Integrand size = 15, antiderivative size = 74 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {3 b \sqrt {a+\frac {b}{x^2}} x^2}{8 a^2}+\frac {\sqrt {a+\frac {b}{x^2}} x^4}{4 a}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Output:
-3/8*b*(a+b/x^2)^(1/2)*x^2/a^2+1/4*(a+b/x^2)^(1/2)*x^4/a+3/8*b^2*arctanh(( a+b/x^2)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.31 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a} x \left (-3 b^2-a b x^2+2 a^2 x^4\right )+6 b^2 \sqrt {b+a x^2} \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{8 a^{5/2} \sqrt {a+\frac {b}{x^2}} x} \] Input:
Integrate[x^3/Sqrt[a + b/x^2],x]
Output:
(Sqrt[a]*x*(-3*b^2 - a*b*x^2 + 2*a^2*x^4) + 6*b^2*Sqrt[b + a*x^2]*ArcTanh[ (Sqrt[a]*x)/(-Sqrt[b] + Sqrt[b + a*x^2])])/(8*a^(5/2)*Sqrt[a + b/x^2]*x)
Time = 0.30 (sec) , antiderivative size = 79, 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 = {798, 52, 52, 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.
\(\displaystyle \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{2} \int \frac {x^6}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}}{4 a}+\frac {x^4 \sqrt {a+\frac {b}{x^2}}}{2 a}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \left (-\frac {b \int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}}{2 a}-\frac {x^2 \sqrt {a+\frac {b}{x^2}}}{a}\right )}{4 a}+\frac {x^4 \sqrt {a+\frac {b}{x^2}}}{2 a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \left (-\frac {\int \frac {1}{\frac {1}{b x^4}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^2}}}{a}-\frac {x^2 \sqrt {a+\frac {b}{x^2}}}{a}\right )}{4 a}+\frac {x^4 \sqrt {a+\frac {b}{x^2}}}{2 a}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^2 \sqrt {a+\frac {b}{x^2}}}{a}\right )}{4 a}+\frac {x^4 \sqrt {a+\frac {b}{x^2}}}{2 a}\right )\) |
Input:
Int[x^3/Sqrt[a + b/x^2],x]
Output:
((Sqrt[a + b/x^2]*x^4)/(2*a) + (3*b*(-((Sqrt[a + b/x^2]*x^2)/a) + (b*ArcTa nh[Sqrt[a + b/x^2]/Sqrt[a]])/a^(3/2)))/(4*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] :> 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]]
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {\left (2 a \,x^{2}-3 b \right ) \left (a \,x^{2}+b \right )}{8 a^{2} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}+\frac {3 b^{2} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) \sqrt {a \,x^{2}+b}}{8 a^{\frac {5}{2}} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\) | \(86\) |
default | \(\frac {\sqrt {a \,x^{2}+b}\, \left (2 x^{3} \sqrt {a \,x^{2}+b}\, a^{\frac {5}{2}}-3 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b}\, b x +3 \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a \,b^{2}\right )}{8 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \,a^{\frac {7}{2}}}\) | \(87\) |
Input:
int(x^3/(a+b/x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/8*(2*a*x^2-3*b)*(a*x^2+b)/a^2/((a*x^2+b)/x^2)^(1/2)+3/8*b^2/a^(5/2)*ln(a ^(1/2)*x+(a*x^2+b)^(1/2))/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.12 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{16 \, a^{3}}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (2 \, a^{2} x^{4} - 3 \, a b x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{8 \, a^{3}}\right ] \] Input:
integrate(x^3/(a+b/x^2)^(1/2),x, algorithm="fricas")
Output:
[1/16*(3*sqrt(a)*b^2*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) + 2*(2*a^2*x^4 - 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2))/a^3, -1/8*(3*sqrt(-a )*b^2*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) - (2*a^2*x^4 - 3*a*b*x^2)*sqrt((a*x^2 + b)/x^2))/a^3]
Time = 3.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.28 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x^{5}}{4 \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {\sqrt {b} x^{3}}{8 a \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {3 b^{\frac {3}{2}} x}{8 a^{2} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{8 a^{\frac {5}{2}}} \] Input:
integrate(x**3/(a+b/x**2)**(1/2),x)
Output:
x**5/(4*sqrt(b)*sqrt(a*x**2/b + 1)) - sqrt(b)*x**3/(8*a*sqrt(a*x**2/b + 1) ) - 3*b**(3/2)*x/(8*a**2*sqrt(a*x**2/b + 1)) + 3*b**2*asinh(sqrt(a)*x/sqrt (b))/(8*a**(5/2))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, a^{\frac {5}{2}}} - \frac {3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {a + \frac {b}{x^{2}}} a b^{2}}{8 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{2} a^{2} - 2 \, {\left (a + \frac {b}{x^{2}}\right )} a^{3} + a^{4}\right )}} \] Input:
integrate(x^3/(a+b/x^2)^(1/2),x, algorithm="maxima")
Output:
-3/16*b^2*log((sqrt(a + b/x^2) - sqrt(a))/(sqrt(a + b/x^2) + sqrt(a)))/a^( 5/2) - 1/8*(3*(a + b/x^2)^(3/2)*b^2 - 5*sqrt(a + b/x^2)*a*b^2)/((a + b/x^2 )^2*a^2 - 2*(a + b/x^2)*a^3 + a^4)
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {1}{8} \, \sqrt {a x^{2} + b} x {\left (\frac {2 \, x^{2}}{a \mathrm {sgn}\left (x\right )} - \frac {3 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {5}{2}}} - \frac {3 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{8 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^3/(a+b/x^2)^(1/2),x, algorithm="giac")
Output:
1/8*sqrt(a*x^2 + b)*x*(2*x^2/(a*sgn(x)) - 3*b/(a^2*sgn(x))) + 3/16*b^2*log (abs(b))*sgn(x)/a^(5/2) - 3/8*b^2*log(abs(-sqrt(a)*x + sqrt(a*x^2 + b)))/( a^(5/2)*sgn(x))
Time = 0.84 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,a^{5/2}}+\frac {5\,x^4\,\sqrt {a+\frac {b}{x^2}}}{8\,a}-\frac {3\,x^4\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}{8\,a^2} \] Input:
int(x^3/(a + b/x^2)^(1/2),x)
Output:
(3*b^2*atanh((a + b/x^2)^(1/2)/a^(1/2)))/(8*a^(5/2)) + (5*x^4*(a + b/x^2)^ (1/2))/(8*a) - (3*x^4*(a + b/x^2)^(3/2))/(8*a^2)
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {2 \sqrt {a \,x^{2}+b}\, a^{2} x^{3}-3 \sqrt {a \,x^{2}+b}\, a b x +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) b^{2}}{8 a^{3}} \] Input:
int(x^3/(a+b/x^2)^(1/2),x)
Output:
(2*sqrt(a*x**2 + b)*a**2*x**3 - 3*sqrt(a*x**2 + b)*a*b*x + 3*sqrt(a)*log(( sqrt(a*x**2 + b) + sqrt(a)*x)/sqrt(b))*b**2)/(8*a**3)