3.169 \(\int \frac{c x^5+d x^7+e x^9+f x^{11}}{\sqrt{a+b x^2}} \, dx\)

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]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^6 - (a*(2*b^3*c - 3*
a*b^2*d + 4*a^2*b*e - 5*a^3*f)*(a + b*x^2)^(3/2))/(3*b^6) + ((b^3*c - 3*a*b^2*d
+ 6*a^2*b*e - 10*a^3*f)*(a + b*x^2)^(5/2))/(5*b^6) + ((b^2*d - 4*a*b*e + 10*a^2*
f)*(a + b*x^2)^(7/2))/(7*b^6) + ((b*e - 5*a*f)*(a + b*x^2)^(9/2))/(9*b^6) + (f*(
a + b*x^2)^(11/2))/(11*b^6)

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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]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^6 - (a*(2*b^3*c - 3*
a*b^2*d + 4*a^2*b*e - 5*a^3*f)*(a + b*x^2)^(3/2))/(3*b^6) + ((b^3*c - 3*a*b^2*d
+ 6*a^2*b*e - 10*a^3*f)*(a + b*x^2)^(5/2))/(5*b^6) + ((b^2*d - 4*a*b*e + 10*a^2*
f)*(a + b*x^2)^(7/2))/(7*b^6) + ((b*e - 5*a*f)*(a + b*x^2)^(9/2))/(9*b^6) + (f*(
a + b*x^2)^(11/2))/(11*b^6)

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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]

-a**2*sqrt(a + b*x**2)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/b**6 + a*(a + b*x
**2)**(3/2)*(5*a**3*f - 4*a**2*b*e + 3*a*b**2*d - 2*b**3*c)/(3*b**6) + f*(a + b*
x**2)**(11/2)/(11*b**6) - (a + b*x**2)**(9/2)*(5*a*f - b*e)/(9*b**6) + (a + b*x*
*2)**(7/2)*(10*a**2*f - 4*a*b*e + b**2*d)/(7*b**6) - (a + b*x**2)**(5/2)*(10*a**
3*f - 6*a**2*b*e + 3*a*b**2*d - b**3*c)/(5*b**6)

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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]

(Sqrt[a + b*x^2]*(-1280*a^5*f + 128*a^4*b*(11*e + 5*f*x^2) - 16*a^3*b^2*(99*d +
44*e*x^2 + 30*f*x^4) + 8*a^2*b^3*(231*c + 99*d*x^2 + 66*e*x^4 + 50*f*x^6) - 2*a*
b^4*x^2*(462*c + 297*d*x^2 + 220*e*x^4 + 175*f*x^6) + b^5*x^4*(693*c + 5*(99*d*x
^2 + 77*e*x^4 + 63*f*x^6))))/(3465*b^6)

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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]

-1/3465*(b*x^2+a)^(1/2)*(-315*b^5*f*x^10+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*b*f*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*b*e+1584*a^3*b^2*d-1848*a^2*b^3*c)/b^6

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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]

Exception raised: ValueError

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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]

1/3465*(315*b^5*f*x^10 + 35*(11*b^5*e - 10*a*b^4*f)*x^8 + 5*(99*b^5*d - 88*a*b^4
*e + 80*a^2*b^3*f)*x^6 + 1848*a^2*b^3*c - 1584*a^3*b^2*d + 1408*a^4*b*e - 1280*a
^5*f + 3*(231*b^5*c - 198*a*b^4*d + 176*a^2*b^3*e - 160*a^3*b^2*f)*x^4 - 4*(231*
a*b^4*c - 198*a^2*b^3*d + 176*a^3*b^2*e - 160*a^4*b*f)*x^2)*sqrt(b*x^2 + a)/b^6

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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]

Piecewise((-256*a**5*f*sqrt(a + b*x**2)/(693*b**6) + 128*a**4*e*sqrt(a + b*x**2)
/(315*b**5) + 128*a**4*f*x**2*sqrt(a + b*x**2)/(693*b**5) - 16*a**3*d*sqrt(a + b
*x**2)/(35*b**4) - 64*a**3*e*x**2*sqrt(a + b*x**2)/(315*b**4) - 32*a**3*f*x**4*s
qrt(a + b*x**2)/(231*b**4) + 8*a**2*c*sqrt(a + b*x**2)/(15*b**3) + 8*a**2*d*x**2
*sqrt(a + b*x**2)/(35*b**3) + 16*a**2*e*x**4*sqrt(a + b*x**2)/(105*b**3) + 80*a*
*2*f*x**6*sqrt(a + b*x**2)/(693*b**3) - 4*a*c*x**2*sqrt(a + b*x**2)/(15*b**2) -
6*a*d*x**4*sqrt(a + b*x**2)/(35*b**2) - 8*a*e*x**6*sqrt(a + b*x**2)/(63*b**2) -
10*a*f*x**8*sqrt(a + b*x**2)/(99*b**2) + c*x**4*sqrt(a + b*x**2)/(5*b) + d*x**6*
sqrt(a + b*x**2)/(7*b) + e*x**8*sqrt(a + b*x**2)/(9*b) + f*x**10*sqrt(a + b*x**2
)/(11*b), Ne(b, 0)), ((c*x**6/6 + d*x**8/8 + e*x**10/10 + f*x**12/12)/sqrt(a), T
rue))

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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]

1/3465*(693*(b*x^2 + a)^(5/2)*b^3*c - 2310*(b*x^2 + a)^(3/2)*a*b^3*c + 3465*sqrt
(b*x^2 + a)*a^2*b^3*c + 495*(b*x^2 + a)^(7/2)*b^2*d - 2079*(b*x^2 + a)^(5/2)*a*b
^2*d + 3465*(b*x^2 + a)^(3/2)*a^2*b^2*d - 3465*sqrt(b*x^2 + a)*a^3*b^2*d + 315*(
b*x^2 + a)^(11/2)*f - 1925*(b*x^2 + a)^(9/2)*a*f + 4950*(b*x^2 + a)^(7/2)*a^2*f
- 6930*(b*x^2 + a)^(5/2)*a^3*f + 5775*(b*x^2 + a)^(3/2)*a^4*f - 3465*sqrt(b*x^2
+ a)*a^5*f + 385*(b*x^2 + a)^(9/2)*b*e - 1980*(b*x^2 + a)^(7/2)*a*b*e + 4158*(b*
x^2 + a)^(5/2)*a^2*b*e - 4620*(b*x^2 + a)^(3/2)*a^3*b*e + 3465*sqrt(b*x^2 + a)*a
^4*b*e)/b^6