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

Optimal. Leaf size=62 \[ \frac{2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{\sqrt{a} c (j-n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.107995, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2031, 2029, 206} \[ \frac{2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{\sqrt{a} c (j-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2031

Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPar
t[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2]
&& NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 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 \frac{(c x)^{-1+\frac{j}{2}}}{\sqrt{a x^j+b x^n}} \, dx &=\frac{\left (x^{-j/2} (c x)^{j/2}\right ) \int \frac{x^{-1+\frac{j}{2}}}{\sqrt{a x^j+b x^n}} \, dx}{c}\\ &=\frac{\left (2 x^{-j/2} (c x)^{j/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}\\ &=\frac{2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{\sqrt{a} c (j-n)}\\ \end{align*}

Mathematica [A]  time = 0.140186, size = 98, normalized size = 1.58 \[ \frac{2 \sqrt{b} (c x)^{j/2} x^{\frac{n-j}{2}} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} c (j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b]*x^((-j + n)/2)*(c*x)^(j/2)*Sqrt[1 + (a*x^(j - n))/b]*ArcSinh[(Sqrt[a]*x^((j - n)/2))/Sqrt[b]])/(Sqr
t[a]*c*(j - n)*Sqrt[a*x^j + b*x^n])

________________________________________________________________________________________

Maple [F]  time = 0.374, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{j}{2}}}{\frac{1}{\sqrt{a{x}^{j}+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{\frac{j}{2} - 1}}{\sqrt{a x^{j} + b x^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*x)**(j/2 - 1)/sqrt(a*x**j + b*x**n), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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