Integrand size = 15, antiderivative size = 132 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}} \]
-1/3/a/x^3/(b*x^2+a)^(7/2)+10/3*b/a^2/x/(b*x^2+a)^(7/2)+80/21*b^2*x/a^3/(b *x^2+a)^(7/2)+32/7*b^2*x/a^4/(b*x^2+a)^(5/2)+128/21*b^2*x/a^5/(b*x^2+a)^(3 /2)+256/21*b^2*x/a^6/(b*x^2+a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {-7 a^5+70 a^4 b x^2+560 a^3 b^2 x^4+1120 a^2 b^3 x^6+896 a b^4 x^8+256 b^5 x^{10}}{21 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]
(-7*a^5 + 70*a^4*b*x^2 + 560*a^3*b^2*x^4 + 1120*a^2*b^3*x^6 + 896*a*b^4*x^ 8 + 256*b^5*x^10)/(21*a^6*x^3*(a + b*x^2)^(7/2))
Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {245, 245, 209, 209, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {10 b \int \frac {1}{x^2 \left (b x^2+a\right )^{9/2}}dx}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {10 b \left (-\frac {8 b \int \frac {1}{\left (b x^2+a\right )^{9/2}}dx}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (\frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -\frac {10 b \left (-\frac {8 b \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\right )}{a}-\frac {1}{a x \left (a+b x^2\right )^{7/2}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}\) |
-1/3*1/(a*x^3*(a + b*x^2)^(7/2)) - (10*b*(-(1/(a*x*(a + b*x^2)^(7/2))) - ( 8*b*(x/(7*a*(a + b*x^2)^(7/2)) + (6*(x/(5*a*(a + b*x^2)^(5/2)) + (4*(x/(3* a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/(5*a)))/(7*a)))/a)) /(3*a)
3.6.31.3.1 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]
Time = 1.96 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.53
method | result | size |
pseudoelliptic | \(-\frac {-\frac {256}{7} b^{5} x^{10}-128 a \,b^{4} x^{8}-160 a^{2} b^{3} x^{6}-80 a^{3} b^{2} x^{4}-10 a^{4} b \,x^{2}+a^{5}}{3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{3} a^{6}}\) | \(70\) |
gosper | \(-\frac {-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}}{21 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(72\) |
trager | \(-\frac {-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}}{21 x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6}}\) | \(72\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-14 b \,x^{2}+a \right )}{3 a^{6} x^{3}}+\frac {\sqrt {b \,x^{2}+a}\, x \left (158 b^{3} x^{6}+511 a \,b^{2} x^{4}+560 a^{2} b \,x^{2}+210 a^{3}\right ) b^{2}}{21 a^{6} \left (x^{8} b^{4}+4 a \,b^{3} x^{6}+6 a^{2} x^{4} b^{2}+4 a^{3} b \,x^{2}+a^{4}\right )}\) | \(119\) |
default | \(-\frac {1}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {10 b \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )}{3 a}\) | \(122\) |
-1/3/(b*x^2+a)^(7/2)*(-256/7*b^5*x^10-128*a*b^4*x^8-160*a^2*b^3*x^6-80*a^3 *b^2*x^4-10*a^4*b*x^2+a^5)/x^3/a^6
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (256 \, b^{5} x^{10} + 896 \, a b^{4} x^{8} + 1120 \, a^{2} b^{3} x^{6} + 560 \, a^{3} b^{2} x^{4} + 70 \, a^{4} b x^{2} - 7 \, a^{5}\right )} \sqrt {b x^{2} + a}}{21 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \]
1/21*(256*b^5*x^10 + 896*a*b^4*x^8 + 1120*a^2*b^3*x^6 + 560*a^3*b^2*x^4 + 70*a^4*b*x^2 - 7*a^5)*sqrt(b*x^2 + a)/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^ 8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (126) = 252\).
Time = 1.91 (sec) , antiderivative size = 668, normalized size of antiderivative = 5.06 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=- \frac {7 a^{6} b^{\frac {51}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {63 a^{5} b^{\frac {53}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {630 a^{4} b^{\frac {55}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1680 a^{3} b^{\frac {57}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {2016 a^{2} b^{\frac {59}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1152 a b^{\frac {61}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {256 b^{\frac {63}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} \]
-7*a**6*b**(51/2)*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b* *26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**1 0 + 21*a**6*b**30*x**12) + 63*a**5*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(21 *a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8* b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 630*a**4*b**(55 /2)*x**4*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a* *6*b**30*x**12) + 1680*a**3*b**(57/2)*x**6*sqrt(a/(b*x**2) + 1)/(21*a**11* b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x **8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 2016*a**2*b**(59/2)*x* *8*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210* a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b** 30*x**12) + 1152*a*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x* *2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 10 5*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 256*b**(63/2)*x**12*sqrt(a/(b* x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x* *6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12)
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {256 \, b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} + \frac {10 \, b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {1}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} \]
256/21*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*b^2*x/((b*x^2 + a)^(5/2)*a^4) + 80/21*b^2*x/((b*x^2 + a)^(7/2)*a^3) + 10/3*b/((b*x^2 + a)^(7/2)*a^2*x) - 1/3/((b*x^2 + a)^(7/2)*a*x^3)
Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {158 \, b^{5} x^{2}}{a^{6}} + \frac {511 \, b^{4}}{a^{5}}\right )} + \frac {560 \, b^{3}}{a^{4}}\right )} x^{2} + \frac {210 \, b^{2}}{a^{3}}\right )} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} + 7 \, a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]
1/21*((x^2*(158*b^5*x^2/a^6 + 511*b^4/a^5) + 560*b^3/a^4)*x^2 + 210*b^2/a^ 3)*x/(b*x^2 + a)^(7/2) - 4/3*(6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) + 7*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^5)
Time = 5.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {128\,b}{21\,a^5}+\frac {256\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {1}{3\,a^2}+\frac {19\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {32\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}} \]