3.2985 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^3} \, dx\)

Optimal. Leaf size=112 \[ \frac{4 a^3 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^4 c^2}-\frac{4 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.162345, antiderivative size = 112, 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 \[ \frac{4 a^3 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^4 c^2}-\frac{4 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*Sqrt[c/x]]*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.1825, size = 99, normalized size = 0.88 \[ \frac{4 a^{3} \sqrt{a + b \sqrt{\frac{c}{x}}}}{b^{4} c^{2}} - \frac{4 a^{2} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{b^{4} c^{2}} + \frac{12 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{5 b^{4} c^{2}} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{7}{2}}}{7 b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.094426, size = 75, normalized size = 0.67 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (16 a^3 x-8 a^2 b x \sqrt{\frac{c}{x}}+6 a b^2 c-5 b^3 c \sqrt{\frac{c}{x}}\right )}{35 b^4 c^2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*Sqrt[c/x]]*x^3),x]

[Out]

(4*Sqrt[a + b*Sqrt[c/x]]*(6*a*b^2*c - 5*b^3*c*Sqrt[c/x] + 16*a^3*x - 8*a^2*b*Sqr
t[c/x]*x))/(35*b^4*c^2*x)

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Maple [C]  time = 0.07, size = 328, normalized size = 2.9 \[{\frac{1}{35\,{b}^{5}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 35\,{a}^{7/2}\sqrt{{\frac{c}{x}}}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){x}^{3}b-35\,{a}^{7/2}\sqrt{{\frac{c}{x}}}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){x}^{3}b-20\,{x}^{3/2} \left ({\frac{c}{x}} \right ) ^{3/2} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{b}^{3}-70\,{x}^{5/2}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{4}-70\,{x}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{4}-76\,{x}^{3/2}\sqrt{{\frac{c}{x}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{2}b+140\,{x}^{3/2} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{3}+44\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}a{b}^{2}c \right ){x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(a+b*(c/x)^(1/2))^(1/2)*(35*a^(7/2)*(c/x)^(1/2)*ln(1/2*(b*(c/x)^(1/2)*x^(1/
2)+2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*x^3*b-35*a^(7/2)*
(c/x)^(1/2)*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(1/2)+
2*a*x^(1/2))/a^(1/2))*x^3*b-20*x^(3/2)*(c/x)^(3/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*b
^3-70*x^(5/2)*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^4-70*x^(5/2)*(a*x+b*(c/x)^(1/2)*x)^(
1/2)*a^4-76*x^(3/2)*(c/x)^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a^2*b+140*x^(3/2)*(a
*x+b*(c/x)^(1/2)*x)^(3/2)*a^3+44*x^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a*b^2*c)/x^
(9/2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^5/(c/x)^(5/2)

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Maxima [A]  time = 1.34328, size = 115, normalized size = 1.03 \[ -\frac{4 \,{\left (\frac{5 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}}}{b^{4}} - \frac{21 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a}{b^{4}} + \frac{35 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{2}}{b^{4}} - \frac{35 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a^{3}}{b^{4}}\right )}}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(c/x) + a)*x^3),x, algorithm="maxima")

[Out]

-4/35*(5*(b*sqrt(c/x) + a)^(7/2)/b^4 - 21*(b*sqrt(c/x) + a)^(5/2)*a/b^4 + 35*(b*
sqrt(c/x) + a)^(3/2)*a^2/b^4 - 35*sqrt(b*sqrt(c/x) + a)*a^3/b^4)/c^2

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Fricas [A]  time = 0.249943, size = 82, normalized size = 0.73 \[ \frac{4 \,{\left (6 \, a b^{2} c + 16 \, a^{3} x -{\left (5 \, b^{3} c + 8 \, a^{2} b x\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{35 \, b^{4} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(c/x) + a)*x^3),x, algorithm="fricas")

[Out]

4/35*(6*a*b^2*c + 16*a^3*x - (5*b^3*c + 8*a^2*b*x)*sqrt(c/x))*sqrt(b*sqrt(c/x) +
a)/(b^4*c^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(a + b*sqrt(c/x))), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(c/x) + a)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError