3.276 \(\int \frac{(a+b x^2)^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=34 \[ 2 a^2 \sqrt{x}+\frac{4}{5} a b x^{5/2}+\frac{2}{9} b^2 x^{9/2} \]

[Out]

2*a^2*Sqrt[x] + (4*a*b*x^(5/2))/5 + (2*b^2*x^(9/2))/9

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Rubi [A]  time = 0.0087443, 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} \[ 2 a^2 \sqrt{x}+\frac{4}{5} a b x^{5/2}+\frac{2}{9} b^2 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] + (4*a*b*x^(5/2))/5 + (2*b^2*x^(9/2))/9

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}{\sqrt{x}} \, dx &=\int \left (\frac{a^2}{\sqrt{x}}+2 a b x^{3/2}+b^2 x^{7/2}\right ) \, dx\\ &=2 a^2 \sqrt{x}+\frac{4}{5} a b x^{5/2}+\frac{2}{9} b^2 x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0075841, size = 30, normalized size = 0.88 \[ \frac{2}{45} \sqrt{x} \left (45 a^2+18 a b x^2+5 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(45*a^2 + 18*a*b*x^2 + 5*b^2*x^4))/45

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{10\,{b}^{2}{x}^{4}+36\,ab{x}^{2}+90\,{a}^{2}}{45}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^(1/2),x)

[Out]

2/45*x^(1/2)*(5*b^2*x^4+18*a*b*x^2+45*a^2)

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Maxima [A]  time = 2.2178, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + 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^(1/2),x, algorithm="maxima")

[Out]

2/9*b^2*x^(9/2) + 4/5*a*b*x^(5/2) + 2*a^2*sqrt(x)

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Fricas [A]  time = 1.16672, size = 65, normalized size = 1.91 \begin{align*} \frac{2}{45} \,{\left (5 \, b^{2} x^{4} + 18 \, a b x^{2} + 45 \, a^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^(1/2),x, algorithm="fricas")

[Out]

2/45*(5*b^2*x^4 + 18*a*b*x^2 + 45*a^2)*sqrt(x)

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Sympy [A]  time = 0.730421, size = 32, normalized size = 0.94 \begin{align*} 2 a^{2} \sqrt{x} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{2 b^{2} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**(1/2),x)

[Out]

2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**(9/2)/9

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Giac [A]  time = 2.61957, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/9*b^2*x^(9/2) + 4/5*a*b*x^(5/2) + 2*a^2*sqrt(x)