3.448 \(\int \frac{(a+b x)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a^3}{3 x^{3/2}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0107336, antiderivative size = 47, 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} \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a^3}{3 x^{3/2}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) - (6*a^2*b)/Sqrt[x] + 6*a*b^2*Sqrt[x] + (2*b^3*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)^3}{x^{5/2}} \, dx &=\int \left (\frac{a^3}{x^{5/2}}+\frac{3 a^2 b}{x^{3/2}}+\frac{3 a b^2}{\sqrt{x}}+b^3 \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^3}{3 x^{3/2}}-\frac{6 a^2 b}{\sqrt{x}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0145666, size = 38, normalized size = 0.81 \[ \frac{2 \left (-9 a^2 b x-a^3+9 a b^2 x^2+b^3 x^3\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^(5/2),x]

[Out]

(2*(-a^3 - 9*a^2*b*x + 9*a*b^2*x^2 + b^3*x^3))/(3*x^(3/2))

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Maple [A]  time = 0.003, size = 34, normalized size = 0.7 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}-18\,a{b}^{2}{x}^{2}+18\,{a}^{2}bx+2\,{a}^{3}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^(5/2),x)

[Out]

-2/3*(-b^3*x^3-9*a*b^2*x^2+9*a^2*b*x+a^3)/x^(3/2)

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Maxima [A]  time = 1.10933, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \,{\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/3*b^3*x^(3/2) + 6*a*b^2*sqrt(x) - 2/3*(9*a^2*b*x + a^3)/x^(3/2)

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Fricas [A]  time = 1.54299, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 9 \, a b^{2} x^{2} - 9 \, a^{2} b x - a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(b^3*x^3 + 9*a*b^2*x^2 - 9*a^2*b*x - a^3)/x^(3/2)

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Sympy [A]  time = 1.19895, size = 46, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 a^{2} b}{\sqrt{x}} + 6 a b^{2} \sqrt{x} + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**(5/2),x)

[Out]

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

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Giac [A]  time = 1.20573, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \,{\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^(5/2),x, algorithm="giac")

[Out]

2/3*b^3*x^(3/2) + 6*a*b^2*sqrt(x) - 2/3*(9*a^2*b*x + a^3)/x^(3/2)