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

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

[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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Rubi [A]  time = 0.039408, 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} \[ \frac{28}{17} a^2 b^6 x^{17}+\frac{56}{15} a^3 b^5 x^{15}+\frac{70}{13} a^4 b^4 x^{13}+\frac{56}{11} a^5 b^3 x^{11}+\frac{28}{9} a^6 b^2 x^9+\frac{8}{7} a^7 b x^7+\frac{a^8 x^5}{5}+\frac{8}{19} a b^7 x^{19}+\frac{b^8 x^{21}}{21} \]

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*}

Mathematica [A]  time = 0.0023435, size = 108, normalized size = 1. \[ \frac{28}{17} a^2 b^6 x^{17}+\frac{56}{15} a^3 b^5 x^{15}+\frac{70}{13} a^4 b^4 x^{13}+\frac{56}{11} a^5 b^3 x^{11}+\frac{28}{9} a^6 b^2 x^9+\frac{8}{7} a^7 b x^7+\frac{a^8 x^5}{5}+\frac{8}{19} a b^7 x^{19}+\frac{b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

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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Maple [A]  time = 0.001, size = 91, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

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 = 2.19793, size = 122, normalized size = 1.13 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.12468, size = 224, normalized size = 2.07 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 0.086483, size = 107, normalized size = 0.99 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 2.86162, size = 122, normalized size = 1.13 \begin{align*} \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} \end{align*}

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