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