3.110 \(\int x^6 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=108 \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{56}{17} a^3 b^5 x^{17}+\frac{14}{3} a^4 b^4 x^{15}+\frac{56}{13} a^5 b^3 x^{13}+\frac{28}{11} a^6 b^2 x^{11}+\frac{8}{9} a^7 b x^9+\frac{a^8 x^7}{7}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{23}}{23} \]

[Out]

(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^
5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23

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Rubi [A]  time = 0.0390258, 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} \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{56}{17} a^3 b^5 x^{17}+\frac{14}{3} a^4 b^4 x^{15}+\frac{56}{13} a^5 b^3 x^{13}+\frac{28}{11} a^6 b^2 x^{11}+\frac{8}{9} a^7 b x^9+\frac{a^8 x^7}{7}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^2)^8,x]

[Out]

(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^
5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23

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

Mathematica [A]  time = 0.0024546, size = 108, normalized size = 1. \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{56}{17} a^3 b^5 x^{17}+\frac{14}{3} a^4 b^4 x^{15}+\frac{56}{13} a^5 b^3 x^{13}+\frac{28}{11} a^6 b^2 x^{11}+\frac{8}{9} a^7 b x^9+\frac{a^8 x^7}{7}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{23}}{23} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^2)^8,x]

[Out]

(a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^
5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23

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Maple [A]  time = 0.001, size = 91, normalized size = 0.8 \begin{align*}{\frac{{a}^{8}{x}^{7}}{7}}+{\frac{8\,{a}^{7}b{x}^{9}}{9}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{11}}{11}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{13}}{13}}+{\frac{14\,{a}^{4}{b}^{4}{x}^{15}}{3}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{17}}{17}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{19}}{19}}+{\frac{8\,a{b}^{7}{x}^{21}}{21}}+{\frac{{b}^{8}{x}^{23}}{23}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^2+a)^8,x)

[Out]

1/7*a^8*x^7+8/9*a^7*b*x^9+28/11*a^6*b^2*x^11+56/13*a^5*b^3*x^13+14/3*a^4*b^4*x^15+56/17*a^3*b^5*x^17+28/19*a^2
*b^6*x^19+8/21*a*b^7*x^21+1/23*b^8*x^23

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Maxima [A]  time = 2.5572, size = 122, normalized size = 1.13 \begin{align*} \frac{1}{23} \, b^{8} x^{23} + \frac{8}{21} \, a b^{7} x^{21} + \frac{28}{19} \, a^{2} b^{6} x^{19} + \frac{56}{17} \, a^{3} b^{5} x^{17} + \frac{14}{3} \, a^{4} b^{4} x^{15} + \frac{56}{13} \, a^{5} b^{3} x^{13} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{8}{9} \, a^{7} b x^{9} + \frac{1}{7} \, a^{8} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*
x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b*x^9 + 1/7*a^8*x^7

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Fricas [A]  time = 1.17847, size = 225, normalized size = 2.08 \begin{align*} \frac{1}{23} x^{23} b^{8} + \frac{8}{21} x^{21} b^{7} a + \frac{28}{19} x^{19} b^{6} a^{2} + \frac{56}{17} x^{17} b^{5} a^{3} + \frac{14}{3} x^{15} b^{4} a^{4} + \frac{56}{13} x^{13} b^{3} a^{5} + \frac{28}{11} x^{11} b^{2} a^{6} + \frac{8}{9} x^{9} b a^{7} + \frac{1}{7} x^{7} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/23*x^23*b^8 + 8/21*x^21*b^7*a + 28/19*x^19*b^6*a^2 + 56/17*x^17*b^5*a^3 + 14/3*x^15*b^4*a^4 + 56/13*x^13*b^3
*a^5 + 28/11*x^11*b^2*a^6 + 8/9*x^9*b*a^7 + 1/7*x^7*a^8

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Sympy [A]  time = 0.084061, size = 107, normalized size = 0.99 \begin{align*} \frac{a^{8} x^{7}}{7} + \frac{8 a^{7} b x^{9}}{9} + \frac{28 a^{6} b^{2} x^{11}}{11} + \frac{56 a^{5} b^{3} x^{13}}{13} + \frac{14 a^{4} b^{4} x^{15}}{3} + \frac{56 a^{3} b^{5} x^{17}}{17} + \frac{28 a^{2} b^{6} x^{19}}{19} + \frac{8 a b^{7} x^{21}}{21} + \frac{b^{8} x^{23}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**2+a)**8,x)

[Out]

a**8*x**7/7 + 8*a**7*b*x**9/9 + 28*a**6*b**2*x**11/11 + 56*a**5*b**3*x**13/13 + 14*a**4*b**4*x**15/3 + 56*a**3
*b**5*x**17/17 + 28*a**2*b**6*x**19/19 + 8*a*b**7*x**21/21 + b**8*x**23/23

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Giac [A]  time = 1.74634, size = 122, normalized size = 1.13 \begin{align*} \frac{1}{23} \, b^{8} x^{23} + \frac{8}{21} \, a b^{7} x^{21} + \frac{28}{19} \, a^{2} b^{6} x^{19} + \frac{56}{17} \, a^{3} b^{5} x^{17} + \frac{14}{3} \, a^{4} b^{4} x^{15} + \frac{56}{13} \, a^{5} b^{3} x^{13} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{8}{9} \, a^{7} b x^{9} + \frac{1}{7} \, a^{8} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/23*b^8*x^23 + 8/21*a*b^7*x^21 + 28/19*a^2*b^6*x^19 + 56/17*a^3*b^5*x^17 + 14/3*a^4*b^4*x^15 + 56/13*a^5*b^3*
x^13 + 28/11*a^6*b^2*x^11 + 8/9*a^7*b*x^9 + 1/7*a^8*x^7