Integrand size = 15, antiderivative size = 69 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 b}{2 a^2 \sqrt {a+b x^2}}-\frac {1}{2 a x^2 \sqrt {a+b x^2}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Output:
-3/2*b/a^2/(b*x^2+a)^(1/2)-1/2/a/x^2/(b*x^2+a)^(1/2)+3/2*b*arctanh((b*x^2+ a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a-3 b x^2}{2 a^2 x^2 \sqrt {a+b x^2}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}} \] Input:
Integrate[1/(x^3*(a + b*x^2)^(3/2)),x]
Output:
(-a - 3*b*x^2)/(2*a^2*x^2*Sqrt[a + b*x^2]) + (3*b*ArcTanh[Sqrt[a + b*x^2]/ Sqrt[a]])/(2*a^(5/2))
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {243, 52, 61, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^{3/2}}dx^2\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 b \int \frac {1}{x^2 \left (b x^2+a\right )^{3/2}}dx^2}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 b \left (\frac {\int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{a}+\frac {2}{a \sqrt {a+b x^2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 b \left (\frac {2 \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a b}+\frac {2}{a \sqrt {a+b x^2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-\frac {3 b \left (\frac {2}{a \sqrt {a+b x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {1}{a x^2 \sqrt {a+b x^2}}\right )\) |
Input:
Int[1/(x^3*(a + b*x^2)^(3/2)),x]
Output:
(-(1/(a*x^2*Sqrt[a + b*x^2])) - (3*b*(2/(a*Sqrt[a + b*x^2]) - (2*ArcTanh[S qrt[a + b*x^2]/Sqrt[a]])/a^(3/2)))/(2*a))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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_)^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]
Time = 0.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 x^{2} b}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {1}{\sqrt {b \,x^{2}+a}}\right )}{a^{2}}\) | \(54\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}}{2 a^{2} x^{2}}-\frac {b}{a^{2} \sqrt {b \,x^{2}+a}}+\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {5}{2}}}\) | \(63\) |
default | \(-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\) | \(67\) |
Input:
int(1/x^3/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
b/a^2*(-1/2*(b*x^2+a)^(1/2)/x^2/b+3/2*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^( 1/2)-1/(b*x^2+a)^(1/2))
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.52 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (3 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {3 \, {\left (b^{2} x^{4} + a b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \] Input:
integrate(1/x^3/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/4*(3*(b^2*x^4 + a*b*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a ) + 2*a)/x^2) - 2*(3*a*b*x^2 + a^2)*sqrt(b*x^2 + a))/(a^3*b*x^4 + a^4*x^2) , -1/2*(3*(b^2*x^4 + a*b*x^2)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (3*a*b*x^2 + a^2)*sqrt(b*x^2 + a))/(a^3*b*x^4 + a^4*x^2)]
Time = 1.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}} \] Input:
integrate(1/x**3/(b*x**2+a)**(3/2),x)
Output:
-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a/(b *x**2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(5/2))
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {3 \, b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {1}{2 \, \sqrt {b x^{2} + a} a x^{2}} \] Input:
integrate(1/x^3/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
3/2*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 3/2*b/(sqrt(b*x^2 + a)*a^2) - 1/2/(sqrt(b*x^2 + a)*a*x^2)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (b x^{2} + a\right )} b - 2 \, a b}{2 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{2} + a} a\right )} a^{2}} \] Input:
integrate(1/x^3/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
-3/2*b*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) - 1/2*(3*(b*x^2 + a )*b - 2*a*b)/(((b*x^2 + a)^(3/2) - sqrt(b*x^2 + a)*a)*a^2)
Time = 0.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {1}{2\,a\,x^2\,\sqrt {b\,x^2+a}}-\frac {3\,b}{2\,a^2\,\sqrt {b\,x^2+a}} \] Input:
int(1/(x^3*(a + b*x^2)^(3/2)),x)
Output:
(3*b*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(5/2)) - 1/(2*a*x^2*(a + b*x^2 )^(1/2)) - (3*b)/(2*a^2*(a + b*x^2)^(1/2))
Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2}-3 \sqrt {b \,x^{2}+a}\, a b \,x^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,x^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{4}}{2 a^{3} x^{2} \left (b \,x^{2}+a \right )} \] Input:
int(1/x^3/(b*x^2+a)^(3/2),x)
Output:
( - sqrt(a + b*x**2)*a**2 - 3*sqrt(a + b*x**2)*a*b*x**2 - 3*sqrt(a)*log((s qrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*x**2 - 3*sqrt(a)*log(( sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4 + 3*sqrt(a)*log ((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*x**2 + 3*sqrt(a)*lo g((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**4)/(2*a**3*x** 2*(a + b*x**2))