3.416 \(\int x (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

[Out]

(a + b*x^2)^(11/2)/(11*b)

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Rubi [A]  time = 0.0034928, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^(9/2),x]

[Out]

(a + b*x^2)^(11/2)/(11*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 )^{9/2} \, dx &=\frac{\left (a+b x^2\right )^{11/2}}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.0055945, size = 18, normalized size = 1. \[ \frac{\left (a+b x^2\right )^{11/2}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^(9/2),x]

[Out]

(a + b*x^2)^(11/2)/(11*b)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{11\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^(9/2),x)

[Out]

1/11*(b*x^2+a)^(11/2)/b

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Maxima [A]  time = 2.12982, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{11}{2}}}{11 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

1/11*(b*x^2 + a)^(11/2)/b

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Fricas [B]  time = 1.64273, size = 139, normalized size = 7.72 \begin{align*} \frac{{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{b x^{2} + a}}{11 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/11*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*sqrt(b*x^2 + a)/b

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Sympy [A]  time = 12.2733, size = 133, normalized size = 7.39 \begin{align*} \begin{cases} \frac{a^{5} \sqrt{a + b x^{2}}}{11 b} + \frac{5 a^{4} x^{2} \sqrt{a + b x^{2}}}{11} + \frac{10 a^{3} b x^{4} \sqrt{a + b x^{2}}}{11} + \frac{10 a^{2} b^{2} x^{6} \sqrt{a + b x^{2}}}{11} + \frac{5 a b^{3} x^{8} \sqrt{a + b x^{2}}}{11} + \frac{b^{4} x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((a**5*sqrt(a + b*x**2)/(11*b) + 5*a**4*x**2*sqrt(a + b*x**2)/11 + 10*a**3*b*x**4*sqrt(a + b*x**2)/11
+ 10*a**2*b**2*x**6*sqrt(a + b*x**2)/11 + 5*a*b**3*x**8*sqrt(a + b*x**2)/11 + b**4*x**10*sqrt(a + b*x**2)/11,
Ne(b, 0)), (a**(9/2)*x**2/2, True))

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Giac [B]  time = 2.59245, size = 267, normalized size = 14.83 \begin{align*} \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 2310 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} + 924 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a^{3} + 198 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{2} + 44 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{3465 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/3465*(315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/2)*a^2 - 2772*(b*x^2 + a)^(5/2
)*a^3 + 2310*(b*x^2 + a)^(3/2)*a^4 + 924*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*a^3 + 198*(15*(b*x^2 +
a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*a^2 + 44*(35*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)
^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*a)/b