*a^2*c^2 - 120*a*b^2*c*d + 35*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sq
rt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(9/2))

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Rubi in Sympy [A]  time = 90.942, size = 211, normalized size = 0.85 \[ \frac{x^{2} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{2 a} - \frac{7 b d^{2} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{12 a^{2} \left (\frac{d}{x}\right )^{\frac{3}{2}}} - \frac{x \left (36 a c - 35 b^{2} d\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{48 a^{3}} + \frac{5 b d \left (44 a c - 21 b^{2} d\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{96 a^{4} \sqrt{\frac{d}{x}}} + \frac{\left (48 a^{2} c^{2} - 120 a b^{2} c d + 35 b^{4} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{64 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b*sqrt(d/x) + c/x)/(2*a) - 7*b*d**2*sqrt(a + b*sqrt(d/x) + c/x)/(1
2*a**2*(d/x)**(3/2)) - x*(36*a*c - 35*b**2*d)*sqrt(a + b*sqrt(d/x) + c/x)/(48*a*
*3) + 5*b*d*(44*a*c - 21*b**2*d)*sqrt(a + b*sqrt(d/x) + c/x)/(96*a**4*sqrt(d/x))
+ (48*a**2*c**2 - 120*a*b**2*c*d + 35*b**4*d**2)*atanh((2*a + b*sqrt(d/x))/(2*s
qrt(a)*sqrt(a + b*sqrt(d/x) + c/x)))/(64*a**(9/2))

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Mathematica [A]  time = 0.252667, size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.046, size = 398, normalized size = 1.6 \[{\frac{1}{192}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 105\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}a{b}^{4}-210\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}+140\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}-112\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3/2}b-360\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{2}{b}^{2}c+96\,{x}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{a}^{9/2}+440\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}bc-144\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}c+144\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{3}{c}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(105*ln(1/2*(b*(d/x)^(1/2)*x^(1/
2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d^2*a*b^4-210*a
^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(3/2)*b^3+140*a^(5/2)*(b*(d/x
)^(1/2)*x+a*x+c)^(1/2)*d*x^(1/2)*b^2-112*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(
d/x)^(1/2)*x^(3/2)*b-360*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)
^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d*a^2*b^2*c+96*x^(3/2)*(b*(d/x)^(1/2)*x+a*x
+c)^(1/2)*a^(9/2)+440*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*
b*c-144*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)*c+144*ln(1/2*(b*(d/x)^(1/2
)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^3*c^2)
/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(b*sqrt(d/x) + a + c/x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(a + b*sqrt(d/x) + c/x), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(b*sqrt(d/x) + a + c/x), x)