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

Optimal. Leaf size=70 \[ -\frac {\tanh ^{-1}\left (\frac {x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt {a} \sqrt {a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt {a} (n-q)} \]

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Rubi [A]  time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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 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])

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]

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*}

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+\frac {q}{2}}}{\sqrt {b x^n+c x^{2 n-q}+a x^q}} \, dx \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+\frac {q}{2}}}{\sqrt {b x^n+c x^{2 n-q}+a x^q}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {1}{2} \, q - 1}}{\sqrt {c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {q}{2}-1}}{\sqrt {a \,x^{q}+b \,x^{n}+c \,x^{2 n -q}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {1}{2} \, q - 1}}{\sqrt {c x^{2 \, n - q} + b x^{n} + a x^{q}}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{\frac {q}{2}-1}}{\sqrt {b\,x^n+a\,x^q+c\,x^{2\,n-q}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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