Optimal. Leaf size=144 \[ -\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]
[Out]
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Rubi [A] time = 0.164832, antiderivative size = 144, 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 \[ -\frac{b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^2}}}{\sqrt{a}}\right )}{8 a^{5/2} x^3}+\frac{b^2 c \sqrt{a+b \sqrt{c x^2}}}{8 a^2 x}-\frac{b \left (c x^2\right )^{3/2} \sqrt{a+b \sqrt{c x^2}}}{12 a c x^5}-\frac{\sqrt{a+b \sqrt{c x^2}}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c*x^2]]/x^4,x]
[Out]
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Rubi in Sympy [A] time = 15.6864, size = 124, normalized size = 0.86 \[ - \frac{\sqrt{a + b \sqrt{c x^{2}}}}{3 x^{3}} - \frac{b \left (c x^{2}\right )^{\frac{3}{2}} \sqrt{a + b \sqrt{c x^{2}}}}{12 a c x^{5}} + \frac{b^{2} c \sqrt{a + b \sqrt{c x^{2}}}}{8 a^{2} x} - \frac{b^{3} \left (c x^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c x^{2}}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0324504, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^2}}}{x^4} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*Sqrt[c*x^2]]/x^4,x]
[Out]
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Maple [A] time = 0.019, size = 97, normalized size = 0.7 \[ -{\frac{1}{24\,{x}^{3}} \left ( 3\,{a}^{9/2}\sqrt{a+b\sqrt{c{x}^{2}}}+8\,{a}^{7/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{3/2}-3\,{a}^{5/2} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{5/2}+3\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{c{x}^{2}}}}{\sqrt{a}}} \right ){a}^{2}{b}^{3} \left ( c{x}^{2} \right ) ^{3/2} \right ){a}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x^2)^(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^2)*b + a)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224765, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} c x^{3} \sqrt{\frac{c}{a}} \log \left (\frac{\sqrt{c x^{2}} b c x + 2 \, a c x - 2 \, \sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a} a \sqrt{\frac{c}{a}}}{x^{2}}\right ) + 2 \,{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{48 \, a^{2} x^{3}}, \frac{3 \, b^{3} c x^{3} \sqrt{-\frac{c}{a}} \arctan \left (\frac{a x \sqrt{-\frac{c}{a}}}{\sqrt{c x^{2}} \sqrt{\sqrt{c x^{2}} b + a}}\right ) +{\left (3 \, b^{2} c x^{2} - 2 \, \sqrt{c x^{2}} a b - 8 \, a^{2}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{24 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{2}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x**2)**(1/2))**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.220577, size = 154, normalized size = 1.07 \[ \frac{\frac{3 \, b^{4} c^{2} \arctan \left (\frac{\sqrt{b \sqrt{c} x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} b^{4} c^{2} - 8 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a b^{4} c^{2} - 3 \, \sqrt{b \sqrt{c} x + a} a^{2} b^{4} c^{2}}{a^{2} b^{3} c^{\frac{3}{2}} x^{3}}}{24 \, b \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)/x^4,x, algorithm="giac")
[Out]