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

Optimal. Leaf size=108 \[ \frac{4}{3} a^2 b^6 x^{21}+\frac{56}{19} a^3 b^5 x^{19}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{15} a^5 b^3 x^{15}+\frac{28}{13} a^6 b^2 x^{13}+\frac{8}{11} a^7 b x^{11}+\frac{a^8 x^9}{9}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25} \]

[Out]

(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3
*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25

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Rubi [A]  time = 0.0481737, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{4}{3} a^2 b^6 x^{21}+\frac{56}{19} a^3 b^5 x^{19}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{15} a^5 b^3 x^{15}+\frac{28}{13} a^6 b^2 x^{13}+\frac{8}{11} a^7 b x^{11}+\frac{a^8 x^9}{9}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^2)^8,x]

[Out]

(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3
*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^8 \left (a+b x^2\right )^8 \, dx &=\int \left (a^8 x^8+8 a^7 b x^{10}+28 a^6 b^2 x^{12}+56 a^5 b^3 x^{14}+70 a^4 b^4 x^{16}+56 a^3 b^5 x^{18}+28 a^2 b^6 x^{20}+8 a b^7 x^{22}+b^8 x^{24}\right ) \, dx\\ &=\frac{a^8 x^9}{9}+\frac{8}{11} a^7 b x^{11}+\frac{28}{13} a^6 b^2 x^{13}+\frac{56}{15} a^5 b^3 x^{15}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{19} a^3 b^5 x^{19}+\frac{4}{3} a^2 b^6 x^{21}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25}\\ \end{align*}

Mathematica [A]  time = 0.002528, size = 108, normalized size = 1. \[ \frac{4}{3} a^2 b^6 x^{21}+\frac{56}{19} a^3 b^5 x^{19}+\frac{70}{17} a^4 b^4 x^{17}+\frac{56}{15} a^5 b^3 x^{15}+\frac{28}{13} a^6 b^2 x^{13}+\frac{8}{11} a^7 b x^{11}+\frac{a^8 x^9}{9}+\frac{8}{23} a b^7 x^{23}+\frac{b^8 x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^2)^8,x]

[Out]

(a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3
*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25

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Maple [A]  time = 0.001, size = 91, normalized size = 0.8 \begin{align*}{\frac{{a}^{8}{x}^{9}}{9}}+{\frac{8\,{a}^{7}b{x}^{11}}{11}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{13}}{13}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{15}}{15}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{17}}{17}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{19}}{19}}+{\frac{4\,{a}^{2}{b}^{6}{x}^{21}}{3}}+{\frac{8\,a{b}^{7}{x}^{23}}{23}}+{\frac{{b}^{8}{x}^{25}}{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a^8*x^9+8/11*a^7*b*x^11+28/13*a^6*b^2*x^13+56/15*a^5*b^3*x^15+70/17*a^4*b^4*x^17+56/19*a^3*b^5*x^19+4/3*a^
2*b^6*x^21+8/23*a*b^7*x^23+1/25*b^8*x^25

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Maxima [A]  time = 2.61743, size = 122, normalized size = 1.13 \begin{align*} \frac{1}{25} \, b^{8} x^{25} + \frac{8}{23} \, a b^{7} x^{23} + \frac{4}{3} \, a^{2} b^{6} x^{21} + \frac{56}{19} \, a^{3} b^{5} x^{19} + \frac{70}{17} \, a^{4} b^{4} x^{17} + \frac{56}{15} \, a^{5} b^{3} x^{15} + \frac{28}{13} \, a^{6} b^{2} x^{13} + \frac{8}{11} \, a^{7} b x^{11} + \frac{1}{9} \, a^{8} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x
^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b*x^11 + 1/9*a^8*x^9

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Fricas [A]  time = 1.1024, size = 227, normalized size = 2.1 \begin{align*} \frac{1}{25} x^{25} b^{8} + \frac{8}{23} x^{23} b^{7} a + \frac{4}{3} x^{21} b^{6} a^{2} + \frac{56}{19} x^{19} b^{5} a^{3} + \frac{70}{17} x^{17} b^{4} a^{4} + \frac{56}{15} x^{15} b^{3} a^{5} + \frac{28}{13} x^{13} b^{2} a^{6} + \frac{8}{11} x^{11} b a^{7} + \frac{1}{9} x^{9} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*b^8 + 8/23*x^23*b^7*a + 4/3*x^21*b^6*a^2 + 56/19*x^19*b^5*a^3 + 70/17*x^17*b^4*a^4 + 56/15*x^15*b^3*
a^5 + 28/13*x^13*b^2*a^6 + 8/11*x^11*b*a^7 + 1/9*x^9*a^8

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Sympy [A]  time = 0.096444, size = 107, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{9}}{9} + \frac{8 a^{7} b x^{11}}{11} + \frac{28 a^{6} b^{2} x^{13}}{13} + \frac{56 a^{5} b^{3} x^{15}}{15} + \frac{70 a^{4} b^{4} x^{17}}{17} + \frac{56 a^{3} b^{5} x^{19}}{19} + \frac{4 a^{2} b^{6} x^{21}}{3} + \frac{8 a b^{7} x^{23}}{23} + \frac{b^{8} x^{25}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**9/9 + 8*a**7*b*x**11/11 + 28*a**6*b**2*x**13/13 + 56*a**5*b**3*x**15/15 + 70*a**4*b**4*x**17/17 + 56*a
**3*b**5*x**19/19 + 4*a**2*b**6*x**21/3 + 8*a*b**7*x**23/23 + b**8*x**25/25

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Giac [A]  time = 2.84994, size = 122, normalized size = 1.13 \begin{align*} \frac{1}{25} \, b^{8} x^{25} + \frac{8}{23} \, a b^{7} x^{23} + \frac{4}{3} \, a^{2} b^{6} x^{21} + \frac{56}{19} \, a^{3} b^{5} x^{19} + \frac{70}{17} \, a^{4} b^{4} x^{17} + \frac{56}{15} \, a^{5} b^{3} x^{15} + \frac{28}{13} \, a^{6} b^{2} x^{13} + \frac{8}{11} \, a^{7} b x^{11} + \frac{1}{9} \, a^{8} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 8/23*a*b^7*x^23 + 4/3*a^2*b^6*x^21 + 56/19*a^3*b^5*x^19 + 70/17*a^4*b^4*x^17 + 56/15*a^5*b^3*x
^15 + 28/13*a^6*b^2*x^13 + 8/11*a^7*b*x^11 + 1/9*a^8*x^9