Optimal. Leaf size=151 \[ -\frac{\sqrt{a+b x^n+c x^{2 n}} F_1\left (-\frac{2}{n};-\frac{1}{2},-\frac{1}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.452266, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\sqrt{a+b x^n+c x^{2 n}} F_1\left (-\frac{2}{n};-\frac{1}{2},-\frac{1}{2};-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^n + c*x^(2*n)]/x^3,x]
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Rubi in Sympy [A] time = 33.8119, size = 131, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (- \frac{2}{n},- \frac{1}{2},- \frac{1}{2},\frac{n - 2}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{2 x^{2} \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n+c*x**(2*n))**(1/2)/x**3,x)
[Out]
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Mathematica [B] time = 8.92062, size = 816, normalized size = 5.4 \[ \frac{\frac{4 a^2 b (n-1) n \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-2}{n};\frac{1}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+\sqrt{b^2-4 a c}\right ) (n-2)^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) n F_1\left (2-\frac{2}{n};\frac{1}{2},\frac{3}{2};3-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-\left (\sqrt{b^2-4 a c}-b\right ) n F_1\left (2-\frac{2}{n};\frac{3}{2},\frac{1}{2};3-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n-8 a (n-1) F_1\left (\frac{n-2}{n};\frac{1}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{\left (\left (c x^n+b\right ) x^n+a\right )^2}{n-2}+\frac{a^2 n \left (-2 c x^n-b+\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{8 a c (n-2) F_1\left (-\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n-2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-2 c n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-2}{n};\frac{1}{2},\frac{3}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{n-2}{n};\frac{3}{2},\frac{1}{2};2-\frac{2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}}{x^2 \left (\left (c x^n+b\right ) x^n+a\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[a + b*x^n + c*x^(2*n)]/x^3,x]
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Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n+c*x^(2*n))^(1/2)/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^(2*n) + b*x^n + a)/x^3,x, algorithm="maxima")
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Fricas [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(sqrt(c*x^(2*n) + b*x^n + a)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{n} + c x^{2 n}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n+c*x**(2*n))**(1/2)/x**3,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^(2*n) + b*x^n + a)/x^3,x, algorithm="giac")
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