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 = 0.0666428, 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, Rules used = {1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt{a} (n-q)} \]

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)))

Rule 1913

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - q), Sub
st[Int[1/(4*a - x^2), x], x, (x^(m + 1)*(2*a + b*x^(n - q)))/Sqrt[a*x^q + b*x^n + c*x^r]], x] /; FreeQ[{a, b,
c, m, n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^{-1+\frac{q}{2}}}{\sqrt{b x^n+c x^{2 n-q}+a x^q}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{\sqrt{b x^n+c x^{2 n-q}+a x^q}}\right )}{n-q}\\ &=-\frac{\tanh ^{-1}\left (\frac{x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt{a} \sqrt{b x^n+c x^{2 n-q}+a x^q}}\right )}{\sqrt{a} (n-q)}\\ \end{align*}

Mathematica [F]  time = 0.392503, 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 = 0.121, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{q}{2}}}{\frac{1}{\sqrt{b{x}^{n}+c{x}^{2\,n-q}+a{x}^{q}}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{x^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{1}{2} \, q - 1}}{\sqrt{c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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