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

Optimal. Leaf size=101 \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{56}{11} a^3 b^5 x^{11}+\frac{70}{9} a^4 b^4 x^9+8 a^5 b^3 x^7+\frac{28}{5} a^6 b^2 x^5+\frac{8}{3} a^7 b x^3+a^8 x+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{17}}{17} \]

[Out]

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

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Rubi [A]  time = 0.0373987, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {194} \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{56}{11} a^3 b^5 x^{11}+\frac{70}{9} a^4 b^4 x^9+8 a^5 b^3 x^7+\frac{28}{5} a^6 b^2 x^5+\frac{8}{3} a^7 b x^3+a^8 x+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8,x]

[Out]

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0014038, size = 101, normalized size = 1. \[ \frac{28}{13} a^2 b^6 x^{13}+\frac{56}{11} a^3 b^5 x^{11}+\frac{70}{9} a^4 b^4 x^9+8 a^5 b^3 x^7+\frac{28}{5} a^6 b^2 x^5+\frac{8}{3} a^7 b x^3+a^8 x+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8,x]

[Out]

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

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Maple [A]  time = 0.001, size = 88, normalized size = 0.9 \begin{align*}{a}^{8}x+{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{5}}{5}}+8\,{a}^{5}{b}^{3}{x}^{7}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{11}}{11}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{13}}{13}}+{\frac{8\,a{b}^{7}{x}^{15}}{15}}+{\frac{{b}^{8}{x}^{17}}{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.86977, size = 117, normalized size = 1.16 \begin{align*} \frac{1}{17} \, b^{8} x^{17} + \frac{8}{15} \, a b^{7} x^{15} + \frac{28}{13} \, a^{2} b^{6} x^{13} + \frac{56}{11} \, a^{3} b^{5} x^{11} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 8 \, a^{5} b^{3} x^{7} + \frac{28}{5} \, a^{6} b^{2} x^{5} + \frac{8}{3} \, a^{7} b x^{3} + a^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12055, size = 207, normalized size = 2.05 \begin{align*} \frac{1}{17} x^{17} b^{8} + \frac{8}{15} x^{15} b^{7} a + \frac{28}{13} x^{13} b^{6} a^{2} + \frac{56}{11} x^{11} b^{5} a^{3} + \frac{70}{9} x^{9} b^{4} a^{4} + 8 x^{7} b^{3} a^{5} + \frac{28}{5} x^{5} b^{2} a^{6} + \frac{8}{3} x^{3} b a^{7} + x a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.080738, size = 102, normalized size = 1.01 \begin{align*} a^{8} x + \frac{8 a^{7} b x^{3}}{3} + \frac{28 a^{6} b^{2} x^{5}}{5} + 8 a^{5} b^{3} x^{7} + \frac{70 a^{4} b^{4} x^{9}}{9} + \frac{56 a^{3} b^{5} x^{11}}{11} + \frac{28 a^{2} b^{6} x^{13}}{13} + \frac{8 a b^{7} x^{15}}{15} + \frac{b^{8} x^{17}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.8359, size = 117, normalized size = 1.16 \begin{align*} \frac{1}{17} \, b^{8} x^{17} + \frac{8}{15} \, a b^{7} x^{15} + \frac{28}{13} \, a^{2} b^{6} x^{13} + \frac{56}{11} \, a^{3} b^{5} x^{11} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 8 \, a^{5} b^{3} x^{7} + \frac{28}{5} \, a^{6} b^{2} x^{5} + \frac{8}{3} \, a^{7} b x^{3} + a^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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