Optimal. Leaf size=172 \[ \frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
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Rubi [A] time = 0.223686, antiderivative size = 172, 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 \[ \frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*Sqrt[c/x]]*x^4),x]
[Out]
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Rubi in Sympy [A] time = 29.6582, size = 153, normalized size = 0.89 \[ \frac{4 a^{5} \sqrt{a + b \sqrt{\frac{c}{x}}}}{b^{6} c^{3}} - \frac{20 a^{4} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{6} c^{3}} + \frac{8 a^{3} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{b^{6} c^{3}} - \frac{40 a^{2} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{7}{2}}}{7 b^{6} c^{3}} + \frac{20 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{9}{2}}}{9 b^{6} c^{3}} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{11}{2}}}{11 b^{6} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(a+b*(c/x)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.120352, size = 111, normalized size = 0.65 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (256 a^5 x^2-128 a^4 b x^2 \sqrt{\frac{c}{x}}+96 a^3 b^2 c x-80 a^2 b^3 c x \sqrt{\frac{c}{x}}+70 a b^4 c^2-63 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{693 b^6 c^3 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*Sqrt[c/x]]*x^4),x]
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Maple [C] time = 0.072, size = 392, normalized size = 2.3 \[ -{\frac{1}{693\,{b}^{7}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{11/2}\sqrt{{\frac{c}{x}}}{x}^{4}b-693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{11/2}\sqrt{{\frac{c}{x}}}{x}^{4}b+252\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2} \left ({\frac{c}{x}} \right ) ^{5/2}{b}^{5}+852\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2} \left ({\frac{c}{x}} \right ) ^{3/2}{a}^{2}{b}^{3}+1748\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2}\sqrt{{\frac{c}{x}}}{a}^{4}b+1386\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{x}^{7/2}{a}^{6}+1386\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{x}^{7/2}{a}^{6}-2772\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2}{a}^{5}-1236\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{3/2}{a}^{3}{b}^{2}c-532\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{x}a{b}^{4}{c}^{2} \right ){x}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(a+b*(c/x)^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 1.33538, size = 171, normalized size = 0.99 \[ -\frac{4 \,{\left (\frac{63 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{11}{2}}}{b^{6}} - \frac{385 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}} a}{b^{6}} + \frac{990 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a^{2}}{b^{6}} - \frac{1386 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{3}}{b^{6}} + \frac{1155 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{4}}{b^{6}} - \frac{693 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a^{5}}{b^{6}}\right )}}{693 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(c/x) + a)*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.247496, size = 120, normalized size = 0.7 \[ \frac{4 \,{\left (70 \, a b^{4} c^{2} + 96 \, a^{3} b^{2} c x + 256 \, a^{5} x^{2} -{\left (63 \, b^{5} c^{2} + 80 \, a^{2} b^{3} c x + 128 \, a^{4} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{693 \, b^{6} c^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(c/x) + a)*x^4),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(a+b*(c/x)**(1/2))**(1/2),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^4),x, algorithm="giac")
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