3.111 \(\int x^4 (a+b x^2)^8 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 108, 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} \[ \frac {28}{17} a^2 b^6 x^{17}+\frac {56}{15} a^3 b^5 x^{15}+\frac {70}{13} a^4 b^4 x^{13}+\frac {56}{11} a^5 b^3 x^{11}+\frac {28}{9} a^6 b^2 x^9+\frac {8}{7} a^7 b x^7+\frac {a^8 x^5}{5}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 108, normalized size = 1.00 \[ \frac {a^8 x^5}{5}+\frac {8}{7} a^7 b x^7+\frac {28}{9} a^6 b^2 x^9+\frac {56}{11} a^5 b^3 x^{11}+\frac {70}{13} a^4 b^4 x^{13}+\frac {56}{15} a^3 b^5 x^{15}+\frac {28}{17} a^2 b^6 x^{17}+\frac {8}{19} a b^7 x^{19}+\frac {b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^8,x]

[Out]

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

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fricas [A]  time = 0.80, size = 90, normalized size = 0.83 \[ \frac {1}{21} x^{21} b^{8} + \frac {8}{19} x^{19} b^{7} a + \frac {28}{17} x^{17} b^{6} a^{2} + \frac {56}{15} x^{15} b^{5} a^{3} + \frac {70}{13} x^{13} b^{4} a^{4} + \frac {56}{11} x^{11} b^{3} a^{5} + \frac {28}{9} x^{9} b^{2} a^{6} + \frac {8}{7} x^{7} b a^{7} + \frac {1}{5} x^{5} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.10, size = 90, normalized size = 0.83 \[ \frac {1}{21} \, b^{8} x^{21} + \frac {8}{19} \, a b^{7} x^{19} + \frac {28}{17} \, a^{2} b^{6} x^{17} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{5} \, a^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^8,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 91, normalized size = 0.84 \[ \frac {1}{21} b^{8} x^{21}+\frac {8}{19} a \,b^{7} x^{19}+\frac {28}{17} a^{2} b^{6} x^{17}+\frac {56}{15} a^{3} b^{5} x^{15}+\frac {70}{13} a^{4} b^{4} x^{13}+\frac {56}{11} a^{5} b^{3} x^{11}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {8}{7} a^{7} b \,x^{7}+\frac {1}{5} a^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^8,x)

[Out]

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

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maxima [A]  time = 1.34, size = 90, normalized size = 0.83 \[ \frac {1}{21} \, b^{8} x^{21} + \frac {8}{19} \, a b^{7} x^{19} + \frac {28}{17} \, a^{2} b^{6} x^{17} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {70}{13} \, a^{4} b^{4} x^{13} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{5} \, a^{8} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.10, size = 90, normalized size = 0.83 \[ \frac {a^8\,x^5}{5}+\frac {8\,a^7\,b\,x^7}{7}+\frac {28\,a^6\,b^2\,x^9}{9}+\frac {56\,a^5\,b^3\,x^{11}}{11}+\frac {70\,a^4\,b^4\,x^{13}}{13}+\frac {56\,a^3\,b^5\,x^{15}}{15}+\frac {28\,a^2\,b^6\,x^{17}}{17}+\frac {8\,a\,b^7\,x^{19}}{19}+\frac {b^8\,x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^8,x)

[Out]

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

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sympy [A]  time = 0.09, size = 107, normalized size = 0.99 \[ \frac {a^{8} x^{5}}{5} + \frac {8 a^{7} b x^{7}}{7} + \frac {28 a^{6} b^{2} x^{9}}{9} + \frac {56 a^{5} b^{3} x^{11}}{11} + \frac {70 a^{4} b^{4} x^{13}}{13} + \frac {56 a^{3} b^{5} x^{15}}{15} + \frac {28 a^{2} b^{6} x^{17}}{17} + \frac {8 a b^{7} x^{19}}{19} + \frac {b^{8} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**8,x)

[Out]

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

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