Integrand size = 21, antiderivative size = 71 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3} \] Output:
-1/3*(a+b*(c*x^2)^(3/2))^(1/2)/x^3-1/3*b*(c*x^2)^(3/2)*arctanh((a+b*(c*x^2 )^(3/2))^(1/2)/a^(1/2))/a^(1/2)/x^3
Time = 1.64 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\frac {-a-b \left (c x^2\right )^{3/2}-b \left (c x^2\right )^{3/2} \sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}} \text {arctanh}\left (\sqrt {1+\frac {b \left (c x^2\right )^{3/2}}{a}}\right )}{3 x^3 \sqrt {a+b \left (c x^2\right )^{3/2}}} \] Input:
Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^4,x]
Output:
(-a - b*(c*x^2)^(3/2) - b*(c*x^2)^(3/2)*Sqrt[1 + (b*(c*x^2)^(3/2))/a]*ArcT anh[Sqrt[1 + (b*(c*x^2)^(3/2))/a]])/(3*x^3*Sqrt[a + b*(c*x^2)^(3/2)])
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {892, 798, 51, 73, 221}
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.
\(\displaystyle \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \frac {\left (c x^2\right )^{3/2} \int \frac {\sqrt {b \left (c x^2\right )^{3/2}+a}}{c^2 x^4}d\sqrt {c x^2}}{x^3}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\left (c x^2\right )^{3/2} \int \frac {\sqrt {b \left (c x^2\right )^{3/2}+a}}{c x^2}d\left (c x^2\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\left (c x^2\right )^{3/2} \left (\frac {1}{2} b \int \frac {1}{\sqrt {c x^2} \sqrt {b \left (c x^2\right )^{3/2}+a}}d\left (c x^2\right )^{3/2}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {c x^2}}\right )}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\left (c x^2\right )^{3/2} \left (\int \frac {1}{\frac {c x^2}{b}-\frac {a}{b}}d\sqrt {b \left (c x^2\right )^{3/2}+a}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {c x^2}}\right )}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (c x^2\right )^{3/2} \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {c x^2}}\right )}{3 x^3}\) |
Input:
Int[Sqrt[a + b*(c*x^2)^(3/2)]/x^4,x]
Output:
((c*x^2)^(3/2)*(-(Sqrt[a + b*(c*x^2)^(3/2)]/Sqrt[c*x^2]) - (b*ArcTanh[Sqrt [a + b*(c*x^2)^(3/2)]/Sqrt[a]])/Sqrt[a]))/(3*x^3)
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(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)]
\[\int \frac {\sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}{x^{4}}d x\]
Input:
int((a+(c*x^2)^(3/2)*b)^(1/2)/x^4,x)
Output:
int((a+(c*x^2)^(3/2)*b)^(1/2)/x^4,x)
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (53) = 106\).
Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.90 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\left [\frac {b c x^{3} \sqrt {\frac {c}{a}} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{6 \, x^{3}}, -\frac {b c x^{3} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c^{2} x^{4} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{b^{2} c^{4} x^{7} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{3 \, x^{3}}\right ] \] Input:
integrate((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x, algorithm="fricas")
Output:
[1/6*(b*c*x^3*sqrt(c/a)*log((b*c^2*x^4 - 2*sqrt(sqrt(c*x^2)*b*c*x^2 + a)*a *x*sqrt(c/a) + 2*sqrt(c*x^2)*a)/x^4) - 2*sqrt(sqrt(c*x^2)*b*c*x^2 + a))/x^ 3, -1/3*(b*c*x^3*sqrt(-c/a)*arctan(-(a*b*c^2*x^4*sqrt(-c/a) - sqrt(c*x^2)* a^2*sqrt(-c/a))*sqrt(sqrt(c*x^2)*b*c*x^2 + a)/(b^2*c^4*x^7 - a^2*c*x)) + s qrt(sqrt(c*x^2)*b*c*x^2 + a))/x^3]
\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x^{4}}\, dx \] Input:
integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**4,x)
Output:
Integral(sqrt(a + b*(c*x**2)**(3/2))/x**4, x)
\[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int { \frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}}{x^{4}} \,d x } \] Input:
integrate((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x, algorithm="maxima")
Output:
integrate(sqrt((c*x^2)^(3/2)*b + a)/x^4, x)
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\frac {1}{3} \, b c^{\frac {3}{2}} {\left (\frac {\arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{b c^{\frac {3}{2}} x^{3}}\right )} \] Input:
integrate((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x, algorithm="giac")
Output:
1/3*b*c^(3/2)*(arctan(sqrt(b*c^(3/2)*x^3 + a)/sqrt(-a))/sqrt(-a) - sqrt(b* c^(3/2)*x^3 + a)/(b*c^(3/2)*x^3))
Timed out. \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x^4} \,d x \] Input:
int((a + b*(c*x^2)^(3/2))^(1/2)/x^4,x)
Output:
int((a + b*(c*x^2)^(3/2))^(1/2)/x^4, x)
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx=\frac {-2 \sqrt {\sqrt {c}\, b c \,x^{3}+a}\, a +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b c \,x^{3}+a}-\sqrt {a}\right ) b c \,x^{3}-\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b c \,x^{3}+a}+\sqrt {a}\right ) b c \,x^{3}}{6 a \,x^{3}} \] Input:
int((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x)
Output:
( - 2*sqrt(sqrt(c)*b*c*x**3 + a)*a + sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*c* x**3 + a) - sqrt(a))*b*c*x**3 - sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*c*x**3 + a) + sqrt(a))*b*c*x**3)/(6*a*x**3)