3.1738 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^4} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]

[Out]

(2*a^2)/(b^3*Sqrt[a + b/x]) + (4*a*Sqrt[a + b/x])/b^3 - (2*(a + b/x)^(3/2))/(3*b^3)

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Rubi [A]  time = 0.0249515, antiderivative size = 55, 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} \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^4),x]

[Out]

(2*a^2)/(b^3*Sqrt[a + b/x]) + (4*a*Sqrt[a + b/x])/b^3 - (2*(a + b/x)^(3/2))/(3*b^3)

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{1}{\left (a+\frac{b}{x}\right )^{3/2} x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.022299, size = 40, normalized size = 0.73 \[ \frac{16 a^2 x^2+8 a b x-2 b^2}{3 b^3 x^2 \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^4),x]

[Out]

(-2*b^2 + 8*a*b*x + 16*a^2*x^2)/(3*b^3*Sqrt[a + b/x]*x^2)

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Maple [A]  time = 0.004, size = 44, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+4\,xab-{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^4,x)

[Out]

2/3*(a*x+b)*(8*a^2*x^2+4*a*b*x-b^2)/x^3/b^3/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 1.00635, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x}} a}{b^{3}} + \frac{2 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="maxima")

[Out]

-2/3*(a + b/x)^(3/2)/b^3 + 4*sqrt(a + b/x)*a/b^3 + 2*a^2/(sqrt(a + b/x)*b^3)

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Fricas [A]  time = 1.48166, size = 96, normalized size = 1.75 \begin{align*} \frac{2 \,{\left (8 \, a^{2} x^{2} + 4 \, a b x - b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a b^{3} x^{2} + b^{4} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="fricas")

[Out]

2/3*(8*a^2*x^2 + 4*a*b*x - b^2)*sqrt((a*x + b)/x)/(a*b^3*x^2 + b^4*x)

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Sympy [B]  time = 2.70181, size = 457, normalized size = 8.31 \begin{align*} \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{7}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{5}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{3}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**4,x)

[Out]

16*a**(9/2)*b**(7/2)*x**3*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b*
*8*x**(3/2)) + 24*a**(7/2)*b**(9/2)*x**2*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2)
+ 3*a**(3/2)*b**8*x**(3/2)) + 6*a**(5/2)*b**(11/2)*x*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b*
*7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 2*a**(3/2)*b**(13/2)*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a
**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 16*a**5*b**3*x**(7/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/
2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 32*a**4*b**4*x**(5/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b*
*7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 16*a**3*b**5*x**(3/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x*
*(5/2) + 3*a**(3/2)*b**8*x**(3/2))

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Giac [A]  time = 1.27384, size = 93, normalized size = 1.69 \begin{align*} \frac{2}{3} \, b{\left (\frac{3 \, a^{2}}{b^{4} \sqrt{\frac{a x + b}{x}}} + \frac{6 \, a b^{8} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{8} \sqrt{\frac{a x + b}{x}}}{x}}{b^{12}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="giac")

[Out]

2/3*b*(3*a^2/(b^4*sqrt((a*x + b)/x)) + (6*a*b^8*sqrt((a*x + b)/x) - (a*x + b)*b^8*sqrt((a*x + b)/x)/x)/b^12)