\(\int \frac {\sqrt {a+b (c x^2)^{3/2}}}{x} \, dx\) [2945]

Optimal result

Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \]

[Out]

-2/3*arctanh((a+b*(c*x^2)^(3/2))^(1/2)/a^(1/2))*a^(1/2)+2/3*(a+b*(c*x^2)^(3/2))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 272, 52, 65, 214} \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \]

[In]

Int[Sqrt[a + b*(c*x^2)^(3/2)]/x,x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x} \, dx,x,\sqrt {c x^2}\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\left (c x^2\right )^{3/2}\right ) \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right ) \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{3 b} \\ & = \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2}{3} \left (\sqrt {a+b \left (c x^2\right )^{3/2}}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )\right ) \]

[In]

Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x,x]

[Out]

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

Maple [F]

\[\int \frac {\sqrt {a +b \left (c \,x^{2}\right )^{\frac {3}{2}}}}{x}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\left [\frac {1}{3} \, \sqrt {a} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {c x^{2}} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}, \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right ] \]

[In]

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

[Out]

[1/3*sqrt(a)*log((b*c^2*x^4 - 2*sqrt(sqrt(c*x^2)*b*c*x^2 + a)*sqrt(c*x^2)*sqrt(a) + 2*sqrt(c*x^2)*a)/x^4) + 2/
3*sqrt(sqrt(c*x^2)*b*c*x^2 + a), 2/3*sqrt(-a)*arctan(sqrt(sqrt(c*x^2)*b*c*x^2 + a)*sqrt(-a)/a) + 2/3*sqrt(sqrt
(c*x^2)*b*c*x^2 + a)]

Sympy [F]

\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x}\, dx \]

[In]

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

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2))/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {1}{3} \, \sqrt {a} \log \left (\frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \]

[In]

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

[Out]

1/3*sqrt(a)*log((sqrt((c*x^2)^(3/2)*b + a) - sqrt(a))/(sqrt((c*x^2)^(3/2)*b + a) + sqrt(a))) + 2/3*sqrt((c*x^2
)^(3/2)*b + a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\frac {2 \, a \arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2}{3} \, \sqrt {b c^{\frac {3}{2}} x^{3} + a} \]

[In]

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

[Out]

2/3*a*arctan(sqrt(b*c^(3/2)*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/3*sqrt(b*c^(3/2)*x^3 + a)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x} \,d x \]

[In]

int((a + b*(c*x^2)^(3/2))^(1/2)/x,x)

[Out]

int((a + b*(c*x^2)^(3/2))^(1/2)/x, x)