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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\left (a+b x^2\right )^{11/2}}{11 b} \]

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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fricas [B]  time = 0.71, size = 65, normalized size = 3.61 \[ \frac {{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {b x^{2} + a}}{11 \, b} \]

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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giac [A]  time = 1.19, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*(b*x^2 + a)^(11/2)/b

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{11 b} \]

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 = 1.32, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{11 \, b} \]

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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mupad [B]  time = 4.81, size = 14, normalized size = 0.78 \[ \frac {{\left (b\,x^2+a\right )}^{11/2}}{11\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(11/2)/(11*b)

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sympy [A]  time = 12.96, size = 133, normalized size = 7.39 \[ \begin {cases} \frac {a^{5} \sqrt {a + b x^{2}}}{11 b} + \frac {5 a^{4} x^{2} \sqrt {a + b x^{2}}}{11} + \frac {10 a^{3} b x^{4} \sqrt {a + b x^{2}}}{11} + \frac {10 a^{2} b^{2} x^{6} \sqrt {a + b x^{2}}}{11} + \frac {5 a b^{3} x^{8} \sqrt {a + b x^{2}}}{11} + \frac {b^{4} x^{10} \sqrt {a + b x^{2}}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]

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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