\(\int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx\) [367]

Optimal result

Integrand size = 22, antiderivative size = 71 \[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x}+\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)} \]

[Out]

2*arctanh(x*a^(1/2)/(a*x^2+b*x^n)^(1/2))*a^(1/2)/c^2/(2-n)-2*(a*x^2+b*x^n)^(1/2)/c^2/(2-n)/x

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2053, 2033, 212} \[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)}-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x} \]

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2033

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 2053

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])

Rubi steps \begin{align*} \text {integral}& = \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) \text {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] (verified)

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\frac {2 \left (a x^2+b x^n-\sqrt {a} \sqrt {b} x^{1+\frac {n}{2}} \sqrt {1+\frac {a x^{2-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{1-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{c^2 (-2+n) x \sqrt {a x^2+b x^n}} \]

[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])

Maple [F]

\[\int \frac {\sqrt {a \,x^{2}+b \,x^{n}}}{c^{2} x^{2}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\frac {\int \frac {\sqrt {a x^{2} + b x^{n}}}{x^{2}}\, dx}{c^{2}} \]

[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

Maxima [F]

\[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\int { \frac {\sqrt {a x^{2} + b x^{n}}}{c^{2} x^{2}} \,d x } \]

[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

Giac [F]

\[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\int { \frac {\sqrt {a x^{2} + b x^{n}}}{c^{2} x^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx=\int \frac {\sqrt {b\,x^n+a\,x^2}}{c^2\,x^2} \,d x \]

[In]

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

[Out]

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