3.3051 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^3} \, dx\)

Optimal. Leaf size=233 \[ -\frac{b \sqrt{d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{9/2}}-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}+\frac{\left (32 a c-35 b^2 d+42 b c \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x} \]

[Out]

-(b*(12*a*c - 7*b^2*d)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(64*c^
4) + ((32*a*c - 35*b^2*d + 42*b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(120
*c^3) - (2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(5*c*x) - (b*Sqrt[d]*(12*a*c - 7*b^2*d
)*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sq
rt[d/x] + c/x])])/(128*c^(9/2))

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Rubi [A]  time = 0.669855, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{b \sqrt{d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{9/2}}-\frac{b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{64 c^4}+\frac{\left (32 a c-35 b^2 d+42 b c \sqrt{\frac{d}{x}}\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{120 c^3}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{5 c x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

-(b*(12*a*c - 7*b^2*d)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(64*c^
4) + ((32*a*c - 35*b^2*d + 42*b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(120
*c^3) - (2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(5*c*x) - (b*Sqrt[d]*(12*a*c - 7*b^2*d
)*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sq
rt[d/x] + c/x])])/(128*c^(9/2))

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Rubi in Sympy [A]  time = 47.7811, size = 201, normalized size = 0.86 \[ - \frac{b \left (12 a c - 7 b^{2} d\right ) \left (b d + 2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{64 c^{4}} - \frac{b \sqrt{d} \left (4 a c - b^{2} d\right ) \left (12 a c - 7 b^{2} d\right ) \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{128 c^{\frac{9}{2}}} - \frac{2 \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{5 c x} + \frac{\left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}} \left (8 a c - \frac{35 b^{2} d}{4} + \frac{21 b c \sqrt{\frac{d}{x}}}{2}\right )}{30 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(12*a*c - 7*b**2*d)*(b*d + 2*c*sqrt(d/x))*sqrt(a + b*sqrt(d/x) + c/x)/(64*c**
4) - b*sqrt(d)*(4*a*c - b**2*d)*(12*a*c - 7*b**2*d)*atanh((b*d + 2*c*sqrt(d/x))/
(2*sqrt(c)*sqrt(d)*sqrt(a + b*sqrt(d/x) + c/x)))/(128*c**(9/2)) - 2*(a + b*sqrt(
d/x) + c/x)**(3/2)/(5*c*x) + (a + b*sqrt(d/x) + c/x)**(3/2)*(8*a*c - 35*b**2*d/4
+ 21*b*c*sqrt(d/x)/2)/(30*c**3)

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Mathematica [A]  time = 0.0745, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3, x]

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Maple [B]  time = 0.04, size = 615, normalized size = 2.6 \[ -{\frac{1}{1920\,{x}^{2}{c}^{5}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 105\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}-210\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}-600\,a\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ){c}^{3/2} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}{b}^{3}-210\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}{x}^{3}{b}^{4}+720\,{a}^{2}\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ){c}^{5/2}\sqrt{{\frac{d}{x}}}{x}^{3}b+780\,a\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}{b}^{3}c+360\,{a}^{2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{3}{b}^{2}c+210\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{d}^{2}{x}^{2}{b}^{4}-420\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}c-720\,{a}^{2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3}b{c}^{2}-360\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}d{x}^{2}{b}^{2}c+720\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}{x}^{2}b{c}^{2}+560\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}dx{b}^{2}{c}^{2}-512\,a \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}x{c}^{3}-672\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}\sqrt{{\frac{d}{x}}}xb{c}^{3}+768\, \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) ^{3/2}{c}^{4} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(105*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)
*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(1/2)*(d/x)^(5/2)*x^5*b^5-210*(b*(d/x
)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^5*b^5-600*a*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/
2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(3/2)*(d/x)^(3/2)*x^4*b^3-210*a*(b*
(d/x)^(1/2)*x+a*x+c)^(1/2)*d^2*x^3*b^4+720*a^2*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)
*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c^(5/2)*(d/x)^(1/2)*x^3*b+780*a*(b*(d/x
)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^4*b^3*c+360*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/
2)*d*x^3*b^2*c+210*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^2*b^4-420*(b*(d/x)^(1/2)*
x+a*x+c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c-720*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)
^(1/2)*x^3*b*c^2-360*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^2*b^2*c+720*a*(b*(d/x)^
(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^2+560*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x
*b^2*c^2-512*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x*c^3-672*(b*(d/x)^(1/2)*x+a*x+c)^(
3/2)*(d/x)^(1/2)*x*b*c^3+768*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^4)/x^2/(b*(d/x)^(1/
2)*x+a*x+c)^(1/2)/c^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^3, 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^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**3, 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(d/x) + a + c/x)/x^3,x, algorithm="giac")

[Out]

Timed out