3.270 \(\int \frac {a+b x^2}{x^{5/2}} \, dx\)

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

[Out]

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

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]

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fricas [A]  time = 0.95, size = 15, normalized size = 0.79 \[ \frac {2 \, {\left (3 \, b x^{2} - a\right )}}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(3*b*x^2 - a)/x^(3/2)

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giac [A]  time = 0.60, size = 13, normalized size = 0.68 \[ 2 \, b \sqrt {x} - \frac {2 \, a}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="giac")

[Out]

2*b*sqrt(x) - 2/3*a/x^(3/2)

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ -\frac {2 \left (-3 b \,x^{2}+a \right )}{3 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^(5/2),x)

[Out]

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

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maxima [A]  time = 1.32, size = 13, normalized size = 0.68 \[ 2 \, b \sqrt {x} - \frac {2 \, a}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="maxima")

[Out]

2*b*sqrt(x) - 2/3*a/x^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^(5/2),x)

[Out]

-(2*a - 6*b*x^2)/(3*x^(3/2))

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sympy [A]  time = 0.56, size = 17, normalized size = 0.89 \[ - \frac {2 a}{3 x^{\frac {3}{2}}} + 2 b \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) + 2*b*sqrt(x)

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