Integrand size = 15, antiderivative size = 64 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=-\frac {1}{2} b \sqrt {a+\frac {b}{x^4}}+\frac {1}{4} a \sqrt {a+\frac {b}{x^4}} x^4+\frac {3}{4} \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \] Output:
-1/2*b*(a+b/x^4)^(1/2)+1/4*a*(a+b/x^4)^(1/2)*x^4+3/4*a^(1/2)*b*arctanh((a+ b/x^4)^(1/2)/a^(1/2))
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.27 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {\sqrt {a+\frac {b}{x^4}} \left (\left (-2 b+a x^4\right ) \sqrt {b+a x^4}+3 \sqrt {a} b x^2 \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )\right )}{4 \sqrt {b+a x^4}} \] Input:
Integrate[(a + b/x^4)^(3/2)*x^3,x]
Output:
(Sqrt[a + b/x^4]*((-2*b + a*x^4)*Sqrt[b + a*x^4] + 3*Sqrt[a]*b*x^2*Log[Sqr t[a]*x^2 + Sqrt[b + a*x^4]]))/(4*Sqrt[b + a*x^4])
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 51, 60, 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.
\(\displaystyle \int x^3 \left (a+\frac {b}{x^4}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{4} \int \left (a+\frac {b}{x^4}\right )^{3/2} x^8d\frac {1}{x^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{2} b \int \sqrt {a+\frac {b}{x^4}} x^4d\frac {1}{x^4}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{2} b \left (a \int \frac {x^4}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x^4}+2 \sqrt {a+\frac {b}{x^4}}\right )\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {1}{b x^8}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^4}}}{b}+2 \sqrt {a+\frac {b}{x^4}}\right )\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (x^4 \left (a+\frac {b}{x^4}\right )^{3/2}-\frac {3}{2} b \left (2 \sqrt {a+\frac {b}{x^4}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )\right )\right )\) |
Input:
Int[(a + b/x^4)^(3/2)*x^3,x]
Output:
((a + b/x^4)^(3/2)*x^4 - (3*b*(2*Sqrt[a + b/x^4] - 2*Sqrt[a]*ArcTanh[Sqrt[ a + b/x^4]/Sqrt[a]]))/2)/4
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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_)^(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 = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {\left (a \,x^{4}-2 b \right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{4}+\frac {3 \sqrt {a}\, b \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{4 \sqrt {a \,x^{4}+b}}\) | \(75\) |
default | \(\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} x^{4} \left (a \,x^{4} \sqrt {a \,x^{4}+b}+3 \sqrt {a}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right ) b \,x^{2}-2 b \sqrt {a \,x^{4}+b}\right )}{4 \left (a \,x^{4}+b \right )^{\frac {3}{2}}}\) | \(82\) |
Input:
int((a+b/x^4)^(3/2)*x^3,x,method=_RETURNVERBOSE)
Output:
1/4*(a*x^4-2*b)*((a*x^4+b)/x^4)^(1/2)+3/4*a^(1/2)*b*ln(a^(1/2)*x^2+(a*x^4+ b)^(1/2))*((a*x^4+b)/x^4)^(1/2)*x^2/(a*x^4+b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.88 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\left [\frac {3}{8} \, \sqrt {a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) + \frac {1}{4} \, {\left (a x^{4} - 2 \, b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}, -\frac {3}{4} \, \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a}\right ) + \frac {1}{4} \, {\left (a x^{4} - 2 \, b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}\right ] \] Input:
integrate((a+b/x^4)^(3/2)*x^3,x, algorithm="fricas")
Output:
