3.277 \(\int \frac{(a+b x^2)^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{2 a^2}{\sqrt{x}}+\frac{4}{3} a b x^{3/2}+\frac{2}{7} b^2 x^{7/2} \]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*b^2*x^(7/2))/7

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Rubi [A]  time = 0.0084777, 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} \[ -\frac{2 a^2}{\sqrt{x}}+\frac{4}{3} a b x^{3/2}+\frac{2}{7} b^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*b^2*x^(7/2))/7

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

Mathematica [A]  time = 0.0091576, size = 30, normalized size = 0.88 \[ \frac{2 \left (-21 a^2+14 a b x^2+3 b^2 x^4\right )}{21 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^(3/2),x]

[Out]

(2*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*Sqrt[x])

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{4}-28\,ab{x}^{2}+42\,{a}^{2}}{21}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(3/2),x)

[Out]

-2/21*(-3*b^2*x^4-14*a*b*x^2+21*a^2)/x^(1/2)

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Maxima [A]  time = 2.22246, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{7} \, b^{2} x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.2249, size = 65, normalized size = 1.91 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{4} + 14 \, a b x^{2} - 21 \, a^{2}\right )}}{21 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*b^2*x^4 + 14*a*b*x^2 - 21*a^2)/sqrt(x)

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Sympy [A]  time = 0.97361, size = 32, normalized size = 0.94 \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{2 b^{2} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(3/2),x)

[Out]

-2*a**2/sqrt(x) + 4*a*b*x**(3/2)/3 + 2*b**2*x**(7/2)/7

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Giac [A]  time = 3.03508, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{7} \, b^{2} x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(3/2),x, algorithm="giac")

[Out]

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