Integrand size = 21, antiderivative size = 133 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{2 x^2}+\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \arctan \left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}}-\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \text {arctanh}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
-1/2*(c*(b*x^2+a)^(1/2))^(3/2)/x^2+3/4*b*arctan((1+b*x^2/a)^(1/4))*(c*(b*x ^2+a)^(1/2))^(3/2)/a/(1+b*x^2/a)^(3/4)-3/4*b*arctanh((1+b*x^2/a)^(1/4))*(c *(b*x^2+a)^(1/2))^(3/2)/a/(1+b*x^2/a)^(3/4)
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (2 \sqrt [4]{a} \left (a+b x^2\right )^{3/4}-3 b x^2 \arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )+3 b x^2 \text {arctanh}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right )}{4 \sqrt [4]{a} x^2 \left (a+b x^2\right )^{3/4}} \]
-1/4*((c*Sqrt[a + b*x^2])^(3/2)*(2*a^(1/4)*(a + b*x^2)^(3/4) - 3*b*x^2*Arc Tan[(a + b*x^2)^(1/4)/a^(1/4)] + 3*b*x^2*ArcTanh[(a + b*x^2)^(1/4)/a^(1/4) ]))/(a^(1/4)*x^2*(a + b*x^2)^(3/4))
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2045, 243, 51, 73, 25, 27, 827, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^3}dx}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^4}dx^2}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {3 b \int \frac {1}{x^2 \sqrt [4]{\frac {b x^2}{a}+1}}dx^2}{4 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (3 \int -\frac {b x^4}{a \left (1-x^8\right )}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (-3 \int \frac {b x^4}{a \left (1-x^8\right )}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (-\frac {3 b \int \frac {x^4}{1-x^8}d\sqrt [4]{\frac {b x^2}{a}+1}}{a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (-\frac {3 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{\frac {b x^2}{a}+1}\right )}{a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (-\frac {3 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {1}{2} \arctan \left (\sqrt [4]{\frac {b x^2}{a}+1}\right )\right )}{a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (-\frac {3 b \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{\frac {b x^2}{a}+1}\right )\right )}{a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
((c*Sqrt[a + b*x^2])^(3/2)*(-((1 + (b*x^2)/a)^(3/4)/x^2) - (3*b*(-1/2*ArcT an[(1 + (b*x^2)/a)^(1/4)] + ArcTanh[(1 + (b*x^2)/a)^(1/4)]/2))/a))/(2*(1 + (b*x^2)/a)^(3/4))
3.3.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
\[\int \frac {\left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{3}}d x\]
Timed out. \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=\int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.04 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=\frac {{\left (3 \, c^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} c}}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}} + \frac {\log \left (\frac {\sqrt {\sqrt {b x^{2} + a} c} - \left (a c^{2}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b x^{2} + a} c} + \left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )} - \frac {4 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} c^{4}}{{\left (b x^{2} + a\right )} c^{2} - a c^{2}}\right )} b}{8 \, c^{2}} \]
1/8*(3*c^4*(2*arctan(sqrt(sqrt(b*x^2 + a)*c)/(a*c^2)^(1/4))/(a*c^2)^(1/4) + log((sqrt(sqrt(b*x^2 + a)*c) - (a*c^2)^(1/4))/(sqrt(sqrt(b*x^2 + a)*c) + (a*c^2)^(1/4)))/(a*c^2)^(1/4)) - 4*(sqrt(b*x^2 + a)*c)^(3/2)*c^4/((b*x^2 + a)*c^2 - a*c^2))*b/c^2
Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.59 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=\frac {{\left (\frac {6 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{a} + \frac {3 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {1}{4}}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} b}{x^{2}}\right )} c^{\frac {3}{2}}}{16 \, b} \]
1/16*(6*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^2 + a) ^(1/4))/(-a)^(1/4))/(-a)^(1/4) + 6*sqrt(2)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2 )*(-a)^(1/4) - 2*(b*x^2 + a)^(1/4))/(-a)^(1/4))/(-a)^(1/4) + 3*sqrt(2)*(-a )^(3/4)*b^2*log(sqrt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^2 + a) + s qrt(-a))/a + 3*sqrt(2)*b^2*log(-sqrt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqr t(b*x^2 + a) + sqrt(-a))/(-a)^(1/4) - 8*(b*x^2 + a)^(3/4)*b/x^2)*c^(3/2)/b
Timed out. \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^3} \,d x \]