Integrand size = 24, antiderivative size = 141 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {a^2}{5 c x^5 \sqrt {c+d x^2}}-\frac {2 a (5 b c-3 a d)}{15 c^2 x^3 \sqrt {c+d x^2}}-\frac {15 b^2 c^2-8 a d (5 b c-3 a d)}{15 c^3 x \sqrt {c+d x^2}}-\frac {2 d \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right ) x}{15 c^4 \sqrt {c+d x^2}} \]
-1/5*a^2/c/x^5/(d*x^2+c)^(1/2)-2/15*a*(-3*a*d+5*b*c)/c^2/x^3/(d*x^2+c)^(1/ 2)+1/15*(-15*b^2*c^2+8*a*d*(-3*a*d+5*b*c))/c^3/x/(d*x^2+c)^(1/2)-2/15*d*(1 5*b^2*c^2-8*a*d*(-3*a*d+5*b*c))*x/c^4/(d*x^2+c)^(1/2)
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-15 b^2 c^2 x^4 \left (c+2 d x^2\right )-10 a b c x^2 \left (c^2-4 c d x^2-8 d^2 x^4\right )-3 a^2 \left (c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right )}{15 c^4 x^5 \sqrt {c+d x^2}} \]
(-15*b^2*c^2*x^4*(c + 2*d*x^2) - 10*a*b*c*x^2*(c^2 - 4*c*d*x^2 - 8*d^2*x^4 ) - 3*a^2*(c^3 - 2*c^2*d*x^2 + 8*c*d^2*x^4 + 16*d^3*x^6))/(15*c^4*x^5*Sqrt [c + d*x^2])
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {365, 359, 245, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 365 |
\(\displaystyle \frac {\int \frac {5 b^2 c x^2+2 a (5 b c-3 a d)}{x^4 \left (d x^2+c\right )^{3/2}}dx}{5 c}-\frac {a^2}{5 c x^5 \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {\left (15 b^2 c^2-8 a d (5 b c-3 a d)\right ) \int \frac {1}{x^2 \left (d x^2+c\right )^{3/2}}dx}{3 c}-\frac {2 a (5 b c-3 a d)}{3 c x^3 \sqrt {c+d x^2}}}{5 c}-\frac {a^2}{5 c x^5 \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {\frac {\left (15 b^2 c^2-8 a d (5 b c-3 a d)\right ) \left (-\frac {2 d \int \frac {1}{\left (d x^2+c\right )^{3/2}}dx}{c}-\frac {1}{c x \sqrt {c+d x^2}}\right )}{3 c}-\frac {2 a (5 b c-3 a d)}{3 c x^3 \sqrt {c+d x^2}}}{5 c}-\frac {a^2}{5 c x^5 \sqrt {c+d x^2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\frac {\left (-\frac {2 d x}{c^2 \sqrt {c+d x^2}}-\frac {1}{c x \sqrt {c+d x^2}}\right ) \left (15 b^2 c^2-8 a d (5 b c-3 a d)\right )}{3 c}-\frac {2 a (5 b c-3 a d)}{3 c x^3 \sqrt {c+d x^2}}}{5 c}-\frac {a^2}{5 c x^5 \sqrt {c+d x^2}}\) |
-1/5*a^2/(c*x^5*Sqrt[c + d*x^2]) + ((-2*a*(5*b*c - 3*a*d))/(3*c*x^3*Sqrt[c + d*x^2]) + ((15*b^2*c^2 - 8*a*d*(5*b*c - 3*a*d))*(-(1/(c*x*Sqrt[c + d*x^ 2])) - (2*d*x)/(c^2*Sqrt[c + d*x^2])))/(3*c))/(5*c)
3.7.58.3.1 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[(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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Time = 2.91 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(\frac {\left (-15 b^{2} x^{4}-10 a b \,x^{2}-3 a^{2}\right ) c^{3}+6 x^{2} d \left (-5 b^{2} x^{4}+\frac {20}{3} a b \,x^{2}+a^{2}\right ) c^{2}-24 x^{4} \left (-\frac {10 b \,x^{2}}{3}+a \right ) d^{2} a c -48 a^{2} d^{3} x^{6}}{15 \sqrt {d \,x^{2}+c}\, x^{5} c^{4}}\) | \(101\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (33 a^{2} d^{2} x^{4}-50 x^{4} a b c d +15 b^{2} c^{2} x^{4}-9 a^{2} c d \,x^{2}+10 a b \,c^{2} x^{2}+3 a^{2} c^{2}\right )}{15 c^{4} x^{5}}-\frac {x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{\sqrt {d \,x^{2}+c}\, c^{4}}\) | \(116\) |
gosper | \(-\frac {48 a^{2} d^{3} x^{6}-80 x^{6} d^{2} a b c +30 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-40 a b \,c^{2} d \,x^{4}+15 b^{2} c^{3} x^{4}-6 a^{2} c^{2} d \,x^{2}+10 a b \,c^{3} x^{2}+3 a^{2} c^{3}}{15 x^{5} \sqrt {d \,x^{2}+c}\, c^{4}}\) | \(117\) |
trager | \(-\frac {48 a^{2} d^{3} x^{6}-80 x^{6} d^{2} a b c +30 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-40 a b \,c^{2} d \,x^{4}+15 b^{2} c^{3} x^{4}-6 a^{2} c^{2} d \,x^{2}+10 a b \,c^{3} x^{2}+3 a^{2} c^{3}}{15 x^{5} \sqrt {d \,x^{2}+c}\, c^{4}}\) | \(117\) |
default | \(b^{2} \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )+a^{2} \left (-\frac {1}{5 c \,x^{5} \sqrt {d \,x^{2}+c}}-\frac {6 d \left (-\frac {1}{3 c \,x^{3} \sqrt {d \,x^{2}+c}}-\frac {4 d \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{3 c}\right )}{5 c}\right )+2 a b \left (-\frac {1}{3 c \,x^{3} \sqrt {d \,x^{2}+c}}-\frac {4 d \left (-\frac {1}{c x \sqrt {d \,x^{2}+c}}-\frac {2 d x}{c^{2} \sqrt {d \,x^{2}+c}}\right )}{3 c}\right )\) | \(188\) |
