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

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^9}{18 b} \]

[Out]

1/18*(b*x^2+a)^9/b

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \[ \frac {\left (a+b x^2\right )^9}{18 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \left (a+b x^2\right )^8 \, dx &=\frac {\left (a+b x^2\right )^9}{18 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {\left (a+b x^2\right )^9}{18 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^8,x]

[Out]

(a + b*x^2)^9/(18*b)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 90, normalized size = 5.62 \[ \frac {1}{18} x^{18} b^{8} + \frac {1}{2} x^{16} b^{7} a + 2 x^{14} b^{6} a^{2} + \frac {14}{3} x^{12} b^{5} a^{3} + 7 x^{10} b^{4} a^{4} + 7 x^{8} b^{3} a^{5} + \frac {14}{3} x^{6} b^{2} a^{6} + 2 x^{4} b a^{7} + \frac {1}{2} x^{2} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18*b^8 + 1/2*x^16*b^7*a + 2*x^14*b^6*a^2 + 14/3*x^12*b^5*a^3 + 7*x^10*b^4*a^4 + 7*x^8*b^3*a^5 + 14/3*x^
6*b^2*a^6 + 2*x^4*b*a^7 + 1/2*x^2*a^8

________________________________________________________________________________________

giac [A]  time = 1.12, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{2} + a\right )}^{9}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b

________________________________________________________________________________________

maple [B]  time = 0.00, size = 91, normalized size = 5.69 \[ \frac {1}{18} b^{8} x^{18}+\frac {1}{2} a \,b^{7} x^{16}+2 a^{2} b^{6} x^{14}+\frac {14}{3} a^{3} b^{5} x^{12}+7 a^{4} b^{4} x^{10}+7 a^{5} b^{3} x^{8}+\frac {14}{3} a^{6} b^{2} x^{6}+2 a^{7} b \,x^{4}+\frac {1}{2} a^{8} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*b^8*x^18+1/2*a*b^7*x^16+2*a^2*b^6*x^14+14/3*a^3*b^5*x^12+7*a^4*b^4*x^10+7*a^5*b^3*x^8+14/3*a^6*b^2*x^6+2*
a^7*b*x^4+1/2*a^8*x^2

________________________________________________________________________________________

maxima [A]  time = 1.28, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{2} + a\right )}^{9}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^2 + a)^9/b

________________________________________________________________________________________

mupad [B]  time = 4.61, size = 14, normalized size = 0.88 \[ \frac {{\left (b\,x^2+a\right )}^9}{18\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^8,x)

[Out]

(a + b*x^2)^9/(18*b)

________________________________________________________________________________________

sympy [B]  time = 0.08, size = 99, normalized size = 6.19 \[ \frac {a^{8} x^{2}}{2} + 2 a^{7} b x^{4} + \frac {14 a^{6} b^{2} x^{6}}{3} + 7 a^{5} b^{3} x^{8} + 7 a^{4} b^{4} x^{10} + \frac {14 a^{3} b^{5} x^{12}}{3} + 2 a^{2} b^{6} x^{14} + \frac {a b^{7} x^{16}}{2} + \frac {b^{8} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**2/2 + 2*a**7*b*x**4 + 14*a**6*b**2*x**6/3 + 7*a**5*b**3*x**8 + 7*a**4*b**4*x**10 + 14*a**3*b**5*x**12/
3 + 2*a**2*b**6*x**14 + a*b**7*x**16/2 + b**8*x**18/18

________________________________________________________________________________________