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

Optimal. Leaf size=19 \[ 2 a \sqrt{x}+\frac{2}{5} b x^{5/2} \]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5

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Rubi [A]  time = 0.0038726, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ 2 a \sqrt{x}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0042909, size = 19, normalized size = 1. \[ 2 a \sqrt{x}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{2\,b{x}^{2}+10\,a}{5}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.89907, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 2*a*sqrt(x)

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Fricas [A]  time = 1.48494, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{5} \,{\left (b x^{2} + 5 \, a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x^2 + 5*a)*sqrt(x)

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Sympy [A]  time = 0.226014, size = 17, normalized size = 0.89 \begin{align*} 2 a \sqrt{x} + \frac{2 b x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*x**(5/2)/5

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Giac [A]  time = 3.11987, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 2*a*sqrt(x)