Integrand size = 15, antiderivative size = 88 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {5 b}{12 a^2 \left (a+\frac {b}{x^4}\right )^{3/2}}+\frac {5 b}{4 a^3 \sqrt {a+\frac {b}{x^4}}}+\frac {x^4}{4 a \left (a+\frac {b}{x^4}\right )^{3/2}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{7/2}} \]
5/12*b/a^2/(a+b/x^4)^(3/2)+1/4*x^4/a/(a+b/x^4)^(3/2)-5/4*b*arctanh((a+b/x^ 4)^(1/2)/a^(1/2))/a^(7/2)+5/4*b/a^3/(a+b/x^4)^(1/2)
Time = 0.50 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {\sqrt {a} x^2 \left (15 b^2+20 a b x^4+3 a^2 x^8\right )-15 b \left (b+a x^4\right )^{3/2} \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{12 a^{7/2} \sqrt {a+\frac {b}{x^4}} x^2 \left (b+a x^4\right )} \]
(Sqrt[a]*x^2*(15*b^2 + 20*a*b*x^4 + 3*a^2*x^8) - 15*b*(b + a*x^4)^(3/2)*Lo g[Sqrt[a]*x^2 + Sqrt[b + a*x^4]])/(12*a^(7/2)*Sqrt[a + b/x^4]*x^2*(b + a*x ^4))
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 52, 61, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\frac {1}{4} \int \frac {x^8}{\left (a+\frac {b}{x^4}\right )^{5/2}}d\frac {1}{x^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{4} \left (\frac {5 b \int \frac {x^4}{\left (a+\frac {b}{x^4}\right )^{5/2}}d\frac {1}{x^4}}{2 a}+\frac {x^4}{a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{4} \left (\frac {5 b \left (\frac {\int \frac {x^4}{\left (a+\frac {b}{x^4}\right )^{3/2}}d\frac {1}{x^4}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )}{2 a}+\frac {x^4}{a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{4} \left (\frac {5 b \left (\frac {\frac {\int \frac {x^4}{\sqrt {a+\frac {b}{x^4}}}d\frac {1}{x^4}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x^4}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )}{2 a}+\frac {x^4}{a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {1}{b x^8}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^4}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x^4}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )}{2 a}+\frac {x^4}{a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+\frac {b}{x^4}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )}{2 a}+\frac {x^4}{a \left (a+\frac {b}{x^4}\right )^{3/2}}\right )\) |
(x^4/(a*(a + b/x^4)^(3/2)) + (5*b*(2/(3*a*(a + b/x^4)^(3/2)) + (2/(a*Sqrt[ a + b/x^4]) - (2*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/a^(3/2))/a))/(2*a))/4
3.21.98.3.1 Defintions of rubi rules used
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_)^(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]]
Leaf count of result is larger than twice the leaf count of optimal. \(139\) vs. \(2(68)=136\).
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(\frac {a \,x^{4}+b}{4 a^{3} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}-\frac {b \left (\frac {5 \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a}}+\frac {\sqrt {a \,x^{4}+b}\, x^{2} \left (2 a \,x^{4}+3 b \right )}{3 a^{2} x^{8}+6 a b \,x^{4}+3 b^{2}}-\frac {3 x^{2}}{\sqrt {a \,x^{4}+b}}\right ) \sqrt {a \,x^{4}+b}}{2 a^{3} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}\) | \(140\) |
| default | \(\frac {\left (a \,x^{4}+b \right )^{\frac {5}{2}} \left (3 \sqrt {a \,x^{4}+b}\, a^{\frac {15}{2}} x^{10}+6 a^{\frac {13}{2}} b \sqrt {a \,x^{4}+b}\, x^{6}+14 a^{\frac {13}{2}} \sqrt {-\frac {\left (a \,x^{2}+\sqrt {-a b}\right ) \left (-a \,x^{2}+\sqrt {-a b}\right )}{a}}\, b \,x^{6}-15 \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) a^{7} b \,x^{8}+3 a^{\frac {11}{2}} b^{2} \sqrt {a \,x^{4}+b}\, x^{2}+12 a^{\frac {11}{2}} b^{2} \sqrt {-\frac {\left (a \,x^{2}+\sqrt {-a b}\right ) \left (-a \,x^{2}+\sqrt {-a b}\right )}{a}}\, x^{2}-30 a^{6} b^{2} \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) x^{4}-15 a^{5} b^{3} \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right )\right )}{12 a^{\frac {13}{2}} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} x^{10} \left (a \,x^{2}+\sqrt {-a b}\right )^{2} \left (-a \,x^{2}+\sqrt {-a b}\right )^{2}}\) | \(282\) |
1/4/a^3*(a*x^4+b)/((a*x^4+b)/x^4)^(1/2)-1/2/a^3*b*(5/2*ln(x^2*a^(1/2)+(a*x ^4+b)^(1/2))/a^(1/2)+1/3*(a*x^4+b)^(1/2)*x^2*(2*a*x^4+3*b)/(a^2*x^8+2*a*b* x^4+b^2)-3*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 = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.95 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} b x^{8} + 2 \, a b^{2} x^{4} + b^{3}\right )} \sqrt {a} \log \left (-2 \, a x^{4} + 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) + 2 \, {\left (3 \, a^{3} x^{12} + 20 \, a^{2} b x^{8} + 15 \, a b^{2} x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{24 \, {\left (a^{6} x^{8} + 2 \, a^{5} b x^{4} + a^{4} b^{2}\right )}}, \frac {15 \, {\left (a^{2} b x^{8} + 2 \, a b^{2} x^{4} + b^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) + {\left (3 \, a^{3} x^{12} + 20 \, a^{2} b x^{8} + 15 \, a b^{2} x^{4}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, {\left (a^{6} x^{8} + 2 \, a^{5} b x^{4} + a^{4} b^{2}\right )}}\right ] \]
[1/24*(15*(a^2*b*x^8 + 2*a*b^2*x^4 + b^3)*sqrt(a)*log(-2*a*x^4 + 2*sqrt(a) *x^4*sqrt((a*x^4 + b)/x^4) - b) + 2*(3*a^3*x^12 + 20*a^2*b*x^8 + 15*a*b^2* x^4)*sqrt((a*x^4 + b)/x^4))/(a^6*x^8 + 2*a^5*b*x^4 + a^4*b^2), 1/12*(15*(a ^2*b*x^8 + 2*a*b^2*x^4 + b^3)*sqrt(-a)*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b )/x^4)/(a*x^4 + b)) + (3*a^3*x^12 + 20*a^2*b*x^8 + 15*a*b^2*x^4)*sqrt((a*x ^4 + b)/x^4))/(a^6*x^8 + 2*a^5*b*x^4 + a^4*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (80) = 160\).
