3.364 \(\int x^{-1-\frac{j}{2}} \sqrt{a x^j+b x^n} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{j-n}-\frac{2 x^{-j/2} \sqrt{a x^j+b x^n}}{j-n} \]

[Out]

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

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Rubi [A]  time = 0.114247, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2028, 2029, 206} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{j-n}-\frac{2 x^{-j/2} \sqrt{a x^j+b x^n}}{j-n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]

[Out]

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

Rule 2028

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*p*(n - j)), x] + Dist[a/c^j, Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c,
j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

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

Mathematica [A]  time = 0.200335, size = 104, normalized size = 1.39 \[ -\frac{2 x^{-j/2} \left (-\sqrt{a} \sqrt{b} x^{\frac{j+n}{2}} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )+a x^j+b x^n\right )}{(j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*x^j + b*x^n - Sqrt[a]*Sqrt[b]*x^((j + n)/2)*Sqrt[1 + (a*x^(j - n))/b]*ArcSinh[(Sqrt[a]*x^((j - n)/2))/S
qrt[b]]))/((j - n)*x^(j/2)*Sqrt[a*x^j + b*x^n])

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Maple [F]  time = 0.532, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-{\frac{j}{2}}}\sqrt{a{x}^{j}+b{x}^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="giac")

[Out]

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