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

Optimal. Leaf size=19 \[ \frac {2}{3} b x^{3/2}-\frac {2 a}{\sqrt {x}} \]

[Out]

2/3*b*x^(3/2)-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} \[ \frac {2}{3} b x^{3/2}-\frac {2 a}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {2}{3} b x^{3/2}-\frac {2 a}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3

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fricas [A]  time = 0.81, size = 14, normalized size = 0.74 \[ \frac {2 \, {\left (b x^{2} - 3 \, a\right )}}{3 \, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x^2 - 3*a)/sqrt(x)

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giac [A]  time = 0.59, size = 13, normalized size = 0.68 \[ \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) - 2*a/sqrt(x)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ -\frac {2 \left (-b \,x^{2}+3 a \right )}{3 \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 13, normalized size = 0.68 \[ \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) - 2*a/sqrt(x)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.79 \[ -\frac {6\,a-2\,b\,x^2}{3\,\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^(3/2),x)

[Out]

-(6*a - 2*b*x^2)/(3*x^(1/2))

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sympy [A]  time = 0.38, size = 17, normalized size = 0.89 \[ - \frac {2 a}{\sqrt {x}} + \frac {2 b x^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) + 2*b*x**(3/2)/3

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