3.3060 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^4} \, dx\)

Optimal. Leaf size=289 \[ \frac{b \sqrt{d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{11/2}}-\frac{\left (1024 a^2 c^2+14 b c \sqrt{\frac{d}{x}} \left (92 a c-45 b^2 d\right )-2940 a b^2 c d+945 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{960 c^5}+\frac{\left (64 a c-63 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{120 c^3 x}+\frac{9 b \left (\frac{d}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{20 c^2 d}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{5 c x^2} \]

[Out]

-((1024*a^2*c^2 - 2940*a*b^2*c*d + 945*b^4*d^2 + 14*b*c*(92*a*c - 45*b^2*d)*Sqrt
[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(960*c^5) + (9*b*Sqrt[a + b*Sqrt[d/x] + c/x]
*(d/x)^(3/2))/(20*c^2*d) - (2*Sqrt[a + b*Sqrt[d/x] + c/x])/(5*c*x^2) + ((64*a*c
- 63*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(120*c^3*x) + (b*Sqrt[d]*(240*a^2*c^2 -
280*a*b^2*c*d + 63*b^4*d^2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sq
rt[a + b*Sqrt[d/x] + c/x])])/(128*c^(11/2))

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Rubi [A]  time = 1.32597, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{b \sqrt{d} \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{128 c^{11/2}}-\frac{\left (1024 a^2 c^2+14 b c \sqrt{\frac{d}{x}} \left (92 a c-45 b^2 d\right )-2940 a b^2 c d+945 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{960 c^5}+\frac{\left (64 a c-63 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{120 c^3 x}+\frac{9 b \left (\frac{d}{x}\right )^{3/2} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{20 c^2 d}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{5 c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((1024*a^2*c^2 - 2940*a*b^2*c*d + 945*b^4*d^2 + 14*b*c*(92*a*c - 45*b^2*d)*Sqrt
[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(960*c^5) + (9*b*Sqrt[a + b*Sqrt[d/x] + c/x]
*(d/x)^(3/2))/(20*c^2*d) - (2*Sqrt[a + b*Sqrt[d/x] + c/x])/(5*c*x^2) + ((64*a*c
- 63*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(120*c^3*x) + (b*Sqrt[d]*(240*a^2*c^2 -
280*a*b^2*c*d + 63*b^4*d^2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sq
rt[a + b*Sqrt[d/x] + c/x])])/(128*c^(11/2))

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Rubi in Sympy [A]  time = 95.5762, size = 257, normalized size = 0.89 \[ \frac{9 b \left (\frac{d}{x}\right )^{\frac{3}{2}} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{20 c^{2} d} + \frac{b \sqrt{d} \left (240 a^{2} c^{2} - 280 a b^{2} c d + 63 b^{4} d^{2}\right ) \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{128 c^{\frac{11}{2}}} - \frac{2 \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{5 c x^{2}} + \frac{\left (64 a c - 63 b^{2} d\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{120 c^{3} x} - \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \left (64 a^{2} c^{2} - \frac{735 a b^{2} c d}{4} + \frac{945 b^{4} d^{2}}{16} + \frac{7 b c \sqrt{\frac{d}{x}} \left (92 a c - 45 b^{2} d\right )}{8}\right )}{60 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*b*(d/x)**(3/2)*sqrt(a + b*sqrt(d/x) + c/x)/(20*c**2*d) + b*sqrt(d)*(240*a**2*c
**2 - 280*a*b**2*c*d + 63*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)))/(128*c**(11/2)) - 2*sqrt(a + b*sqrt(d/x) + c/x
)/(5*c*x**2) + (64*a*c - 63*b**2*d)*sqrt(a + b*sqrt(d/x) + c/x)/(120*c**3*x) - s
qrt(a + b*sqrt(d/x) + c/x)*(64*a**2*c**2 - 735*a*b**2*c*d/4 + 945*b**4*d**2/16 +
7*b*c*sqrt(d/x)*(92*a*c - 45*b**2*d)/8)/(60*c**5)

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Mathematica [A]  time = 0.22737, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4), x]

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Maple [A]  time = 0.046, size = 487, normalized size = 1.7 \[{\frac{1}{1920\,{x}^{2}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 945\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5}{b}^{5}c-1890\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}{c}^{3/2}{x}^{2}{b}^{4}-4200\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{4}a{b}^{3}{c}^{2}+1260\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{5/2} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}+5880\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{c}^{5/2}{x}^{2}a{b}^{2}+3600\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{{\frac{d}{x}}}{x}^{3}{a}^{2}b{c}^{3}-1008\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{c}^{7/2}x{b}^{2}-2576\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{7/2}\sqrt{{\frac{d}{x}}}{x}^{2}ab-2048\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{7/2}{x}^{2}{a}^{2}+864\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{9/2}\sqrt{{\frac{d}{x}}}xb+1024\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{9/2}xa-768\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{11/2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{c}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x^2*(945*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))*(d/x)^(5/2)*x^5*b^5*c-1890*(b*(d/x)^
(1/2)*x+a*x+c)^(1/2)*d^2*c^(3/2)*x^2*b^4-4200*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))*(d/x)^(3/2)*x^4*a*b^3*c^2+1260*(b*(d/x)^
(1/2)*x+a*x+c)^(1/2)*c^(5/2)*(d/x)^(3/2)*x^3*b^3+5880*(b*(d/x)^(1/2)*x+a*x+c)^(1
/2)*d*c^(5/2)*x^2*a*b^2+3600*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))*(d/x)^(1/2)*x^3*a^2*b*c^3-1008*(b*(d/x)^(1/2)*x+a*x+c)^(1
/2)*d*c^(7/2)*x*b^2-2576*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(7/2)*(d/x)^(1/2)*x^2*a
*b-2048*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(7/2)*x^2*a^2+864*(b*(d/x)^(1/2)*x+a*x+c
)^(1/2)*c^(9/2)*(d/x)^(1/2)*x*b+1024*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(9/2)*x*a-7
68*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(11/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(13/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^4),x, algorithm="maxima")

[Out]

integrate(1/(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(1/(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. \[ \int \frac{1}{x^{4} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*sqrt(a + b*sqrt(d/x) + c/x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^4), x)