1/15*((-15*b^2*x^4-10*a*b*x^2-3*a^2)*c^3+6*x^2*d*(-5*b^2*x^4+20/3*a*b*x^2+ a^2)*c^2-24*x^4*(-10/3*b*x^2+a)*d^2*a*c-48*a^2*d^3*x^6)/(d*x^2+c)^(1/2)/x^ 5/c^4
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, {\left (15 \, b^{2} c^{2} d - 40 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{6} + 3 \, a^{2} c^{3} + {\left (15 \, b^{2} c^{3} - 40 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x^{4} + 2 \, {\left (5 \, a b c^{3} - 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, {\left (c^{4} d x^{7} + c^{5} x^{5}\right )}} \]
-1/15*(2*(15*b^2*c^2*d - 40*a*b*c*d^2 + 24*a^2*d^3)*x^6 + 3*a^2*c^3 + (15* b^2*c^3 - 40*a*b*c^2*d + 24*a^2*c*d^2)*x^4 + 2*(5*a*b*c^3 - 3*a^2*c^2*d)*x ^2)*sqrt(d*x^2 + c)/(c^4*d*x^7 + c^5*x^5)
\[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{6} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {2 \, b^{2} d x}{\sqrt {d x^{2} + c} c^{2}} + \frac {16 \, a b d^{2} x}{3 \, \sqrt {d x^{2} + c} c^{3}} - \frac {16 \, a^{2} d^{3} x}{5 \, \sqrt {d x^{2} + c} c^{4}} - \frac {b^{2}}{\sqrt {d x^{2} + c} c x} + \frac {8 \, a b d}{3 \, \sqrt {d x^{2} + c} c^{2} x} - \frac {8 \, a^{2} d^{2}}{5 \, \sqrt {d x^{2} + c} c^{3} x} - \frac {2 \, a b}{3 \, \sqrt {d x^{2} + c} c x^{3}} + \frac {2 \, a^{2} d}{5 \, \sqrt {d x^{2} + c} c^{2} x^{3}} - \frac {a^{2}}{5 \, \sqrt {d x^{2} + c} c x^{5}} \]
-2*b^2*d*x/(sqrt(d*x^2 + c)*c^2) + 16/3*a*b*d^2*x/(sqrt(d*x^2 + c)*c^3) - 16/5*a^2*d^3*x/(sqrt(d*x^2 + c)*c^4) - b^2/(sqrt(d*x^2 + c)*c*x) + 8/3*a*b *d/(sqrt(d*x^2 + c)*c^2*x) - 8/5*a^2*d^2/(sqrt(d*x^2 + c)*c^3*x) - 2/3*a*b /(sqrt(d*x^2 + c)*c*x^3) + 2/5*a^2*d/(sqrt(d*x^2 + c)*c^2*x^3) - 1/5*a^2/( sqrt(d*x^2 + c)*c*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (125) = 250\).
Time = 0.31 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{\sqrt {d x^{2} + c} c^{4}} + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt {d} - 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c d^{\frac {3}{2}} + 15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt {d} + 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac {3}{2}} - 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c d^{\frac {5}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt {d} - 320 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac {3}{2}} + 240 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt {d} + 220 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac {3}{2}} - 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac {5}{2}} + 15 \, b^{2} c^{6} \sqrt {d} - 50 \, a b c^{5} d^{\frac {3}{2}} + 33 \, a^{2} c^{4} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5} c^{3}} \]
-(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x/(sqrt(d*x^2 + c)*c^4) + 2/15*(15*(s qrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^2*sqrt(d) - 30*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c*d^(3/2) + 15*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^3*sqrt(d) + 180*(sqrt(d)*x - sqrt (d*x^2 + c))^6*a*b*c^2*d^(3/2) - 90*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c* d^(5/2) + 90*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^4*sqrt(d) - 320*(sqrt(d )*x - sqrt(d*x^2 + c))^4*a*b*c^3*d^(3/2) + 240*(sqrt(d)*x - sqrt(d*x^2 + c ))^4*a^2*c^2*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^5*sqrt(d) + 220*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^4*d^(3/2) - 150*(sqrt(d)*x - s qrt(d*x^2 + c))^2*a^2*c^3*d^(5/2) + 15*b^2*c^6*sqrt(d) - 50*a*b*c^5*d^(3/2 ) + 33*a^2*c^4*d^(5/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^5*c^3)
Time = 5.85 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{x^6 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {3\,a^2\,c^3-6\,a^2\,c^2\,d\,x^2+24\,a^2\,c\,d^2\,x^4+48\,a^2\,d^3\,x^6+10\,a\,b\,c^3\,x^2-40\,a\,b\,c^2\,d\,x^4-80\,a\,b\,c\,d^2\,x^6+15\,b^2\,c^3\,x^4+30\,b^2\,c^2\,d\,x^6}{15\,c^4\,x^5\,\sqrt {d\,x^2+c}} \]