3.393 \(\int \frac{1}{\sqrt{c x} \sqrt{a x+b x^n}} \, dx\)

Optimal. Leaf size=51 \[ \frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{\sqrt{a} (1-n) \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.0852557, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2031, 2029, 206} \[ \frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{\sqrt{a} (1-n) \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.119446, size = 87, normalized size = 1.71 \[ -\frac{2 \sqrt{b} x^{\frac{n+1}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} (n-1) \sqrt{c x} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.338, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{ax+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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