3.410 \(\int x^{13} \left (a+b x^2\right )^{9/2} \, dx\)

Optimal. Leaf size=140 \[ \frac{a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}-\frac{6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac{a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac{20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}+\frac{\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac{2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

[Out]

(a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(
a + b*x^2)^(15/2))/b^7 - (20*a^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x
^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a + b*x^2)^(23/2)/(23
*b^7)

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Rubi [A]  time = 0.202669, antiderivative size = 140, 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 \[ \frac{a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}-\frac{6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac{a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac{20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}+\frac{\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac{2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

Antiderivative was successfully verified.

[In]  Int[x^13*(a + b*x^2)^(9/2),x]

[Out]

(a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(
a + b*x^2)^(15/2))/b^7 - (20*a^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x
^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a + b*x^2)^(23/2)/(23
*b^7)

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Rubi in Sympy [A]  time = 27.5785, size = 129, normalized size = 0.92 \[ \frac{a^{6} \left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b^{7}} - \frac{6 a^{5} \left (a + b x^{2}\right )^{\frac{13}{2}}}{13 b^{7}} + \frac{a^{4} \left (a + b x^{2}\right )^{\frac{15}{2}}}{b^{7}} - \frac{20 a^{3} \left (a + b x^{2}\right )^{\frac{17}{2}}}{17 b^{7}} + \frac{15 a^{2} \left (a + b x^{2}\right )^{\frac{19}{2}}}{19 b^{7}} - \frac{2 a \left (a + b x^{2}\right )^{\frac{21}{2}}}{7 b^{7}} + \frac{\left (a + b x^{2}\right )^{\frac{23}{2}}}{23 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13*(b*x**2+a)**(9/2),x)

[Out]

a**6*(a + b*x**2)**(11/2)/(11*b**7) - 6*a**5*(a + b*x**2)**(13/2)/(13*b**7) + a*
*4*(a + b*x**2)**(15/2)/b**7 - 20*a**3*(a + b*x**2)**(17/2)/(17*b**7) + 15*a**2*
(a + b*x**2)**(19/2)/(19*b**7) - 2*a*(a + b*x**2)**(21/2)/(7*b**7) + (a + b*x**2
)**(23/2)/(23*b**7)

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Mathematica [A]  time = 0.0669657, size = 83, normalized size = 0.59 \[ \frac{\left (a+b x^2\right )^{11/2} \left (1024 a^6-5632 a^5 b x^2+18304 a^4 b^2 x^4-45760 a^3 b^3 x^6+97240 a^2 b^4 x^8-184756 a b^5 x^{10}+323323 b^6 x^{12}\right )}{7436429 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(1024*a^6 - 5632*a^5*b*x^2 + 18304*a^4*b^2*x^4 - 45760*a^3*b
^3*x^6 + 97240*a^2*b^4*x^8 - 184756*a*b^5*x^10 + 323323*b^6*x^12))/(7436429*b^7)

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Maple [A]  time = 0.01, size = 80, normalized size = 0.6 \[{\frac{323323\,{x}^{12}{b}^{6}-184756\,a{x}^{10}{b}^{5}+97240\,{a}^{2}{x}^{8}{b}^{4}-45760\,{a}^{3}{x}^{6}{b}^{3}+18304\,{a}^{4}{x}^{4}{b}^{2}-5632\,{a}^{5}{x}^{2}b+1024\,{a}^{6}}{7436429\,{b}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13*(b*x^2+a)^(9/2),x)

[Out]

1/7436429*(b*x^2+a)^(11/2)*(323323*b^6*x^12-184756*a*b^5*x^10+97240*a^2*b^4*x^8-
45760*a^3*b^3*x^6+18304*a^4*b^2*x^4-5632*a^5*b*x^2+1024*a^6)/b^7

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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((b*x^2 + a)^(9/2)*x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2542, size = 181, normalized size = 1.29 \[ \frac{{\left (323323 \, b^{11} x^{22} + 1431859 \, a b^{10} x^{20} + 2406690 \, a^{2} b^{9} x^{18} + 1826110 \, a^{3} b^{8} x^{16} + 530959 \, a^{4} b^{7} x^{14} + 231 \, a^{5} b^{6} x^{12} - 252 \, a^{6} b^{5} x^{10} + 280 \, a^{7} b^{4} x^{8} - 320 \, a^{8} b^{3} x^{6} + 384 \, a^{9} b^{2} x^{4} - 512 \, a^{10} b x^{2} + 1024 \, a^{11}\right )} \sqrt{b x^{2} + a}}{7436429 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)*x^13,x, algorithm="fricas")

[Out]

1/7436429*(323323*b^11*x^22 + 1431859*a*b^10*x^20 + 2406690*a^2*b^9*x^18 + 18261
10*a^3*b^8*x^16 + 530959*a^4*b^7*x^14 + 231*a^5*b^6*x^12 - 252*a^6*b^5*x^10 + 28
0*a^7*b^4*x^8 - 320*a^8*b^3*x^6 + 384*a^9*b^2*x^4 - 512*a^10*b*x^2 + 1024*a^11)*
sqrt(b*x^2 + a)/b^7

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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**13*(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224934, 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^13,x, algorithm="giac")

[Out]

Done