[3/8*sqrt(a)*b*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) + 1 /4*(a*x^4 - 2*b)*sqrt((a*x^4 + b)/x^4), -3/4*sqrt(-a)*b*arctan(sqrt(-a)*sq rt((a*x^4 + b)/x^4)/a) + 1/4*(a*x^4 - 2*b)*sqrt((a*x^4 + b)/x^4)]
Time = 1.79 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.48 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {3 \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4} + \frac {a^{2} x^{6}}{4 \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {a \sqrt {b} x^{2}}{4 \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {b^{\frac {3}{2}}}{2 x^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \] Input:
integrate((a+b/x**4)**(3/2)*x**3,x)
Output:
3*sqrt(a)*b*asinh(sqrt(a)*x**2/sqrt(b))/4 + a**2*x**6/(4*sqrt(b)*sqrt(a*x* *4/b + 1)) - a*sqrt(b)*x**2/(4*sqrt(a*x**4/b + 1)) - b**(3/2)/(2*x**2*sqrt (a*x**4/b + 1))
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {1}{4} \, \sqrt {a + \frac {b}{x^{4}}} a x^{4} - \frac {3}{8} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} b \] Input:
integrate((a+b/x^4)^(3/2)*x^3,x, algorithm="maxima")
Output:
1/4*sqrt(a + b/x^4)*a*x^4 - 3/8*sqrt(a)*b*log((sqrt(a + b/x^4) - sqrt(a))/ (sqrt(a + b/x^4) + sqrt(a))) - 1/2*sqrt(a + b/x^4)*b
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {1}{4} \, \sqrt {a x^{4} + b} a x^{2} - \frac {3}{8} \, \sqrt {a} b \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {\sqrt {a} b^{2}}{{\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b} \] Input:
integrate((a+b/x^4)^(3/2)*x^3,x, algorithm="giac")
Output:
1/4*sqrt(a*x^4 + b)*a*x^2 - 3/8*sqrt(a)*b*log((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2) + sqrt(a)*b^2/((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2 - b)
Time = 1.00 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {a\,x^4\,\sqrt {a+\frac {b}{x^4}}}{4}-\frac {b\,\sqrt {a+\frac {b}{x^4}}}{2}+\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4} \] Input:
int(x^3*(a + b/x^4)^(3/2),x)
Output:
(a*x^4*(a + b/x^4)^(1/2))/4 - (b*(a + b/x^4)^(1/2))/2 + (3*a^(1/2)*b*atanh ((a + b/x^4)^(1/2)/a^(1/2)))/4
Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.22 \[ \int \left (a+\frac {b}{x^4}\right )^{3/2} x^3 \, dx=\frac {48 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a b \,x^{6}+12 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) b^{2} x^{2}+16 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a^{2} x^{10}-56 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, a b \,x^{6}-33 \sqrt {a}\, \sqrt {a \,x^{4}+b}\, b^{2} x^{2}+48 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a^{2} b \,x^{8}+36 \,\mathrm {log}\left (\frac {\sqrt {a \,x^{4}+b}+\sqrt {a}\, x^{2}}{\sqrt {b}}\right ) a \,b^{2} x^{4}+16 a^{3} x^{12}-48 a^{2} b \,x^{8}-63 a \,b^{2} x^{4}-8 b^{3}}{16 x^{2} \left (4 \sqrt {a \,x^{4}+b}\, a \,x^{4}+\sqrt {a \,x^{4}+b}\, b +4 \sqrt {a}\, a \,x^{6}+3 \sqrt {a}\, b \,x^{2}\right )} \] Input:
int((a+b/x^4)^(3/2)*x^3,x)
Output:
(48*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b) )*a*b*x**6 + 12*sqrt(a)*sqrt(a*x**4 + b)*log((sqrt(a*x**4 + b) + sqrt(a)*x **2)/sqrt(b))*b**2*x**2 + 16*sqrt(a)*sqrt(a*x**4 + b)*a**2*x**10 - 56*sqrt (a)*sqrt(a*x**4 + b)*a*b*x**6 - 33*sqrt(a)*sqrt(a*x**4 + b)*b**2*x**2 + 48 *log((sqrt(a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a**2*b*x**8 + 36*log((sqrt (a*x**4 + b) + sqrt(a)*x**2)/sqrt(b))*a*b**2*x**4 + 16*a**3*x**12 - 48*a** 2*b*x**8 - 63*a*b**2*x**4 - 8*b**3)/(16*x**2*(4*sqrt(a*x**4 + b)*a*x**4 + sqrt(a*x**4 + b)*b + 4*sqrt(a)*a*x**6 + 3*sqrt(a)*b*x**2))