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]

1/7*(b*x^2+a)^(7/2)/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} \[ \frac {\left (a+b x^2\right )^{7/2}}{7 b} \]

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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (a+b x^2\right )^{7/2}}{7 b} \]

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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fricas [B]  time = 0.46, size = 43, normalized size = 2.39 \[ \frac {{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {b x^{2} + a}}{7 \, b} \]

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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giac [A]  time = 1.20, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{7 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(b*x^2 + a)^(7/2)/b

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b} \]

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 = 1.34, size = 14, normalized size = 0.78 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{7 \, b} \]

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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mupad [B]  time = 4.60, size = 14, normalized size = 0.78 \[ \frac {{\left (b\,x^2+a\right )}^{7/2}}{7\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^(5/2),x)

[Out]

(a + b*x^2)^(7/2)/(7*b)

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sympy [A]  time = 2.10, size = 85, normalized size = 4.72 \[ \begin {cases} \frac {a^{3} \sqrt {a + b x^{2}}}{7 b} + \frac {3 a^{2} x^{2} \sqrt {a + b x^{2}}}{7} + \frac {3 a b x^{4} \sqrt {a + b x^{2}}}{7} + \frac {b^{2} x^{6} \sqrt {a + b x^{2}}}{7} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]

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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