3.390 \(\int x (a+b x^2)^{5/2} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b} \]

[Out]

(a + b*x^2)^(7/2)/(7*b)

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Rubi [A]  time = 0.0034692, 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} \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^(5/2),x]

[Out]

(a + b*x^2)^(7/2)/(7*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 )^{5/2} \, dx &=\frac{\left (a+b x^2\right )^{7/2}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0045158, size = 18, normalized size = 1. \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^(5/2),x]

[Out]

(a + b*x^2)^(7/2)/(7*b)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^(5/2),x)

[Out]

1/7*(b*x^2+a)^(7/2)/b

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Maxima [A]  time = 1.94114, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

1/7*(b*x^2 + a)^(7/2)/b

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Fricas [B]  time = 1.58128, size = 90, normalized size = 5. \begin{align*} \frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/7*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)/b

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Sympy [A]  time = 1.94809, size = 85, normalized size = 4.72 \begin{align*} \begin{cases} \frac{a^{3} \sqrt{a + b x^{2}}}{7 b} + \frac{3 a^{2} x^{2} \sqrt{a + b x^{2}}}{7} + \frac{3 a b x^{4} \sqrt{a + b x^{2}}}{7} + \frac{b^{2} x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{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)**(5/2),x)

[Out]

Piecewise((a**3*sqrt(a + b*x**2)/(7*b) + 3*a**2*x**2*sqrt(a + b*x**2)/7 + 3*a*b*x**4*sqrt(a + b*x**2)/7 + b**2
*x**6*sqrt(a + b*x**2)/7, Ne(b, 0)), (a**(5/2)*x**2/2, True))

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Giac [B]  time = 1.67839, size = 95, normalized size = 5.28 \begin{align*} \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 14 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a}{105 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/105*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 70*(b*x^2 + a)^(3/2)*a^2 + 14*(3*(b*x^2 + a)^(5/2) - 5*
(b*x^2 + a)^(3/2)*a)*a)/b