Optimal. Leaf size=70 \[ -\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt{a} (n-q)} \]
[Out]
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Rubi [A] time = 0.103628, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt{a} (n-q)} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + q/2)/Sqrt[b*x^n + c*x^(2*n - q) + a*x^q],x]
[Out]
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Rubi in Sympy [A] time = 24.4809, size = 99, normalized size = 1.41 \[ - \frac{x^{\frac{q}{2}} \sqrt{a + b x^{n - q} + c x^{2 n - 2 q}} \operatorname{atanh}{\left (\frac{2 a + b x^{n - q}}{2 \sqrt{a} \sqrt{a + b x^{n - q} + c x^{2 n - 2 q}}} \right )}}{\sqrt{a} \left (n - q\right ) \sqrt{a x^{q} + b x^{n} + c x^{2 n - q}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+1/2*q)/(b*x**n+c*x**(2*n-q)+a*x**q)**(1/2),x)
[Out]
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Mathematica [A] time = 0.346561, size = 0, normalized size = 0. \[ \int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x^(-1 + q/2)/Sqrt[b*x^n + c*x^(2*n - q) + a*x^q],x]
[Out]
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Maple [F] time = 0.2, size = 0, normalized size = 0. \[ \int{1{x}^{-1+{\frac{q}{2}}}{\frac{1}{\sqrt{b{x}^{n}+c{x}^{2\,n-q}+a{x}^{q}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+1/2*q)/(b*x^n+c*x^(2*n-q)+a*x^q)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/2*q - 1)/sqrt(c*x^(2*n - q) + b*x^n + a*x^q),x, algorithm="maxima")
[Out]
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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(x^(1/2*q - 1)/sqrt(c*x^(2*n - q) + b*x^n + a*x^q),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+1/2*q)/(b*x**n+c*x**(2*n-q)+a*x**q)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(1/2*q - 1)/sqrt(c*x^(2*n - q) + b*x^n + a*x^q),x, algorithm="giac")
[Out]