3.1763 \(\int (a+\frac {b}{x})^{3/2} x^{3/2} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {264} \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x^(3/2),x]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \[ \frac {2 x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x^(3/2),x]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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fricas [B]  time = 0.79, size = 35, normalized size = 1.52 \[ \frac {2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 64, normalized size = 2.78 \[ \frac {2}{3} \, b {\left (\frac {{\left (a x + b\right )}^{\frac {3}{2}}}{a} - \frac {b^{\frac {3}{2}}}{a}\right )} \mathrm {sgn}\relax (x) + \frac {2}{15} \, a {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 25, normalized size = 1.09 \[ \frac {2 \left (a x +b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 17, normalized size = 0.74 \[ \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(a + b/x)^(5/2)*x^(5/2)/a

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mupad [B]  time = 1.38, size = 34, normalized size = 1.48 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{5/2}}{5}+\frac {4\,b\,x^{3/2}}{5}+\frac {2\,b^2\,\sqrt {x}}{5\,a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + b/x)^(3/2),x)

[Out]

(a + b/x)^(1/2)*((2*a*x^(5/2))/5 + (4*b*x^(3/2))/5 + (2*b^2*x^(1/2))/(5*a))

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sympy [B]  time = 9.46, size = 63, normalized size = 2.74 \[ \frac {2 a \sqrt {b} x^{2} \sqrt {\frac {a x}{b} + 1}}{5} + \frac {4 b^{\frac {3}{2}} x \sqrt {\frac {a x}{b} + 1}}{5} + \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(b)*x**2*sqrt(a*x/b + 1)/5 + 4*b**(3/2)*x*sqrt(a*x/b + 1)/5 + 2*b**(5/2)*sqrt(a*x/b + 1)/(5*a)

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