3.2980 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^4} \, dx\)

Optimal. Leaf size=174 \[ \frac{4 a^5 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac{4 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}+\frac{40 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3} \]

[Out]

(4*a^5*(a + b*Sqrt[c/x])^(3/2))/(3*b^6*c^3) - (4*a^4*(a + b*Sqrt[c/x])^(5/2))/(b
^6*c^3) + (40*a^3*(a + b*Sqrt[c/x])^(7/2))/(7*b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x
])^(9/2))/(9*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(11/2))/(11*b^6*c^3) - (4*(a + b
*Sqrt[c/x])^(13/2))/(13*b^6*c^3)

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Rubi [A]  time = 0.231799, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 a^5 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac{4 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}+\frac{40 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^5*(a + b*Sqrt[c/x])^(3/2))/(3*b^6*c^3) - (4*a^4*(a + b*Sqrt[c/x])^(5/2))/(b
^6*c^3) + (40*a^3*(a + b*Sqrt[c/x])^(7/2))/(7*b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x
])^(9/2))/(9*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(11/2))/(11*b^6*c^3) - (4*(a + b
*Sqrt[c/x])^(13/2))/(13*b^6*c^3)

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Rubi in Sympy [A]  time = 30.0696, size = 155, normalized size = 0.89 \[ \frac{4 a^{5} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{6} c^{3}} - \frac{4 a^{4} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{b^{6} c^{3}} + \frac{40 a^{3} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{7}{2}}}{7 b^{6} c^{3}} - \frac{40 a^{2} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{9}{2}}}{9 b^{6} c^{3}} + \frac{20 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{11}{2}}}{11 b^{6} c^{3}} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{13}{2}}}{13 b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**5*(a + b*sqrt(c/x))**(3/2)/(3*b**6*c**3) - 4*a**4*(a + b*sqrt(c/x))**(5/2)/
(b**6*c**3) + 40*a**3*(a + b*sqrt(c/x))**(7/2)/(7*b**6*c**3) - 40*a**2*(a + b*sq
rt(c/x))**(9/2)/(9*b**6*c**3) + 20*a*(a + b*sqrt(c/x))**(11/2)/(11*b**6*c**3) -
4*(a + b*sqrt(c/x))**(13/2)/(13*b**6*c**3)

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Mathematica [A]  time = 0.0731644, size = 111, normalized size = 0.64 \[ \frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (256 a^5 x^2-384 a^4 b x^2 \sqrt{\frac{c}{x}}+480 a^3 b^2 c x-560 a^2 b^3 c x \sqrt{\frac{c}{x}}+630 a b^4 c^2-693 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{9009 b^6 c^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*(a + b*Sqrt[c/x])^(3/2)*(630*a*b^4*c^2 + 480*a^3*b^2*c*x - 560*a^2*b^3*c*Sqrt
[c/x]*x - 693*b^5*c*(c/x)^(3/2)*x + 256*a^5*x^2 - 384*a^4*b*Sqrt[c/x]*x^2))/(900
9*b^6*c^3*x^2)

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Maple [A]  time = 0.031, size = 133, normalized size = 0.8 \[ -{\frac{4}{9009\,{x}^{3}{c}^{3}{b}^{6}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 693\,{x}^{2} \left ({\frac{c}{x}} \right ) ^{5/2}{b}^{5}+560\,{x}^{2} \left ({\frac{c}{x}} \right ) ^{3/2}{a}^{2}{b}^{3}+384\,{x}^{2}\sqrt{{\frac{c}{x}}}{a}^{4}b-256\,{a}^{5}{x}^{2}-480\,xc{a}^{3}{b}^{2}-630\,{c}^{2}a{b}^{4} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9009*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/x^3/c^3*(693*x^2*(c/
x)^(5/2)*b^5+560*x^2*(c/x)^(3/2)*a^2*b^3+384*x^2*(c/x)^(1/2)*a^4*b-256*a^5*x^2-4
80*x*c*a^3*b^2-630*c^2*a*b^4)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^6

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Maxima [A]  time = 1.34587, size = 171, normalized size = 0.98 \[ -\frac{4 \,{\left (\frac{693 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{13}{2}}}{b^{6}} - \frac{4095 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{11}{2}} a}{b^{6}} + \frac{10010 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}} a^{2}}{b^{6}} - \frac{12870 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a^{3}}{b^{6}} + \frac{9009 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{3003 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9009*(693*(b*sqrt(c/x) + a)^(13/2)/b^6 - 4095*(b*sqrt(c/x) + a)^(11/2)*a/b^6
+ 10010*(b*sqrt(c/x) + a)^(9/2)*a^2/b^6 - 12870*(b*sqrt(c/x) + a)^(7/2)*a^3/b^6
+ 9009*(b*sqrt(c/x) + a)^(5/2)*a^4/b^6 - 3003*(b*sqrt(c/x) + a)^(3/2)*a^5/b^6)/c
^3

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Fricas [A]  time = 0.252676, size = 142, normalized size = 0.82 \[ -\frac{4 \,{\left (693 \, b^{6} c^{3} - 70 \, a^{2} b^{4} c^{2} x - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} x^{3} +{\left (63 \, a b^{5} c^{2} x + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b x^{3}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{9009 \, b^{6} c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/9009*(693*b^6*c^3 - 70*a^2*b^4*c^2*x - 96*a^4*b^2*c*x^2 - 256*a^6*x^3 + (63*a
*b^5*c^2*x + 80*a^3*b^3*c*x^2 + 128*a^5*b*x^3)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a)/
(b^6*c^3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out