Optimal. Leaf size=214 \[ \frac{\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac{\left (a+b x^2\right )^{5/2} \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac{a \left (a+b x^2\right )^{3/2} \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}+\frac{\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left (a+b x^2\right )^{11/2}}{11 b^6} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.463463, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac{\left (a+b x^2\right )^{5/2} \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac{a \left (a+b x^2\right )^{3/2} \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac{a^2 \sqrt{a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}+\frac{\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac{f \left (a+b x^2\right )^{11/2}}{11 b^6} \]
Antiderivative was successfully verified.
[In] Int[(c*x^5 + d*x^7 + e*x^9 + f*x^11)/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 86.4853, size = 206, normalized size = 0.96 \[ - \frac{a^{2} \sqrt{a + b x^{2}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{6}} + \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \left (5 a^{3} f - 4 a^{2} b e + 3 a b^{2} d - 2 b^{3} c\right )}{3 b^{6}} + \frac{f \left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b^{6}} - \frac{\left (a + b x^{2}\right )^{\frac{9}{2}} \left (5 a f - b e\right )}{9 b^{6}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \left (10 a^{2} f - 4 a b e + b^{2} d\right )}{7 b^{6}} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right )}{5 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**11+e*x**9+d*x**7+c*x**5)/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.222092, size = 158, normalized size = 0.74 \[ \frac{\sqrt{a+b x^2} \left (-1280 a^5 f+128 a^4 b \left (11 e+5 f x^2\right )-16 a^3 b^2 \left (99 d+44 e x^2+30 f x^4\right )+8 a^2 b^3 \left (231 c+99 d x^2+66 e x^4+50 f x^6\right )-2 a b^4 x^2 \left (462 c+297 d x^2+220 e x^4+175 f x^6\right )+b^5 x^4 \left (693 c+5 \left (99 d x^2+77 e x^4+63 f x^6\right )\right )\right )}{3465 b^6} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^5 + d*x^7 + e*x^9 + f*x^11)/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 193, normalized size = 0.9 \[ -{\frac{-315\,f{x}^{10}{b}^{5}+350\,a{b}^{4}f{x}^{8}-385\,{b}^{5}e{x}^{8}-400\,{a}^{2}{b}^{3}f{x}^{6}+440\,a{b}^{4}e{x}^{6}-495\,{b}^{5}d{x}^{6}+480\,{a}^{3}{b}^{2}f{x}^{4}-528\,{a}^{2}{b}^{3}e{x}^{4}+594\,a{b}^{4}d{x}^{4}-693\,{b}^{5}c{x}^{4}-640\,{a}^{4}bf{x}^{2}+704\,{a}^{3}{b}^{2}e{x}^{2}-792\,{a}^{2}{b}^{3}d{x}^{2}+924\,a{b}^{4}c{x}^{2}+1280\,{a}^{5}f-1408\,{a}^{4}be+1584\,{a}^{3}{b}^{2}d-1848\,{a}^{2}{b}^{3}c}{3465\,{b}^{6}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^11+e*x^9+d*x^7+c*x^5)/(b*x^2+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^11 + e*x^9 + d*x^7 + c*x^5)/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.262701, size = 239, normalized size = 1.12 \[ \frac{{\left (315 \, b^{5} f x^{10} + 35 \,{\left (11 \, b^{5} e - 10 \, a b^{4} f\right )} x^{8} + 5 \,{\left (99 \, b^{5} d - 88 \, a b^{4} e + 80 \, a^{2} b^{3} f\right )} x^{6} + 1848 \, a^{2} b^{3} c - 1584 \, a^{3} b^{2} d + 1408 \, a^{4} b e - 1280 \, a^{5} f + 3 \,{\left (231 \, b^{5} c - 198 \, a b^{4} d + 176 \, a^{2} b^{3} e - 160 \, a^{3} b^{2} f\right )} x^{4} - 4 \,{\left (231 \, a b^{4} c - 198 \, a^{2} b^{3} d + 176 \, a^{3} b^{2} e - 160 \, a^{4} b f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3465 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^11 + e*x^9 + d*x^7 + c*x^5)/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.76848, size = 442, normalized size = 2.07 \[ \begin{cases} - \frac{256 a^{5} f \sqrt{a + b x^{2}}}{693 b^{6}} + \frac{128 a^{4} e \sqrt{a + b x^{2}}}{315 b^{5}} + \frac{128 a^{4} f x^{2} \sqrt{a + b x^{2}}}{693 b^{5}} - \frac{16 a^{3} d \sqrt{a + b x^{2}}}{35 b^{4}} - \frac{64 a^{3} e x^{2} \sqrt{a + b x^{2}}}{315 b^{4}} - \frac{32 a^{3} f x^{4} \sqrt{a + b x^{2}}}{231 b^{4}} + \frac{8 a^{2} c \sqrt{a + b x^{2}}}{15 b^{3}} + \frac{8 a^{2} d x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} + \frac{16 a^{2} e x^{4} \sqrt{a + b x^{2}}}{105 b^{3}} + \frac{80 a^{2} f x^{6} \sqrt{a + b x^{2}}}{693 b^{3}} - \frac{4 a c x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} - \frac{6 a d x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} - \frac{8 a e x^{6} \sqrt{a + b x^{2}}}{63 b^{2}} - \frac{10 a f x^{8} \sqrt{a + b x^{2}}}{99 b^{2}} + \frac{c x^{4} \sqrt{a + b x^{2}}}{5 b} + \frac{d x^{6} \sqrt{a + b x^{2}}}{7 b} + \frac{e x^{8} \sqrt{a + b x^{2}}}{9 b} + \frac{f x^{10} \sqrt{a + b x^{2}}}{11 b} & \text{for}\: b \neq 0 \\\frac{\frac{c x^{6}}{6} + \frac{d x^{8}}{8} + \frac{e x^{10}}{10} + \frac{f x^{12}}{12}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**11+e*x**9+d*x**7+c*x**5)/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.230981, size = 387, normalized size = 1.81 \[ \frac{693 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3} c - 2310 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{3} c + 3465 \, \sqrt{b x^{2} + a} a^{2} b^{3} c + 495 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2} d - 2079 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b^{2} d + 3465 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b^{2} d - 3465 \, \sqrt{b x^{2} + a} a^{3} b^{2} d + 315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} f - 1925 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a f + 4950 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} f - 6930 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} f + 5775 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} f - 3465 \, \sqrt{b x^{2} + a} a^{5} f + 385 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} b e - 1980 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a b e + 4158 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} b e - 4620 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} b e + 3465 \, \sqrt{b x^{2} + a} a^{4} b e}{3465 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^11 + e*x^9 + d*x^7 + c*x^5)/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]