3.1653 \(\int (a+\frac{b}{x})^2 x^{5/2} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]

[Out]

(2*b^2*x^(3/2))/3 + (4*a*b*x^(5/2))/5 + (2*a^2*x^(7/2))/7

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Rubi [A]  time = 0.0098707, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2*x^(5/2),x]

[Out]

(2*b^2*x^(3/2))/3 + (4*a*b*x^(5/2))/5 + (2*a^2*x^(7/2))/7

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 \left (a+\frac{b}{x}\right )^2 x^{5/2} \, dx &=\int \sqrt{x} (b+a x)^2 \, dx\\ &=\int \left (b^2 \sqrt{x}+2 a b x^{3/2}+a^2 x^{5/2}\right ) \, dx\\ &=\frac{2}{3} b^2 x^{3/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{7} a^2 x^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0074049, size = 28, normalized size = 0.78 \[ \frac{2}{105} x^{3/2} \left (15 a^2 x^2+42 a b x+35 b^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2*x^(5/2),x]

[Out]

(2*x^(3/2)*(35*b^2 + 42*a*b*x + 15*a^2*x^2))/105

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*}{\frac{30\,{a}^{2}{x}^{2}+84\,xab+70\,{b}^{2}}{105}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^2*x^2+42*a*b*x+35*b^2)*x^(3/2)

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Maxima [A]  time = 0.962952, size = 35, normalized size = 0.97 \begin{align*} \frac{2}{105} \,{\left (15 \, a^{2} + \frac{42 \, a b}{x} + \frac{35 \, b^{2}}{x^{2}}\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^2 + 42*a*b/x + 35*b^2/x^2)*x^(7/2)

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Fricas [A]  time = 1.5976, size = 70, normalized size = 1.94 \begin{align*} \frac{2}{105} \,{\left (15 \, a^{2} x^{3} + 42 \, a b x^{2} + 35 \, b^{2} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^2*x^3 + 42*a*b*x^2 + 35*b^2*x)*sqrt(x)

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Sympy [A]  time = 2.67022, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(5/2)/5 + 2*b**2*x**(3/2)/3

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Giac [A]  time = 1.09946, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{7} \, a^{2} x^{\frac{7}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*a^2*x^(7/2) + 4/5*a*b*x^(5/2) + 2/3*b^2*x^(3/2)