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

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

[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

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Rubi [A]  time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 106, normalized size = 1.00 \[ \frac {a^8 x^3}{3}+\frac {8}{5} a^7 b x^5+4 a^6 b^2 x^7+\frac {56}{9} a^5 b^3 x^9+\frac {70}{11} a^4 b^4 x^{11}+\frac {56}{13} a^3 b^5 x^{13}+\frac {28}{15} a^2 b^6 x^{15}+\frac {8}{17} a b^7 x^{17}+\frac {b^8 x^{19}}{19} \]

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

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fricas [A]  time = 0.59, size = 90, normalized size = 0.85 \[ \frac {1}{19} x^{19} b^{8} + \frac {8}{17} x^{17} b^{7} a + \frac {28}{15} x^{15} b^{6} a^{2} + \frac {56}{13} x^{13} b^{5} a^{3} + \frac {70}{11} x^{11} b^{4} a^{4} + \frac {56}{9} x^{9} b^{3} a^{5} + 4 x^{7} b^{2} a^{6} + \frac {8}{5} x^{5} b a^{7} + \frac {1}{3} x^{3} a^{8} \]

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

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giac [A]  time = 0.91, size = 90, normalized size = 0.85 \[ \frac {1}{19} \, b^{8} x^{19} + \frac {8}{17} \, a b^{7} x^{17} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {56}{13} \, a^{3} b^{5} x^{13} + \frac {70}{11} \, a^{4} b^{4} x^{11} + \frac {56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{3} \, a^{8} x^{3} \]

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

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maple [A]  time = 0.00, size = 91, normalized size = 0.86 \[ \frac {1}{19} b^{8} x^{19}+\frac {8}{17} a \,b^{7} x^{17}+\frac {28}{15} a^{2} b^{6} x^{15}+\frac {56}{13} a^{3} b^{5} x^{13}+\frac {70}{11} a^{4} b^{4} x^{11}+\frac {56}{9} a^{5} b^{3} x^{9}+4 a^{6} b^{2} x^{7}+\frac {8}{5} a^{7} b \,x^{5}+\frac {1}{3} a^{8} x^{3} \]

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

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maxima [A]  time = 1.37, size = 90, normalized size = 0.85 \[ \frac {1}{19} \, b^{8} x^{19} + \frac {8}{17} \, a b^{7} x^{17} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {56}{13} \, a^{3} b^{5} x^{13} + \frac {70}{11} \, a^{4} b^{4} x^{11} + \frac {56}{9} \, a^{5} b^{3} x^{9} + 4 \, a^{6} b^{2} x^{7} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{3} \, a^{8} x^{3} \]

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

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mupad [B]  time = 4.97, size = 90, normalized size = 0.85 \[ \frac {a^8\,x^3}{3}+\frac {8\,a^7\,b\,x^5}{5}+4\,a^6\,b^2\,x^7+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {70\,a^4\,b^4\,x^{11}}{11}+\frac {56\,a^3\,b^5\,x^{13}}{13}+\frac {28\,a^2\,b^6\,x^{15}}{15}+\frac {8\,a\,b^7\,x^{17}}{17}+\frac {b^8\,x^{19}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^2)^8,x)

[Out]

(a^8*x^3)/3 + (b^8*x^19)/19 + (8*a^7*b*x^5)/5 + (8*a*b^7*x^17)/17 + 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

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sympy [A]  time = 0.09, size = 105, normalized size = 0.99 \[ \frac {a^{8} x^{3}}{3} + \frac {8 a^{7} b x^{5}}{5} + 4 a^{6} b^{2} x^{7} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {70 a^{4} b^{4} x^{11}}{11} + \frac {56 a^{3} b^{5} x^{13}}{13} + \frac {28 a^{2} b^{6} x^{15}}{15} + \frac {8 a b^{7} x^{17}}{17} + \frac {b^{8} x^{19}}{19} \]

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

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