3.3052 \(\int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^4} \, dx\)

Optimal. Leaf size=371 \[ \frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{1024 c^{13/2}}+\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}-\frac{\left (1024 a^2 c^2+18 b c \sqrt{\frac{d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{11 b \left (\frac{d}{x}\right )^{3/2} \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2} \]

[Out]

(b*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sq
rt[d/x] + c/x])/(512*c^6) - ((1024*a^2*c^2 - 3276*a*b^2*c*d + 1155*b^4*d^2 + 18*
b*c*(148*a*c - 77*b^2*d)*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(6720*c^5) +
(11*b*(a + b*Sqrt[d/x] + c/x)^(3/2)*(d/x)^(3/2))/(42*c^2*d) - (2*(a + b*Sqrt[d/x
] + c/x)^(3/2))/(7*c*x^2) + ((32*a*c - 33*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/
(140*c^3*x) + (b*Sqrt[d]*(4*a*c - b^2*d)*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^
2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])
])/(1024*c^(13/2))

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Rubi [A]  time = 1.58708, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{b \sqrt{d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{1024 c^{13/2}}+\frac{b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{512 c^6}-\frac{\left (1024 a^2 c^2+18 b c \sqrt{\frac{d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{6720 c^5}+\frac{\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{140 c^3 x}+\frac{11 b \left (\frac{d}{x}\right )^{3/2} \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{42 c^2 d}-\frac{2 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{7 c x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

(b*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sq
rt[d/x] + c/x])/(512*c^6) - ((1024*a^2*c^2 - 3276*a*b^2*c*d + 1155*b^4*d^2 + 18*
b*c*(148*a*c - 77*b^2*d)*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(6720*c^5) +
(11*b*(a + b*Sqrt[d/x] + c/x)^(3/2)*(d/x)^(3/2))/(42*c^2*d) - (2*(a + b*Sqrt[d/x
] + c/x)^(3/2))/(7*c*x^2) + ((32*a*c - 33*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/
(140*c^3*x) + (b*Sqrt[d]*(4*a*c - b^2*d)*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^
2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])
])/(1024*c^(13/2))

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Rubi in Sympy [A]  time = 109.339, size = 332, normalized size = 0.89 \[ \frac{11 b \left (\frac{d}{x}\right )^{\frac{3}{2}} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{42 c^{2} d} + \frac{b \left (b d + 2 c \sqrt{\frac{d}{x}}\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \left (80 a^{2} c^{2} - 120 a b^{2} c d + 33 b^{4} d^{2}\right )}{512 c^{6}} + \frac{b \sqrt{d} \left (4 a c - b^{2} d\right ) \left (80 a^{2} c^{2} - 120 a b^{2} c d + 33 b^{4} d^{2}\right ) \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{1024 c^{\frac{13}{2}}} - \frac{2 \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{7 c x^{2}} + \frac{\left (32 a c - 33 b^{2} d\right ) \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{140 c^{3} x} - \frac{\left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}} \left (192 a^{2} c^{2} - \frac{2457 a b^{2} c d}{4} + \frac{3465 b^{4} d^{2}}{16} + \frac{27 b c \sqrt{\frac{d}{x}} \left (148 a c - 77 b^{2} d\right )}{8}\right )}{1260 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x**4,x)

[Out]

11*b*(d/x)**(3/2)*(a + b*sqrt(d/x) + c/x)**(3/2)/(42*c**2*d) + b*(b*d + 2*c*sqrt
(d/x))*sqrt(a + b*sqrt(d/x) + c/x)*(80*a**2*c**2 - 120*a*b**2*c*d + 33*b**4*d**2
)/(512*c**6) + b*sqrt(d)*(4*a*c - b**2*d)*(80*a**2*c**2 - 120*a*b**2*c*d + 33*b*
*4*d**2)*atanh((b*d + 2*c*sqrt(d/x))/(2*sqrt(c)*sqrt(d)*sqrt(a + b*sqrt(d/x) + c
/x)))/(1024*c**(13/2)) - 2*(a + b*sqrt(d/x) + c/x)**(3/2)/(7*c*x**2) + (32*a*c -
33*b**2*d)*(a + b*sqrt(d/x) + c/x)**(3/2)/(140*c**3*x) - (a + b*sqrt(d/x) + c/x
)**(3/2)*(192*a**2*c**2 - 2457*a*b**2*c*d/4 + 3465*b**4*d**2/16 + 27*b*c*sqrt(d/
x)*(148*a*c - 77*b**2*d)/8)/(1260*c**5)

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Mathematica [A]  time = 0.07649, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4, x]

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Maple [B]  time = 0.046, size = 979, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+c/x+b*(d/x)^(1/2))^(1/2)/x^4,x)

[Out]

1/107520*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(-39060*a*(b*(d/x)^(1/2)*x+a*x+c)^(1/
2)*(d/x)^(5/2)*x^6*b^5*c+33600*a^3*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^(7/2)*(d/x)^(1/2)*x^4*b+67200*a^2*(b*(d/x)^(1/2)*
x+a*x+c)^(1/2)*(d/x)^(3/2)*x^5*b^3*c^2-25200*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d
^2*x^4*b^4*c+16800*a^3*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^4*b^2*c^2+25200*a*(b*(d
/x)^(1/2)*x+a*x+c)^(3/2)*d^2*x^3*b^4*c-16800*a^2*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d
*x^3*b^2*c^2+52416*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^2*b^2*c^3-42624*a*(b*(d/x
)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^4+26460*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)^(5/2)*x^6*b^5-58800
*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)^(3/2)*x^5*b^3-50400*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^4
*b^3*c^2-33600*a^3*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^4*b*c^3+33600*a^2
*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^3*b*c^3-16384*a^2*(b*(d/x)^(1/2)*x+
a*x+c)^(3/2)*x^2*c^4+24576*a*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x*c^5-6930*(b*(d/x)^(
1/2)*x+a*x+c)^(3/2)*d^3*x^3*b^6+6930*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(7/2)*x
^7*b^7-3465*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)^(7/2)*x^7*b^7+13860*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(5/2
)*x^5*b^5*c+22176*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c^3+6930*a*(
b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d^3*x^4*b^6-18480*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d^2
*x^2*b^4*c^2-25344*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x*b^2*c^4+28160*(b*(d/x)^(1/2
)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^5-30720*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*c^6)/x^
3/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^4, 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^4,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+c/x+b*(d/x)**(1/2))**(1/2)/x**4,x)

[Out]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**4, 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^4,x, algorithm="giac")

[Out]

Timed out