3.1707 \(\int \frac{(a+\frac{b}{x})^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0175401, antiderivative size = 38, 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 = {266, 43} \[ \frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)/x^3,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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 \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^3} \, dx &=-\operatorname{Subst}\left (\int x (a+b x)^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0154076, size = 38, normalized size = 1. \[ \frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)/x^3,x]

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-5\,b \right ) }{35\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(a*x+b)*(2*a*x-5*b)*((a*x+b)/x)^(3/2)/b^2/x^2

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Maxima [A]  time = 1.29173, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{5 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.84527, size = 105, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (2 \, a^{3} x^{3} - a^{2} b x^{2} - 8 \, a b^{2} x - 5 \, b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{35 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(2*a^3*x^3 - a^2*b*x^2 - 8*a*b^2*x - 5*b^3)*sqrt((a*x + b)/x)/(b^2*x^3)

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Sympy [B]  time = 1.30367, size = 360, normalized size = 9.47 \begin{align*} \frac{4 a^{\frac{15}{2}} b^{\frac{3}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} + \frac{2 a^{\frac{13}{2}} b^{\frac{5}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{18 a^{\frac{11}{2}} b^{\frac{7}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{26 a^{\frac{9}{2}} b^{\frac{9}{2}} x \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{10 a^{\frac{7}{2}} b^{\frac{11}{2}} \sqrt{\frac{a x}{b} + 1}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{4 a^{8} b x^{\frac{9}{2}}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} - \frac{4 a^{7} b^{2} x^{\frac{7}{2}}}{35 a^{\frac{9}{2}} b^{3} x^{\frac{9}{2}} + 35 a^{\frac{7}{2}} b^{4} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a**(15/2)*b**(3/2)*x**4*sqrt(a*x/b + 1)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2)) + 2*a**(13/2
)*b**(5/2)*x**3*sqrt(a*x/b + 1)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2)) - 18*a**(11/2)*b**(7/2
)*x**2*sqrt(a*x/b + 1)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2)) - 26*a**(9/2)*b**(9/2)*x*sqrt(a
*x/b + 1)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2)) - 10*a**(7/2)*b**(11/2)*sqrt(a*x/b + 1)/(35*
a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2)) - 4*a**8*b*x**(9/2)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/
2)*b**4*x**(7/2)) - 4*a**7*b**2*x**(7/2)/(35*a**(9/2)*b**3*x**(9/2) + 35*a**(7/2)*b**4*x**(7/2))

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Giac [B]  time = 1.17603, size = 239, normalized size = 6.29 \begin{align*} \frac{2 \,{\left (35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b \mathrm{sgn}\left (x\right ) + 140 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 98 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{3} \mathrm{sgn}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{4} \mathrm{sgn}\left (x\right ) + 5 \, b^{5} \mathrm{sgn}\left (x\right )\right )}}{35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(35*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*sgn(x) + 105*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b*sgn(x)
+ 140*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^2*sgn(x) + 98*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^3*sgn(x
) + 35*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^4*sgn(x) + 5*b^5*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^7