3.85 \(\int x^{13} \left (a+b x^2\right )^8 \, dx\)

Optimal. Leaf size=129 \[ \frac{a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac{3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac{15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac{5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}+\frac{\left (a+b x^2\right )^{15}}{30 b^7}-\frac{3 a \left (a+b x^2\right )^{14}}{14 b^7} \]

[Out]

(a^6*(a + b*x^2)^9)/(18*b^7) - (3*a^5*(a + b*x^2)^10)/(10*b^7) + (15*a^4*(a + b*
x^2)^11)/(22*b^7) - (5*a^3*(a + b*x^2)^12)/(6*b^7) + (15*a^2*(a + b*x^2)^13)/(26
*b^7) - (3*a*(a + b*x^2)^14)/(14*b^7) + (a + b*x^2)^15/(30*b^7)

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Rubi [A]  time = 0.469701, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac{3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac{15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac{5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}+\frac{\left (a+b x^2\right )^{15}}{30 b^7}-\frac{3 a \left (a+b x^2\right )^{14}}{14 b^7} \]

Antiderivative was successfully verified.

[In]  Int[x^13*(a + b*x^2)^8,x]

[Out]

(a^6*(a + b*x^2)^9)/(18*b^7) - (3*a^5*(a + b*x^2)^10)/(10*b^7) + (15*a^4*(a + b*
x^2)^11)/(22*b^7) - (5*a^3*(a + b*x^2)^12)/(6*b^7) + (15*a^2*(a + b*x^2)^13)/(26
*b^7) - (3*a*(a + b*x^2)^14)/(14*b^7) + (a + b*x^2)^15/(30*b^7)

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Rubi in Sympy [A]  time = 26.6533, size = 105, normalized size = 0.81 \[ \frac{a^{8} x^{14}}{14} + \frac{a^{7} b x^{16}}{2} + \frac{14 a^{6} b^{2} x^{18}}{9} + \frac{14 a^{5} b^{3} x^{20}}{5} + \frac{35 a^{4} b^{4} x^{22}}{11} + \frac{7 a^{3} b^{5} x^{24}}{3} + \frac{14 a^{2} b^{6} x^{26}}{13} + \frac{2 a b^{7} x^{28}}{7} + \frac{b^{8} x^{30}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13*(b*x**2+a)**8,x)

[Out]

a**8*x**14/14 + a**7*b*x**16/2 + 14*a**6*b**2*x**18/9 + 14*a**5*b**3*x**20/5 + 3
5*a**4*b**4*x**22/11 + 7*a**3*b**5*x**24/3 + 14*a**2*b**6*x**26/13 + 2*a*b**7*x*
*28/7 + b**8*x**30/30

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Mathematica [A]  time = 0.00495942, size = 108, normalized size = 0.84 \[ \frac{a^8 x^{14}}{14}+\frac{1}{2} a^7 b x^{16}+\frac{14}{9} a^6 b^2 x^{18}+\frac{14}{5} a^5 b^3 x^{20}+\frac{35}{11} a^4 b^4 x^{22}+\frac{7}{3} a^3 b^5 x^{24}+\frac{14}{13} a^2 b^6 x^{26}+\frac{2}{7} a b^7 x^{28}+\frac{b^8 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13*(a + b*x^2)^8,x]

[Out]

(a^8*x^14)/14 + (a^7*b*x^16)/2 + (14*a^6*b^2*x^18)/9 + (14*a^5*b^3*x^20)/5 + (35
*a^4*b^4*x^22)/11 + (7*a^3*b^5*x^24)/3 + (14*a^2*b^6*x^26)/13 + (2*a*b^7*x^28)/7
+ (b^8*x^30)/30

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Maple [A]  time = 0.002, size = 91, normalized size = 0.7 \[{\frac{{b}^{8}{x}^{30}}{30}}+{\frac{2\,a{b}^{7}{x}^{28}}{7}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{26}}{13}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{24}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{22}}{11}}+{\frac{14\,{a}^{5}{b}^{3}{x}^{20}}{5}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{18}}{9}}+{\frac{{a}^{7}b{x}^{16}}{2}}+{\frac{{a}^{8}{x}^{14}}{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13*(b*x^2+a)^8,x)

[Out]

