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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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} \end {gather*}

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 = 0.01, size = 38, normalized size = 0.84 \begin {gather*} \frac {2 \left (-5 a^3+15 a^2 b x+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt {x}} \end {gather*}

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])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 38, normalized size = 0.84 \begin {gather*} \frac {2 \left (-5 a^3+15 a^2 b x+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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])

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 34, normalized size = 0.76 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.03, size = 35, normalized size = 0.78 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 36, normalized size = 0.80 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 35, normalized size = 0.78 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 35, normalized size = 0.78 \begin {gather*} \frac {2\,b^3\,x^{5/2}}{5}-\frac {2\,a^3}{\sqrt {x}}+6\,a^2\,b\,\sqrt {x}+2\,a\,b^2\,x^{3/2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^(3/2),x)

[Out]

(2*b^3*x^(5/2))/5 - (2*a^3)/x^(1/2) + 6*a^2*b*x^(1/2) + 2*a*b^2*x^(3/2)

________________________________________________________________________________________

sympy [A]  time = 0.64, size = 44, normalized size = 0.98 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________