Optimal. Leaf size=97 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]
[Out]
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Rubi [A] time = 0.155149, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c*x^3]]/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**3)**(1/2))**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0334027, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^4} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^4,x]
[Out]
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Maple [A] time = 0.181, size = 81, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{3}} \left ( -{b}^{2}{\it Artanh} \left ({1\sqrt{a+b\sqrt{c{x}^{3}}}{\frac{1}{\sqrt{a}}}} \right ) c{x}^{3}a+\sqrt{c{x}^{3}}b\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{{\frac{3}{2}}}+2\,\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{5/2} \right ){a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^3)^(1/2))^(1/2)/x^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a)/x^4,x, algorithm="maxima")
[Out]
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Fricas [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(sqrt(c*x^3)*b + a)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**3)**(1/2))**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.242669, size = 127, normalized size = 1.31 \[ -\frac{1}{6} \, b^{2} c^{\frac{3}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{\sqrt{c x} b c x + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a c} + \frac{\sqrt{\sqrt{c x} b c x + a c} a c +{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}}}{a b^{2} c^{4} x^{3}}\right )}{\left | c \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^3)*b + a)/x^4,x, algorithm="giac")
[Out]