3.279 \(\int \frac{(a+b x^2)^2}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.008181, 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} \[ -\frac{2 a^2}{5 x^{5/2}}-\frac{4 a b}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^(7/2),x]

[Out]

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

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^{7/2}} \, dx &=\int \left (\frac{a^2}{x^{7/2}}+\frac{2 a b}{x^{3/2}}+b^2 \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^2}{5 x^{5/2}}-\frac{4 a b}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0087118, size = 30, normalized size = 0.88 \[ \frac{2 \left (-3 a^2-30 a b x^2+5 b^2 x^4\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-10\,{b}^{2}{x}^{4}+60\,ab{x}^{2}+6\,{a}^{2}}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.93498, size = 34, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2),x, algorithm="maxima")

[Out]

2/3*b^2*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)

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Fricas [A]  time = 1.30228, size = 63, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (5 \, b^{2} x^{4} - 30 \, a b x^{2} - 3 \, a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.82759, size = 32, normalized size = 0.94 \begin{align*} - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(7/2),x)

[Out]

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

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Giac [A]  time = 2.65104, size = 34, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(7/2),x, algorithm="giac")

[Out]

2/3*b^2*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)