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]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ 2 a \sqrt {x}+\frac {2}{5} b x^{5/2} \]

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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fricas [A]  time = 0.63, size = 14, normalized size = 0.74 \[ \frac {2}{5} \, {\left (b x^{2} + 5 \, a\right )} \sqrt {x} \]

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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giac [A]  time = 0.63, size = 13, normalized size = 0.68 \[ \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \]

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)

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maple [A]  time = 0.00, size = 15, normalized size = 0.79 \[ \frac {2 \left (b \,x^{2}+5 a \right ) \sqrt {x}}{5} \]

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 = 1.30, size = 13, normalized size = 0.68 \[ \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \]

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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mupad [B]  time = 0.03, size = 14, normalized size = 0.74 \[ \frac {2\,\sqrt {x}\,\left (b\,x^2+5\,a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^(1/2),x)

[Out]

(2*x^(1/2)*(5*a + b*x^2))/5

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sympy [A]  time = 0.23, size = 17, normalized size = 0.89 \[ 2 a \sqrt {x} + \frac {2 b x^{\frac {5}{2}}}{5} \]

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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