Integrand size = 22, antiderivative size = 108 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac {2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac {4 b (2 A b-a B) x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {8 b (2 A b-a B) x}{3 a^4 \sqrt {a+b x^2}} \]
-1/3*A/a/x^3/(b*x^2+a)^(3/2)+(2*A*b-B*a)/a^2/x/(b*x^2+a)^(3/2)+4/3*b*(2*A* b-B*a)*x/a^3/(b*x^2+a)^(3/2)+8/3*b*(2*A*b-B*a)*x/a^4/(b*x^2+a)^(1/2)
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=\frac {16 A b^3 x^6+6 a^2 b x^2 \left (A-2 B x^2\right )-8 a b^2 x^4 \left (-3 A+B x^2\right )-a^3 \left (A+3 B x^2\right )}{3 a^4 x^3 \left (a+b x^2\right )^{3/2}} \]
(16*A*b^3*x^6 + 6*a^2*b*x^2*(A - 2*B*x^2) - 8*a*b^2*x^4*(-3*A + B*x^2) - a ^3*(A + 3*B*x^2))/(3*a^4*x^3*(a + b*x^2)^(3/2))
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {359, 245, 209, 208}
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.
\(\displaystyle \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {(2 A b-a B) \int \frac {1}{x^2 \left (b x^2+a\right )^{5/2}}dx}{a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {(2 A b-a B) \left (-\frac {4 b \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {(2 A b-a B) \left (-\frac {4 b \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -\frac {\left (-\frac {4 b \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{3/2}}\right ) (2 A b-a B)}{a}-\frac {A}{3 a x^3 \left (a+b x^2\right )^{3/2}}\) |
-1/3*A/(a*x^3*(a + b*x^2)^(3/2)) - ((2*A*b - a*B)*(-(1/(a*x*(a + b*x^2)^(3 /2))) - (4*b*(x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/ a))/a
3.6.95.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Time = 2.92 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {\left (3 x^{2} B +A \right ) a^{3}-6 b \,x^{2} \left (-2 x^{2} B +A \right ) a^{2}-24 x^{4} b^{2} \left (-\frac {x^{2} B}{3}+A \right ) a -16 x^{6} b^{3} A}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} x^{3} a^{4}}\) | \(74\) |
gosper | \(-\frac {-16 x^{6} b^{3} A +8 x^{6} a \,b^{2} B -24 A a \,b^{2} x^{4}+12 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+3 B \,a^{3} x^{2}+a^{3} A}{3 x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}\) | \(82\) |
trager | \(-\frac {-16 x^{6} b^{3} A +8 x^{6} a \,b^{2} B -24 A a \,b^{2} x^{4}+12 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+3 B \,a^{3} x^{2}+a^{3} A}{3 x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}\) | \(82\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-8 A b \,x^{2}+3 B a \,x^{2}+A a \right )}{3 a^{4} x^{3}}+\frac {\sqrt {b \,x^{2}+a}\, x \left (8 A \,b^{2} x^{2}-5 B a b \,x^{2}+9 a b A -6 a^{2} B \right ) b}{3 a^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(102\) |
default | \(A \left (-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )}{a}\right )+B \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )\) | \(140\) |
-1/3/(b*x^2+a)^(3/2)*((3*B*x^2+A)*a^3-6*b*x^2*(-2*B*x^2+A)*a^2-24*x^4*b^2* (-1/3*x^2*B+A)*a-16*x^6*b^3*A)/x^3/a^4
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{4} + A a^{3} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \]
-1/3*(8*(B*a*b^2 - 2*A*b^3)*x^6 + 12*(B*a^2*b - 2*A*a*b^2)*x^4 + A*a^3 + 3 *(B*a^3 - 2*A*a^2*b)*x^2)*sqrt(b*x^2 + a)/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6 *x^3)
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (99) = 198\).
Time = 10.06 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.85 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {5 a^{3} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {30 a^{2} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {40 a b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac {16 b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) + B \left (- \frac {3 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {12 a b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac {8 b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) \]
A*(-a**4*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x **4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 5*a**3*b**(21/2)*x**2*sqrt( a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 30*a**2*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**7 *b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 40*a*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10* x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 16*b**(27/2)*x**8*sqrt(a/( b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3 *a**4*b**12*x**8)) + B*(-3*a**2*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4) - 12*a*b**(11/2)*x**2*sqrt(a/(b*x* *2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4) - 8*b**(13/2) *x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x **4))
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {8 \, B b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, B b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {16 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A b^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} + \frac {2 \, A b}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}} \]
-8/3*B*b*x/(sqrt(b*x^2 + a)*a^3) - 4/3*B*b*x/((b*x^2 + a)^(3/2)*a^2) + 16/ 3*A*b^2*x/(sqrt(b*x^2 + a)*a^4) + 8/3*A*b^2*x/((b*x^2 + a)^(3/2)*a^3) - B/ ((b*x^2 + a)^(3/2)*a*x) + 2*A*b/((b*x^2 + a)^(3/2)*a^2*x) - 1/3*A/((b*x^2 + a)^(3/2)*a*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (92) = 184\).
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.07 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {x {\left (\frac {{\left (5 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{2}}{a^{7} b} + \frac {3 \, {\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )}}{a^{7} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} + 3 \, B a^{3} \sqrt {b} - 8 \, A a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{3}} \]
-1/3*x*((5*B*a^4*b^3 - 8*A*a^3*b^4)*x^2/(a^7*b) + 3*(2*B*a^5*b^2 - 3*A*a^4 *b^3)/(a^7*b))/(b*x^2 + a)^(3/2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4* B*a*sqrt(b) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b) + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a *b^(3/2) + 3*B*a^3*sqrt(b) - 8*A*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^3)
Time = 5.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {16\,A\,{\left (b\,x^2+a\right )}^3+A\,a^3+B\,a^3\,x^2-24\,A\,a\,{\left (b\,x^2+a\right )}^2+6\,A\,a^2\,\left (b\,x^2+a\right )-8\,B\,a\,x^2\,{\left (b\,x^2+a\right )}^2+4\,B\,a^2\,x^2\,\left (b\,x^2+a\right )}{{\left (b\,x^2+a\right )}^{3/2}\,\left (\frac {3\,a^5\,x}{b}-\frac {3\,a^4\,x\,\left (b\,x^2+a\right )}{b}\right )} \]