1/30*b^8*x^30+2/7*a*b^7*x^28+14/13*a^2*b^6*x^26+7/3*a^3*b^5*x^24+35/11*a^4*b^4*x
^22+14/5*a^5*b^3*x^20+14/9*a^6*b^2*x^18+1/2*a^7*b*x^16+1/14*a^8*x^14

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Maxima [A]  time = 1.3442, size = 122, normalized size = 0.95 \[ \frac{1}{30} \, b^{8} x^{30} + \frac{2}{7} \, a b^{7} x^{28} + \frac{14}{13} \, a^{2} b^{6} x^{26} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{35}{11} \, a^{4} b^{4} x^{22} + \frac{14}{5} \, a^{5} b^{3} x^{20} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{1}{2} \, a^{7} b x^{16} + \frac{1}{14} \, a^{8} x^{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^13,x, algorithm="maxima")

[Out]

1/30*b^8*x^30 + 2/7*a*b^7*x^28 + 14/13*a^2*b^6*x^26 + 7/3*a^3*b^5*x^24 + 35/11*a
^4*b^4*x^22 + 14/5*a^5*b^3*x^20 + 14/9*a^6*b^2*x^18 + 1/2*a^7*b*x^16 + 1/14*a^8*
x^14

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Fricas [A]  time = 0.177387, size = 1, normalized size = 0.01 \[ \frac{1}{30} x^{30} b^{8} + \frac{2}{7} x^{28} b^{7} a + \frac{14}{13} x^{26} b^{6} a^{2} + \frac{7}{3} x^{24} b^{5} a^{3} + \frac{35}{11} x^{22} b^{4} a^{4} + \frac{14}{5} x^{20} b^{3} a^{5} + \frac{14}{9} x^{18} b^{2} a^{6} + \frac{1}{2} x^{16} b a^{7} + \frac{1}{14} x^{14} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^13,x, algorithm="fricas")

[Out]

1/30*x^30*b^8 + 2/7*x^28*b^7*a + 14/13*x^26*b^6*a^2 + 7/3*x^24*b^5*a^3 + 35/11*x
^22*b^4*a^4 + 14/5*x^20*b^3*a^5 + 14/9*x^18*b^2*a^6 + 1/2*x^16*b*a^7 + 1/14*x^14
*a^8

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Sympy [A]  time = 0.164373, size = 105, normalized size = 0.81 \[ \frac{a^{8} x^{14}}{14} + \frac{a^{7} b x^{16}}{2} + \frac{14 a^{6} b^{2} x^{18}}{9} + \frac{14 a^{5} b^{3} x^{20}}{5} + \frac{35 a^{4} b^{4} x^{22}}{11} + \frac{7 a^{3} b^{5} x^{24}}{3} + \frac{14 a^{2} b^{6} x^{26}}{13} + \frac{2 a b^{7} x^{28}}{7} + \frac{b^{8} x^{30}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13*(b*x**2+a)**8,x)

[Out]

a**8*x**14/14 + a**7*b*x**16/2 + 14*a**6*b**2*x**18/9 + 14*a**5*b**3*x**20/5 + 3
5*a**4*b**4*x**22/11 + 7*a**3*b**5*x**24/3 + 14*a**2*b**6*x**26/13 + 2*a*b**7*x*
*28/7 + b**8*x**30/30

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GIAC/XCAS [A]  time = 0.210322, size = 122, normalized size = 0.95 \[ \frac{1}{30} \, b^{8} x^{30} + \frac{2}{7} \, a b^{7} x^{28} + \frac{14}{13} \, a^{2} b^{6} x^{26} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{35}{11} \, a^{4} b^{4} x^{22} + \frac{14}{5} \, a^{5} b^{3} x^{20} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{1}{2} \, a^{7} b x^{16} + \frac{1}{14} \, a^{8} x^{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^13,x, algorithm="giac")

[Out]

1/30*b^8*x^30 + 2/7*a*b^7*x^28 + 14/13*a^2*b^6*x^26 + 7/3*a^3*b^5*x^24 + 35/11*a
^4*b^4*x^22 + 14/5*a^5*b^3*x^20 + 14/9*a^6*b^2*x^18 + 1/2*a^7*b*x^16 + 1/14*a^8*
x^14