3.2979 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^3} \, dx\)

Optimal. Leaf size=116 \[ \frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

(4*a^3*(a + b*Sqrt[c/x])^(3/2))/(3*b^4*c^2) - (12*a^2*(a + b*Sqrt[c/x])^(5/2))/(
5*b^4*c^2) + (12*a*(a + b*Sqrt[c/x])^(7/2))/(7*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(
9/2))/(9*b^4*c^2)

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Rubi [A]  time = 0.164548, antiderivative size = 116, 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 \[ \frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^3*(a + b*Sqrt[c/x])^(3/2))/(3*b^4*c^2) - (12*a^2*(a + b*Sqrt[c/x])^(5/2))/(
5*b^4*c^2) + (12*a*(a + b*Sqrt[c/x])^(7/2))/(7*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(
9/2))/(9*b^4*c^2)

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Rubi in Sympy [A]  time = 20.2083, size = 102, normalized size = 0.88 \[ \frac{4 a^{3} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{4} c^{2}} - \frac{12 a^{2} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{5 b^{4} c^{2}} + \frac{12 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{7}{2}}}{7 b^{4} c^{2}} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{9}{2}}}{9 b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**3*(a + b*sqrt(c/x))**(3/2)/(3*b**4*c**2) - 12*a**2*(a + b*sqrt(c/x))**(5/2)
/(5*b**4*c**2) + 12*a*(a + b*sqrt(c/x))**(7/2)/(7*b**4*c**2) - 4*(a + b*sqrt(c/x
))**(9/2)/(9*b**4*c**2)

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Mathematica [A]  time = 0.0565637, size = 75, normalized size = 0.65 \[ \frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (16 a^3 x-24 a^2 b x \sqrt{\frac{c}{x}}+30 a b^2 c-35 b^3 c \sqrt{\frac{c}{x}}\right )}{315 b^4 c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

(4*(a + b*Sqrt[c/x])^(3/2)*(30*a*b^2*c - 35*b^3*c*Sqrt[c/x] + 16*a^3*x - 24*a^2*
b*Sqrt[c/x]*x))/(315*b^4*c^2*x)

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Maple [A]  time = 0.03, size = 97, normalized size = 0.8 \[ -{\frac{4}{315\,{x}^{2}{c}^{2}{b}^{4}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 35\,x \left ({\frac{c}{x}} \right ) ^{3/2}{b}^{3}+24\,x\sqrt{{\frac{c}{x}}}{a}^{2}b-16\,{a}^{3}x-30\,ac{b}^{2} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/315*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/x^2/c^2*(35*x*(c/x)^(
3/2)*b^3+24*x*(c/x)^(1/2)*a^2*b-16*a^3*x-30*a*c*b^2)/(x*(a+b*(c/x)^(1/2)))^(1/2)
/b^4

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Maxima [A]  time = 1.42676, size = 115, normalized size = 0.99 \[ -\frac{4 \,{\left (\frac{35 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/315*(35*(b*sqrt(c/x) + a)^(9/2)/b^4 - 135*(b*sqrt(c/x) + a)^(7/2)*a/b^4 + 189
*(b*sqrt(c/x) + a)^(5/2)*a^2/b^4 - 105*(b*sqrt(c/x) + a)^(3/2)*a^3/b^4)/c^2

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Fricas [A]  time = 0.255793, size = 104, normalized size = 0.9 \[ -\frac{4 \,{\left (35 \, b^{4} c^{2} - 6 \, a^{2} b^{2} c x - 16 \, a^{4} x^{2} +{\left (5 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{315 \, b^{4} c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/315*(35*b^4*c^2 - 6*a^2*b^2*c*x - 16*a^4*x^2 + (5*a*b^3*c*x + 8*a^3*b*x^2)*sq
rt(c/x))*sqrt(b*sqrt(c/x) + a)/(b^4*c^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out