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

Optimal. Leaf size=45 \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]

[Out]

(-2*a^3)/Sqrt[x] + 6*a^2*b*Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(5/2))/5

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Rubi [A]  time = 0.0111342, antiderivative size = 45, 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} \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + 6*a^2*b*Sqrt[x] + 2*a*b^2*x^(3/2) + (2*b^3*x^(5/2))/5

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

Mathematica [A]  time = 0.0114963, size = 38, normalized size = 0.84 \[ \frac{2 \left (15 a^2 b x-5 a^3+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^(3/2),x]

[Out]

(2*(-5*a^3 + 15*a^2*b*x + 5*a*b^2*x^2 + b^3*x^3))/(5*Sqrt[x])

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}-10\,a{b}^{2}{x}^{2}-30\,{a}^{2}bx+10\,{a}^{3}}{5}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(-b^3*x^3-5*a*b^2*x^2-15*a^2*b*x+5*a^3)/x^(1/2)

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Maxima [A]  time = 1.09566, size = 47, normalized size = 1.04 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b^3*x^(5/2) + 2*a*b^2*x^(3/2) + 6*a^2*b*sqrt(x) - 2*a^3/sqrt(x)

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Fricas [A]  time = 1.51226, size = 78, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 5 \, a b^{2} x^{2} + 15 \, a^{2} b x - 5 \, a^{3}\right )}}{5 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b^3*x^3 + 5*a*b^2*x^2 + 15*a^2*b*x - 5*a^3)/sqrt(x)

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Sympy [A]  time = 0.897178, size = 44, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + 6 a^{2} b \sqrt{x} + 2 a b^{2} x^{\frac{3}{2}} + \frac{2 b^{3} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/sqrt(x) + 6*a**2*b*sqrt(x) + 2*a*b**2*x**(3/2) + 2*b**3*x**(5/2)/5

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Giac [A]  time = 1.22818, size = 47, normalized size = 1.04 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b^3*x^(5/2) + 2*a*b^2*x^(3/2) + 6*a^2*b*sqrt(x) - 2*a^3/sqrt(x)