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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]