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

Optimal. Leaf size=106 \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{56}{13} a^3 b^5 x^{13}+\frac{70}{11} a^4 b^4 x^{11}+\frac{56}{9} a^5 b^3 x^9+4 a^6 b^2 x^7+\frac{8}{5} a^7 b x^5+\frac{a^8 x^3}{3}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{19}}{19} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0402238, antiderivative size = 106, 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} \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{56}{13} a^3 b^5 x^{13}+\frac{70}{11} a^4 b^4 x^{11}+\frac{56}{9} a^5 b^3 x^9+4 a^6 b^2 x^7+\frac{8}{5} a^7 b x^5+\frac{a^8 x^3}{3}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0023477, size = 106, normalized size = 1. \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{56}{13} a^3 b^5 x^{13}+\frac{70}{11} a^4 b^4 x^{11}+\frac{56}{9} a^5 b^3 x^9+4 a^6 b^2 x^7+\frac{8}{5} a^7 b x^5+\frac{a^8 x^3}{3}+\frac{8}{17} a b^7 x^{17}+\frac{b^8 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2)^8,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 91, normalized size = 0.9 \begin{align*}{\frac{{a}^{8}{x}^{3}}{3}}+{\frac{8\,{a}^{7}b{x}^{5}}{5}}+4\,{a}^{6}{b}^{2}{x}^{7}+{\frac{56\,{a}^{5}{b}^{3}{x}^{9}}{9}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{11}}{11}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{13}}{13}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{15}}{15}}+{\frac{8\,a{b}^{7}{x}^{17}}{17}}+{\frac{{b}^{8}{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.37976, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{19} \, b^{8} x^{19} + \frac{8}{17} \, a b^{7} x^{17} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{56}{13} \, a^{3} b^{5} x^{13} + \frac{70}{11} \, a^{4} b^{4} x^{11} + \frac{56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac{8}{5} \, a^{7} b x^{5} + \frac{1}{3} \, a^{8} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.01202, size = 217, normalized size = 2.05 \begin{align*} \frac{1}{19} x^{19} b^{8} + \frac{8}{17} x^{17} b^{7} a + \frac{28}{15} x^{15} b^{6} a^{2} + \frac{56}{13} x^{13} b^{5} a^{3} + \frac{70}{11} x^{11} b^{4} a^{4} + \frac{56}{9} x^{9} b^{3} a^{5} + 4 x^{7} b^{2} a^{6} + \frac{8}{5} x^{5} b a^{7} + \frac{1}{3} x^{3} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.087928, size = 105, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{3}}{3} + \frac{8 a^{7} b x^{5}}{5} + 4 a^{6} b^{2} x^{7} + \frac{56 a^{5} b^{3} x^{9}}{9} + \frac{70 a^{4} b^{4} x^{11}}{11} + \frac{56 a^{3} b^{5} x^{13}}{13} + \frac{28 a^{2} b^{6} x^{15}}{15} + \frac{8 a b^{7} x^{17}}{17} + \frac{b^{8} x^{19}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.36252, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{19} \, b^{8} x^{19} + \frac{8}{17} \, a b^{7} x^{17} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{56}{13} \, a^{3} b^{5} x^{13} + \frac{70}{11} \, a^{4} b^{4} x^{11} + \frac{56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac{8}{5} \, a^{7} b x^{5} + \frac{1}{3} \, a^{8} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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