3.87 \(\int x^9 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=91 \[ \frac{3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}-\frac{a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac{a^4 \left (a+b x^2\right )^9}{18 b^5}+\frac{\left (a+b x^2\right )^{13}}{26 b^5}-\frac{a \left (a+b x^2\right )^{12}}{6 b^5} \]

[Out]

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

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Rubi [A]  time = 0.141735, antiderivative size = 91, 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, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}-\frac{a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac{a^4 \left (a+b x^2\right )^9}{18 b^5}+\frac{\left (a+b x^2\right )^{13}}{26 b^5}-\frac{a \left (a+b x^2\right )^{12}}{6 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^9*(a + b*x^2)^8,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^9 \left (a+b x^2\right )^8 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^4 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^4 (a+b x)^8}{b^4}-\frac{4 a^3 (a+b x)^9}{b^4}+\frac{6 a^2 (a+b x)^{10}}{b^4}-\frac{4 a (a+b x)^{11}}{b^4}+\frac{(a+b x)^{12}}{b^4}\right ) \, dx,x,x^2\right )\\ &=\frac{a^4 \left (a+b x^2\right )^9}{18 b^5}-\frac{a^3 \left (a+b x^2\right )^{10}}{5 b^5}+\frac{3 a^2 \left (a+b x^2\right )^{11}}{11 b^5}-\frac{a \left (a+b x^2\right )^{12}}{6 b^5}+\frac{\left (a+b x^2\right )^{13}}{26 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0024927, size = 106, normalized size = 1.16 \[ \frac{14}{11} a^2 b^6 x^{22}+\frac{14}{5} a^3 b^5 x^{20}+\frac{35}{9} a^4 b^4 x^{18}+\frac{7}{2} a^5 b^3 x^{16}+2 a^6 b^2 x^{14}+\frac{2}{3} a^7 b x^{12}+\frac{a^8 x^{10}}{10}+\frac{1}{3} a b^7 x^{24}+\frac{b^8 x^{26}}{26} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a + b*x^2)^8,x]

[Out]

(a^8*x^10)/10 + (2*a^7*b*x^12)/3 + 2*a^6*b^2*x^14 + (7*a^5*b^3*x^16)/2 + (35*a^4*b^4*x^18)/9 + (14*a^3*b^5*x^2
0)/5 + (14*a^2*b^6*x^22)/11 + (a*b^7*x^24)/3 + (b^8*x^26)/26

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Maple [A]  time = 0., size = 91, normalized size = 1. \begin{align*}{\frac{{b}^{8}{x}^{26}}{26}}+{\frac{a{b}^{7}{x}^{24}}{3}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{22}}{11}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{20}}{5}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{18}}{9}}+{\frac{7\,{a}^{5}{b}^{3}{x}^{16}}{2}}+2\,{a}^{6}{b}^{2}{x}^{14}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{{a}^{8}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(b*x^2+a)^8,x)

[Out]

1/26*b^8*x^26+1/3*a*b^7*x^24+14/11*a^2*b^6*x^22+14/5*a^3*b^5*x^20+35/9*a^4*b^4*x^18+7/2*a^5*b^3*x^16+2*a^6*b^2
*x^14+2/3*a^7*b*x^12+1/10*a^8*x^10

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Maxima [A]  time = 2.0037, size = 122, normalized size = 1.34 \begin{align*} \frac{1}{26} \, b^{8} x^{26} + \frac{1}{3} \, a b^{7} x^{24} + \frac{14}{11} \, a^{2} b^{6} x^{22} + \frac{14}{5} \, a^{3} b^{5} x^{20} + \frac{35}{9} \, a^{4} b^{4} x^{18} + \frac{7}{2} \, a^{5} b^{3} x^{16} + 2 \, a^{6} b^{2} x^{14} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{10} \, a^{8} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/26*b^8*x^26 + 1/3*a*b^7*x^24 + 14/11*a^2*b^6*x^22 + 14/5*a^3*b^5*x^20 + 35/9*a^4*b^4*x^18 + 7/2*a^5*b^3*x^16
+ 2*a^6*b^2*x^14 + 2/3*a^7*b*x^12 + 1/10*a^8*x^10

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Fricas [A]  time = 1.17211, size = 219, normalized size = 2.41 \begin{align*} \frac{1}{26} x^{26} b^{8} + \frac{1}{3} x^{24} b^{7} a + \frac{14}{11} x^{22} b^{6} a^{2} + \frac{14}{5} x^{20} b^{5} a^{3} + \frac{35}{9} x^{18} b^{4} a^{4} + \frac{7}{2} x^{16} b^{3} a^{5} + 2 x^{14} b^{2} a^{6} + \frac{2}{3} x^{12} b a^{7} + \frac{1}{10} x^{10} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/26*x^26*b^8 + 1/3*x^24*b^7*a + 14/11*x^22*b^6*a^2 + 14/5*x^20*b^5*a^3 + 35/9*x^18*b^4*a^4 + 7/2*x^16*b^3*a^5
+ 2*x^14*b^2*a^6 + 2/3*x^12*b*a^7 + 1/10*x^10*a^8

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Sympy [A]  time = 0.082466, size = 104, normalized size = 1.14 \begin{align*} \frac{a^{8} x^{10}}{10} + \frac{2 a^{7} b x^{12}}{3} + 2 a^{6} b^{2} x^{14} + \frac{7 a^{5} b^{3} x^{16}}{2} + \frac{35 a^{4} b^{4} x^{18}}{9} + \frac{14 a^{3} b^{5} x^{20}}{5} + \frac{14 a^{2} b^{6} x^{22}}{11} + \frac{a b^{7} x^{24}}{3} + \frac{b^{8} x^{26}}{26} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(b*x**2+a)**8,x)

[Out]

a**8*x**10/10 + 2*a**7*b*x**12/3 + 2*a**6*b**2*x**14 + 7*a**5*b**3*x**16/2 + 35*a**4*b**4*x**18/9 + 14*a**3*b*
*5*x**20/5 + 14*a**2*b**6*x**22/11 + a*b**7*x**24/3 + b**8*x**26/26

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Giac [A]  time = 1.99636, size = 122, normalized size = 1.34 \begin{align*} \frac{1}{26} \, b^{8} x^{26} + \frac{1}{3} \, a b^{7} x^{24} + \frac{14}{11} \, a^{2} b^{6} x^{22} + \frac{14}{5} \, a^{3} b^{5} x^{20} + \frac{35}{9} \, a^{4} b^{4} x^{18} + \frac{7}{2} \, a^{5} b^{3} x^{16} + 2 \, a^{6} b^{2} x^{14} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{10} \, a^{8} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/26*b^8*x^26 + 1/3*a*b^7*x^24 + 14/11*a^2*b^6*x^22 + 14/5*a^3*b^5*x^20 + 35/9*a^4*b^4*x^18 + 7/2*a^5*b^3*x^16
+ 2*a^6*b^2*x^14 + 2/3*a^7*b*x^12 + 1/10*a^8*x^10