3.411 \(\int x^{11} \left (a+b x^2\right )^{9/2} \, dx\)

Optimal. Leaf size=122 \[ -\frac{a^5 \left (a+b x^2\right )^{11/2}}{11 b^6}+\frac{5 a^4 \left (a+b x^2\right )^{13/2}}{13 b^6}-\frac{2 a^3 \left (a+b x^2\right )^{15/2}}{3 b^6}+\frac{10 a^2 \left (a+b x^2\right )^{17/2}}{17 b^6}+\frac{\left (a+b x^2\right )^{21/2}}{21 b^6}-\frac{5 a \left (a+b x^2\right )^{19/2}}{19 b^6} \]

[Out]

-(a^5*(a + b*x^2)^(11/2))/(11*b^6) + (5*a^4*(a + b*x^2)^(13/2))/(13*b^6) - (2*a^
3*(a + b*x^2)^(15/2))/(3*b^6) + (10*a^2*(a + b*x^2)^(17/2))/(17*b^6) - (5*a*(a +
b*x^2)^(19/2))/(19*b^6) + (a + b*x^2)^(21/2)/(21*b^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.174919, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^5 \left (a+b x^2\right )^{11/2}}{11 b^6}+\frac{5 a^4 \left (a+b x^2\right )^{13/2}}{13 b^6}-\frac{2 a^3 \left (a+b x^2\right )^{15/2}}{3 b^6}+\frac{10 a^2 \left (a+b x^2\right )^{17/2}}{17 b^6}+\frac{\left (a+b x^2\right )^{21/2}}{21 b^6}-\frac{5 a \left (a+b x^2\right )^{19/2}}{19 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^2)^(9/2),x]

[Out]

-(a^5*(a + b*x^2)^(11/2))/(11*b^6) + (5*a^4*(a + b*x^2)^(13/2))/(13*b^6) - (2*a^
3*(a + b*x^2)^(15/2))/(3*b^6) + (10*a^2*(a + b*x^2)^(17/2))/(17*b^6) - (5*a*(a +
b*x^2)^(19/2))/(19*b^6) + (a + b*x^2)^(21/2)/(21*b^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.0092, size = 112, normalized size = 0.92 \[ - \frac{a^{5} \left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b^{6}} + \frac{5 a^{4} \left (a + b x^{2}\right )^{\frac{13}{2}}}{13 b^{6}} - \frac{2 a^{3} \left (a + b x^{2}\right )^{\frac{15}{2}}}{3 b^{6}} + \frac{10 a^{2} \left (a + b x^{2}\right )^{\frac{17}{2}}}{17 b^{6}} - \frac{5 a \left (a + b x^{2}\right )^{\frac{19}{2}}}{19 b^{6}} + \frac{\left (a + b x^{2}\right )^{\frac{21}{2}}}{21 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x**2+a)**(9/2),x)

[Out]

-a**5*(a + b*x**2)**(11/2)/(11*b**6) + 5*a**4*(a + b*x**2)**(13/2)/(13*b**6) - 2
*a**3*(a + b*x**2)**(15/2)/(3*b**6) + 10*a**2*(a + b*x**2)**(17/2)/(17*b**6) - 5
*a*(a + b*x**2)**(19/2)/(19*b**6) + (a + b*x**2)**(21/2)/(21*b**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0746111, size = 72, normalized size = 0.59 \[ \frac{\left (a+b x^2\right )^{11/2} \left (-256 a^5+1408 a^4 b x^2-4576 a^3 b^2 x^4+11440 a^2 b^3 x^6-24310 a b^4 x^8+46189 b^5 x^{10}\right )}{969969 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(-256*a^5 + 1408*a^4*b*x^2 - 4576*a^3*b^2*x^4 + 11440*a^2*b^
3*x^6 - 24310*a*b^4*x^8 + 46189*b^5*x^10))/(969969*b^6)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 69, normalized size = 0.6 \[ -{\frac{-46189\,{b}^{5}{x}^{10}+24310\,a{b}^{4}{x}^{8}-11440\,{a}^{2}{b}^{3}{x}^{6}+4576\,{a}^{3}{b}^{2}{x}^{4}-1408\,{a}^{4}b{x}^{2}+256\,{a}^{5}}{969969\,{b}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^2+a)^(9/2),x)

[Out]

-1/969969*(b*x^2+a)^(11/2)*(-46189*b^5*x^10+24310*a*b^4*x^8-11440*a^2*b^3*x^6+45
76*a^3*b^2*x^4-1408*a^4*b*x^2+256*a^5)/b^6

_______________________________________________________________________________________

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((b*x^2 + a)^(9/2)*x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.252354, size = 166, normalized size = 1.36 \[ \frac{{\left (46189 \, b^{10} x^{20} + 206635 \, a b^{9} x^{18} + 351780 \, a^{2} b^{8} x^{16} + 271414 \, a^{3} b^{7} x^{14} + 80773 \, a^{4} b^{6} x^{12} + 63 \, a^{5} b^{5} x^{10} - 70 \, a^{6} b^{4} x^{8} + 80 \, a^{7} b^{3} x^{6} - 96 \, a^{8} b^{2} x^{4} + 128 \, a^{9} b x^{2} - 256 \, a^{10}\right )} \sqrt{b x^{2} + a}}{969969 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)*x^11,x, algorithm="fricas")

[Out]

1/969969*(46189*b^10*x^20 + 206635*a*b^9*x^18 + 351780*a^2*b^8*x^16 + 271414*a^3
*b^7*x^14 + 80773*a^4*b^6*x^12 + 63*a^5*b^5*x^10 - 70*a^6*b^4*x^8 + 80*a^7*b^3*x
^6 - 96*a^8*b^2*x^4 + 128*a^9*b*x^2 - 256*a^10)*sqrt(b*x^2 + a)/b^6

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b*x**2+a)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218227, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)*x^11,x, algorithm="giac")

[Out]

Done