3.391 \(\int \frac{\sqrt{c x}}{\sqrt{a x^3+b x^n}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{\sqrt{a} (3-n) \sqrt{x}} \]

[Out]

(2*Sqrt[c*x]*ArcTanh[(Sqrt[a]*x^(3/2))/Sqrt[a*x^3 + b*x^n]])/(Sqrt[a]*(3 - n)*Sqrt[x])

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Rubi [A]  time = 0.0987987, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2031, 2029, 206} \[ \frac{2 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{\sqrt{a} (3-n) \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]/Sqrt[a*x^3 + b*x^n],x]

[Out]

(2*Sqrt[c*x]*ArcTanh[(Sqrt[a]*x^(3/2))/Sqrt[a*x^3 + b*x^n]])/(Sqrt[a]*(3 - n)*Sqrt[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{\sqrt{c x}}{\sqrt{a x^3+b x^n}} \, dx &=\frac{\sqrt{c x} \int \frac{\sqrt{x}}{\sqrt{a x^3+b x^n}} \, dx}{\sqrt{x}}\\ &=\frac{\left (2 \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{(3-n) \sqrt{x}}\\ &=\frac{2 \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{\sqrt{a} (3-n) \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.153705, size = 89, normalized size = 1.68 \[ -\frac{2 \sqrt{b} \sqrt{c x} x^{\frac{n-1}{2}} \sqrt{\frac{a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{3}{2}-\frac{n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} (n-3) \sqrt{a x^3+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x]/Sqrt[a*x^3 + b*x^n],x]

[Out]

(-2*Sqrt[b]*x^((-1 + n)/2)*Sqrt[c*x]*Sqrt[1 + (a*x^(3 - n))/b]*ArcSinh[(Sqrt[a]*x^(3/2 - n/2))/Sqrt[b]])/(Sqrt
[a]*(-3 + n)*Sqrt[a*x^3 + b*x^n])

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Maple [F]  time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{cx}{\frac{1}{\sqrt{a{x}^{3}+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x)

[Out]

int((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x}}{\sqrt{a x^{3} + b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x)/sqrt(a*x^3 + b*x^n), 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((c*x)^(1/2)/(a*x^3+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{\sqrt{c x}}{\sqrt{a x^{3} + b x^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(1/2)/(a*x**3+b*x**n)**(1/2),x)

[Out]

Integral(sqrt(c*x)/sqrt(a*x**3 + b*x**n), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x}}{\sqrt{a x^{3} + b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x)/sqrt(a*x^3 + b*x^n), x)