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

Optimal. Leaf size=34 \[ -\frac{2 a^2}{3 x^{3/2}}+4 a b \sqrt{x}+\frac{2}{5} b^2 x^{5/2} \]

[Out]

(-2*a^2)/(3*x^(3/2)) + 4*a*b*Sqrt[x] + (2*b^2*x^(5/2))/5

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Rubi [A]  time = 0.0082254, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ -\frac{2 a^2}{3 x^{3/2}}+4 a b \sqrt{x}+\frac{2}{5} b^2 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) + 4*a*b*Sqrt[x] + (2*b^2*x^(5/2))/5

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac{a^2}{x^{5/2}}+\frac{2 a b}{\sqrt{x}}+b^2 x^{3/2}\right ) \, dx\\ &=-\frac{2 a^2}{3 x^{3/2}}+4 a b \sqrt{x}+\frac{2}{5} b^2 x^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0091456, size = 30, normalized size = 0.88 \[ \frac{2 \left (-5 a^2+30 a b x^2+3 b^2 x^4\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^(5/2),x]

[Out]

(2*(-5*a^2 + 30*a*b*x^2 + 3*b^2*x^4))/(15*x^(3/2))

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{4}-60\,ab{x}^{2}+10\,{a}^{2}}{15}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*b^2*x^4-30*a*b*x^2+5*a^2)/x^(3/2)

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Maxima [A]  time = 1.22346, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b^2*x^(5/2) + 4*a*b*sqrt(x) - 2/3*a^2/x^(3/2)

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Fricas [A]  time = 1.1983, size = 63, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{4} + 30 \, a b x^{2} - 5 \, a^{2}\right )}}{15 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*b^2*x^4 + 30*a*b*x^2 - 5*a^2)/x^(3/2)

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Sympy [A]  time = 1.21377, size = 32, normalized size = 0.94 \begin{align*} - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} + 4 a b \sqrt{x} + \frac{2 b^{2} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(5/2),x)

[Out]

-2*a**2/(3*x**(3/2)) + 4*a*b*sqrt(x) + 2*b**2*x**(5/2)/5

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Giac [A]  time = 2.34887, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b^2*x^(5/2) + 4*a*b*sqrt(x) - 2/3*a^2/x^(3/2)