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

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

[Out]

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

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Rubi [A]  time = 0.0074047, 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} \[ -\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 b^2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) - (4*a*b)/Sqrt[x] + 2*b^2*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)^2}{x^{5/2}} \, dx &=\int \left (\frac{a^2}{x^{5/2}}+\frac{2 a b}{x^{3/2}}+\frac{b^2}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 b^2 \sqrt{x}\\ \end{align*}

Mathematica [A]  time = 0.0087331, size = 26, normalized size = 0.81 \[ -\frac{2 \left (a^2+6 a b x-3 b^2 x^2\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.7 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{2}+12\,abx+2\,{a}^{2}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05362, size = 31, normalized size = 0.97 \begin{align*} 2 \, b^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.40446, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.883137, size = 31, normalized size = 0.97 \begin{align*} - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 a b}{\sqrt{x}} + 2 b^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21539, size = 31, normalized size = 0.97 \begin{align*} 2 \, b^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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