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

Optimal. Leaf size=32 \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]

[Out]

(-2*a^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*b^2*x^(3/2))/3

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Rubi [A]  time = 0.0067661, antiderivative size = 32, normalized size of antiderivative = 1., 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 = {43} \[ -\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*b^2*x^(3/2))/3

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)^2}{x^{3/2}} \, dx &=\int \left (\frac{a^2}{x^{3/2}}+\frac{2 a b}{\sqrt{x}}+b^2 \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} b^2 x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0094845, size = 27, normalized size = 0.84 \[ \frac{2 \left (-3 a^2+6 a b x+b^2 x^2\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^(3/2),x]

[Out]

(2*(-3*a^2 + 6*a*b*x + b^2*x^2))/(3*Sqrt[x])

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}-12\,abx+6\,{a}^{2}}{3}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-b^2*x^2-6*a*b*x+3*a^2)/x^(1/2)

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Maxima [A]  time = 1.02959, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2) + 4*a*b*sqrt(x) - 2*a^2/sqrt(x)

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Fricas [A]  time = 1.46386, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b^2*x^2 + 6*a*b*x - 3*a^2)/sqrt(x)

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Sympy [A]  time = 0.730087, size = 31, normalized size = 0.97 \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} + 4 a b \sqrt{x} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/sqrt(x) + 4*a*b*sqrt(x) + 2*b**2*x**(3/2)/3

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Giac [A]  time = 1.20122, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2) + 4*a*b*sqrt(x) - 2*a^2/sqrt(x)