3.3047 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x \, dx\)

Optimal. Leaf size=209 \[ -\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]

[Out]

(-5*b*d^2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(12*a^2*(d/x)^(3/2)) - ((4*a*c - 5*b^2*
d)*(2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(32*a^3) + ((a + b*Sqrt[d/
x] + c/x)^(3/2)*x^2)/(2*a) - ((4*a*c - 5*b^2*d)*(4*a*c - b^2*d)*ArcTanh[(2*a + b
*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(7/2))

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Rubi [A]  time = 0.691702, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\left (4 a c-5 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{64 a^{7/2}}-\frac{x \left (4 a c-5 b^2 d\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{32 a^3}-\frac{5 b d^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{12 a^2 \left (\frac{d}{x}\right )^{3/2}}+\frac{x^2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

(-5*b*d^2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(12*a^2*(d/x)^(3/2)) - ((4*a*c - 5*b^2*
d)*(2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(32*a^3) + ((a + b*Sqrt[d/
x] + c/x)^(3/2)*x^2)/(2*a) - ((4*a*c - 5*b^2*d)*(4*a*c - b^2*d)*ArcTanh[(2*a + b
*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(64*a^(7/2))

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Rubi in Sympy [A]  time = 49.0968, size = 172, normalized size = 0.82 \[ \frac{x^{2} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{2 a} - \frac{5 b d^{2} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{12 a^{2} \left (\frac{d}{x}\right )^{\frac{3}{2}}} - \frac{x \left (2 a + b \sqrt{\frac{d}{x}}\right ) \left (4 a c - 5 b^{2} d\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{32 a^{3}} - \frac{\left (4 a c - 5 b^{2} d\right ) \left (4 a c - b^{2} d\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{64 a^{\frac{7}{2}}} \]

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*(a + b*sqrt(d/x) + c/x)**(3/2)/(2*a) - 5*b*d**2*(a + b*sqrt(d/x) + c/x)**(3
/2)/(12*a**2*(d/x)**(3/2)) - x*(2*a + b*sqrt(d/x))*(4*a*c - 5*b**2*d)*sqrt(a + b
*sqrt(d/x) + c/x)/(32*a**3) - (4*a*c - 5*b**2*d)*(4*a*c - b**2*d)*atanh((2*a + b
*sqrt(d/x))/(2*sqrt(a)*sqrt(a + b*sqrt(d/x) + c/x)))/(64*a**(7/2))

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Mathematica [A]  time = 0.188334, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.038, size = 398, normalized size = 1.9 \[{\frac{1}{192}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 30\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}+60\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}-15\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}a{b}^{4}+96\,\sqrt{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{a}^{7/2}-80\,{a}^{5/2} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}\sqrt{x}b-48\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}c-24\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}bc+72\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{2}{b}^{2}c-48\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{3}{c}^{2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{9}{2}}}} \]

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)*(30*a^(3/2)*(b*(d/x)^(1/2)*x+a*x
+c)^(1/2)*(d/x)^(3/2)*x^(3/2)*b^3+60*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(
1/2)*b^2-15*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+96*x^(1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a^(7/
2)-80*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^(1/2)*b-48*a^(7/2)*(b*
(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)*c-24*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d
/x)^(1/2)*x^(1/2)*b*c+72*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-48*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^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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 [A]  time = 0.332584, size = 360, normalized size = 1.72 \[ \frac{1}{192} \,{\left (2 \, \sqrt{b \sqrt{d} \sqrt{x} + a x + c}{\left (2 \,{\left (4 \, \sqrt{x}{\left (\frac{b \sqrt{d}}{a} + 6 \, \sqrt{x}\right )} - \frac{5 \, a b^{2} d - 12 \, a^{2} c}{a^{3}}\right )} \sqrt{x} + \frac{15 \, b^{3} d^{\frac{3}{2}} - 52 \, a b c \sqrt{d}}{a^{3}}\right )} + \frac{3 \,{\left (5 \, b^{4} d^{2} - 24 \, a b^{2} c d + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -b \sqrt{d} - 2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{b \sqrt{d} \sqrt{x} + a x + c}\right )} \right |}\right )}{a^{\frac{7}{2}}}\right )}{\rm sign}\left (x\right ) - \frac{{\left (15 \, b^{4} d^{2}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 72 \, a b^{2} c d{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{3} \sqrt{c} d^{\frac{3}{2}} + 48 \, a^{2} c^{2}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 104 \, a^{\frac{3}{2}} b c^{\frac{3}{2}} \sqrt{d}\right )}{\rm sign}\left (x\right )}{192 \, a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(2*sqrt(b*sqrt(d)*sqrt(x) + a*x + c)*(2*(4*sqrt(x)*(b*sqrt(d)/a + 6*sqrt(x
)) - (5*a*b^2*d - 12*a^2*c)/a^3)*sqrt(x) + (15*b^3*d^(3/2) - 52*a*b*c*sqrt(d))/a
^3) + 3*(5*b^4*d^2 - 24*a*b^2*c*d + 16*a^2*c^2)*ln(abs(-b*sqrt(d) - 2*sqrt(a)*(s
qrt(a)*sqrt(x) - sqrt(b*sqrt(d)*sqrt(x) + a*x + c))))/a^(7/2))*sign(x) - 1/192*(
15*b^4*d^2*ln(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 72*a*b^2*c*d*ln(abs(-b*sqrt
(d) + 2*sqrt(a)*sqrt(c))) + 30*sqrt(a)*b^3*sqrt(c)*d^(3/2) + 48*a^2*c^2*ln(abs(-
b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 104*a^(3/2)*b*c^(3/2)*sqrt(d))*sign(x)/a^(7/2)