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

Optimal. Leaf size=17 \[ 2 b \sqrt {x}-\frac {2 a}{\sqrt {x}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 17, 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*} 2 b \sqrt {x}-\frac {2 a}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(3/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 (b x-a)}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(3/2),x]

[Out]

(2*(-a + b*x))/Sqrt[x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 (b x-a)}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)/x^(3/2),x]

[Out]

(2*(-a + b*x))/Sqrt[x]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 12, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (b x - a\right )}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x - a)/sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.92, size = 13, normalized size = 0.76 \begin {gather*} 2 \, b \sqrt {x} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 12, normalized size = 0.71 \begin {gather*} -\frac {2 \left (-b x +a \right )}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 13, normalized size = 0.76 \begin {gather*} 2 \, b \sqrt {x} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 11, normalized size = 0.65 \begin {gather*} -\frac {2\,\left (a-b\,x\right )}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(3/2),x)

[Out]

-(2*(a - b*x))/x^(1/2)

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 15, normalized size = 0.88 \begin {gather*} - \frac {2 a}{\sqrt {x}} + 2 b \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) + 2*b*sqrt(x)

________________________________________________________________________________________