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

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

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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) - (2*b)/Sqrt[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2 (a+3 b x)}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(5/2),x]

[Out]

(-2*(a + 3*b*x))/(3*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2 (a+3 b x)}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)/x^(5/2),x]

[Out]

(-2*(a + 3*b*x))/(3*x^(3/2))

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fricas [A]  time = 0.83, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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giac [A]  time = 0.97, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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maple [A]  time = 0.00, size = 12, normalized size = 0.63 \begin {gather*} -\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} -\frac {2\,a+6\,b\,x}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(5/2),x)

[Out]

-(2*a + 6*b*x)/(3*x^(3/2))

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sympy [A]  time = 0.56, size = 19, normalized size = 1.00 \begin {gather*} - \frac {2 a}{3 x^{\frac {3}{2}}} - \frac {2 b}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(3*x**(3/2)) - 2*b/sqrt(x)

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