3.140 \(\int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx\)

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]

-(ArcTanh[(x^(q/2)*(2*a + b*x^(n - q)))/(2*Sqrt[a]*Sqrt[b*x^n + c*x^(2*n - q) +
a*x^q])]/(Sqrt[a]*(n - q)))

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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]

-(ArcTanh[(x^(q/2)*(2*a + b*x^(n - q)))/(2*Sqrt[a]*Sqrt[b*x^n + c*x^(2*n - q) +
a*x^q])]/(Sqrt[a]*(n - q)))

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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]

-x**(q/2)*sqrt(a + b*x**(n - q) + c*x**(2*n - 2*q))*atanh((2*a + b*x**(n - q))/(
2*sqrt(a)*sqrt(a + b*x**(n - q) + c*x**(2*n - 2*q))))/(sqrt(a)*(n - q)*sqrt(a*x*
*q + b*x**n + c*x**(2*n - q)))

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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]

Integrate[x^(-1 + q/2)/Sqrt[b*x^n + c*x^(2*n - q) + a*x^q], x]

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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]

int(x^(-1+1/2*q)/(b*x^n+c*x^(2*n-q)+a*x^q)^(1/2),x)

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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]

integrate(x^(1/2*q - 1)/sqrt(c*x^(2*n - q) + b*x^n + a*x^q), x)

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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]

Exception raised: TypeError

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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]

Timed 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]

integrate(x^(1/2*q - 1)/sqrt(c*x^(2*n - q) + b*x^n + a*x^q), x)