3.367 \(\int \frac{\sqrt{a x^2+b x^n}}{c^2 x^2} \, dx\)

Optimal. Leaf size=71 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^2 (2-n)}-\frac{2 \sqrt{a x^2+b x^n}}{c^2 (2-n) x} \]

[Out]

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

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Rubi [A]  time = 0.0755826, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 2028, 2008, 206} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^n}}\right )}{c^2 (2-n)}-\frac{2 \sqrt{a x^2+b x^n}}{c^2 (2-n) x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^2 + b*x^n]/(c^2*x^2),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2008

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

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

Mathematica [A]  time = 0.144159, size = 99, normalized size = 1.39 \[ \frac{2 \left (-\sqrt{a} \sqrt{b} x^{\frac{n}{2}+1} \sqrt{\frac{a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{1-\frac{n}{2}}}{\sqrt{b}}\right )+a x^2+b x^n\right )}{c^2 (n-2) x \sqrt{a x^2+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{2}{x}^{2}}\sqrt{a{x}^{2}+b{x}^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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