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

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

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Rubi [A]  time = 0.01, antiderivative size = 47, 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*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 0.81 \begin {gather*} \frac {2 \left (-a^3-9 a^2 b x+9 a b^2 x^2+b^3 x^3\right )}{3 x^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 38, normalized size = 0.81 \begin {gather*} \frac {2 \left (-a^3-9 a^2 b x+9 a b^2 x^2+b^3 x^3\right )}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 0.90, size = 34, normalized size = 0.72 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.07, size = 34, normalized size = 0.72 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 34, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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 = 1.37, size = 34, normalized size = 0.72 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 35, normalized size = 0.74 \begin {gather*} -\frac {2\,a^3+18\,a^2\,b\,x-18\,a\,b^2\,x^2-2\,b^3\,x^3}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^(5/2),x)

[Out]

-(2*a^3 - 2*b^3*x^3 - 18*a*b^2*x^2 + 18*a^2*b*x)/(3*x^(3/2))

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sympy [A]  time = 0.78, size = 46, normalized size = 0.98 \begin {gather*} - \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 {gather*}

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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