Optimal. Leaf size=434 \[ \frac{21 b^2 c \sqrt{a+b \sqrt{c x^3}}}{160 a^2 x}+\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{8/3} c^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{160 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{3 b c^3 x^5 \sqrt{a+b \sqrt{c x^3}}}{40 a \left (c x^3\right )^{5/2}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{4 x^4} \]
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Rubi [A] time = 0.676885, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{21 b^2 c \sqrt{a+b \sqrt{c x^3}}}{160 a^2 x}+\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{8/3} c^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{160 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{3 b c^3 x^5 \sqrt{a+b \sqrt{c x^3}}}{40 a \left (c x^3\right )^{5/2}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{4 x^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c*x^3]]/x^5,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**3)**(1/2))**(1/2)/x**5,x)
[Out]
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Mathematica [A] time = 0.0317759, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^5} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^5,x]
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Maple [A] time = 0.188, size = 346, normalized size = 0.8 \[ -{\frac{1}{320\,{x}^{4}{a}^{2}} \left ( 7\,i{b}^{2}\sqrt{3}\sqrt [3]{-ac{b}^{2}}\sqrt{2}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{1}{x \left ( i\sqrt{3}-3 \right ) } \left ( b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}+2\,b\sqrt{c{x}^{3}}+\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) c{x}^{4}-42\,\sqrt{c{x}^{3}}{x}^{3}{b}^{3}c-18\,{x}^{3}a{b}^{2}c+104\,\sqrt{c{x}^{3}}{a}^{2}b+80\,{a}^{3} \right ){\frac{1}{\sqrt{a+b\sqrt{c{x}^{3}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^3)^(1/2))^(1/2)/x^5,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a)/x^5,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a)/x^5,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**3)**(1/2))**(1/2)/x**5,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{c x^{3}} b + a}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a)/x^5,x, algorithm="giac")
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