Time = 2.78 (sec) , antiderivative size = 819, normalized size of antiderivative = 9.31 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {6 a^{17} x^{16} \sqrt {1 + \frac {b}{a x^{4}}}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {46 a^{16} b x^{12} \sqrt {1 + \frac {b}{a x^{4}}}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{16} b x^{12} \log {\left (\frac {b}{a x^{4}} \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{16} b x^{12} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {70 a^{15} b^{2} x^{8} \sqrt {1 + \frac {b}{a x^{4}}}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{15} b^{2} x^{8} \log {\left (\frac {b}{a x^{4}} \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{15} b^{2} x^{8} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {30 a^{14} b^{3} x^{4} \sqrt {1 + \frac {b}{a x^{4}}}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {45 a^{14} b^{3} x^{4} \log {\left (\frac {b}{a x^{4}} \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} - \frac {90 a^{14} b^{3} x^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} + \frac {15 a^{13} b^{4} \log {\left (\frac {b}{a x^{4}} \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} - \frac {30 a^{13} b^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{24 a^{\frac {39}{2}} x^{12} + 72 a^{\frac {37}{2}} b x^{8} + 72 a^{\frac {35}{2}} b^{2} x^{4} + 24 a^{\frac {33}{2}} b^{3}} \]
6*a**17*x**16*sqrt(1 + b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x* *8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 46*a**16*b*x**12*sqrt(1 + b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b* *2*x**4 + 24*a**(33/2)*b**3) + 15*a**16*b*x**12*log(b/(a*x**4))/(24*a**(39 /2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b* *3) - 30*a**16*b*x**12*log(sqrt(1 + b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 70*a* *15*b**2*x**8*sqrt(1 + b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x* *8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 45*a**15*b**2*x**8*log( b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2* x**4 + 24*a**(33/2)*b**3) - 90*a**15*b**2*x**8*log(sqrt(1 + b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24 *a**(33/2)*b**3) + 30*a**14*b**3*x**4*sqrt(1 + b/(a*x**4))/(24*a**(39/2)*x **12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 45*a**14*b**3*x**4*log(b/(a*x**4))/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x **8 + 72*a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) - 90*a**14*b**3*x**4*log (sqrt(1 + b/(a*x**4)) + 1)/(24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72* a**(35/2)*b**2*x**4 + 24*a**(33/2)*b**3) + 15*a**13*b**4*log(b/(a*x**4))/( 24*a**(39/2)*x**12 + 72*a**(37/2)*b*x**8 + 72*a**(35/2)*b**2*x**4 + 24*a** (33/2)*b**3) - 30*a**13*b**4*log(sqrt(1 + b/(a*x**4)) + 1)/(24*a**(39/2...
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.15 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {15 \, {\left (a + \frac {b}{x^{4}}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x^{4}}\right )} a b - 2 \, a^{2} b}{12 \, {\left ({\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}} a^{4}\right )}} + \frac {5 \, b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{8 \, a^{\frac {7}{2}}} \]
1/12*(15*(a + b/x^4)^2*b - 10*(a + b/x^4)*a*b - 2*a^2*b)/((a + b/x^4)^(5/2 )*a^3 - (a + b/x^4)^(3/2)*a^4) + 5/8*b*log((sqrt(a + b/x^4) - sqrt(a))/(sq rt(a + b/x^4) + sqrt(a)))/a^(7/2)
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {3 \, x^{4}}{a} + \frac {20 \, b}{a^{2}}\right )} x^{4} + \frac {15 \, b^{2}}{a^{3}}\right )} x^{2}}{12 \, {\left (a x^{4} + b\right )}^{\frac {3}{2}}} + \frac {5 \, b \log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{4 \, a^{\frac {7}{2}}} \]
1/12*((3*x^4/a + 20*b/a^2)*x^4 + 15*b^2/a^3)*x^2/(a*x^4 + b)^(3/2) + 5/4*b *log(abs(-sqrt(a)*x^2 + sqrt(a*x^4 + b)))/a^(7/2)
Time = 6.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{\left (a+\frac {b}{x^4}\right )^{5/2}} \, dx=\frac {5\,b}{3\,a^2\,{\left (a+\frac {b}{x^4}\right )}^{3/2}}+\frac {x^4}{4\,a\,{\left (a+\frac {b}{x^4}\right )}^{3/2}}-\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4\,a^{7/2}}+\frac {5\,b^2}{4\,a^3\,x^4\,{\left (a+\frac {b}{x^4}\right )}^{3/2}} \]