3.18.48 \(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^4} \, dx\) [1748]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \frac {2 a^2}{3 b^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a}{b^3 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^4),x]

[Out]

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

Rule 45

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])

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^4} \, dx &=-\text {Subst}\left (\int \frac {x^2}{(a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{5/2}}-\frac {2 a}{b^2 (a+b x)^{3/2}}+\frac {1}{b^2 \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 a^2}{3 b^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a}{b^3 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 46, normalized size = 0.84 \begin {gather*} -\frac {2 \sqrt {\frac {b+a x}{x}} \left (3 b^2+12 a b x+8 a^2 x^2\right )}{3 b^3 (b+a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^4),x]

[Out]

(-2*Sqrt[(b + a*x)/x]*(3*b^2 + 12*a*b*x + 8*a^2*x^2))/(3*b^3*(b + a*x)^2)

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.05, size = 607, normalized size = 11.04

method	result	size
gosper	\(-\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}+12 a b x +3 b^{2}\right )}{3 x^{3} b^{3} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\)	\(44\)
trager	\(-\frac {2 \left (8 a^{2} x^{2}+12 a b x +3 b^{2}\right ) \sqrt {-\frac {-a x -b}{x}}}{3 b^{3} \left (a x +b \right )^{2}}\)	\(47\)
risch	\(-\frac {2 \left (a x +b \right )}{b^{3} x \sqrt {\frac {a x +b}{x}}}-\frac {2 a \left (5 a x +6 b \right )}{3 \left (a x +b \right ) b^{3} \sqrt {\frac {a x +b}{x}}}\)	\(58\)
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, \left (-6 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, x^{5}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} x^{5}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}+24 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}-18 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} b \,x^{4}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} x^{4}-12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}-18 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b \,x^{4}-9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} x^{4}+28 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b \,x^{2}-18 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{2} x^{3}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} x^{3}-36 a^{\frac {5}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,x^{2}-18 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{2} x^{3}-9 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} x^{3}-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{3} x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} x^{2}-36 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} x -6 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x}\, b^{3} x^{2}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} x^{2}-12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{3}\right )}{6 x \sqrt {x \left (a x +b \right )}\, b^{4} \left (a x +b \right )^{3} \sqrt {a}}\)	\(607\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x*b)^(5/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

1/6*((a*x+b)/x)^(1/2)/x*(-6*a^(9/2)*(x*(a*x+b))^(1/2)*x^5+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/
2))*a^4*b*x^5-6*(a*x^2+b*x)^(1/2)*a^(9/2)*x^5-3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*b*x^
5+24*(x*(a*x+b))^(3/2)*a^(7/2)*x^3-18*(x*(a*x+b))^(1/2)*a^(7/2)*b*x^4+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*a^3*b^2*x^4-12*(a*x^2+b*x)^(3/2)*a^(7/2)*x^3-18*(a*x^2+b*x)^(1/2)*a^(7/2)*b*x^4-9*ln(1/2*(2*(a
*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b^2*x^4+28*a^(5/2)*(x*(a*x+b))^(3/2)*b*x^2-18*(x*(a*x+b))^(1/2)*
a^(5/2)*b^2*x^3+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^3*x^3-36*a^(5/2)*(a*x^2+b*x)^(3/
2)*b*x^2-18*(a*x^2+b*x)^(1/2)*a^(5/2)*b^2*x^3-9*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^3*
x^3-6*(x*(a*x+b))^(1/2)*a^(3/2)*b^3*x^2+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^4*x^2-36*(
a*x^2+b*x)^(3/2)*a^(3/2)*b^2*x-6*a^(3/2)*(a*x^2+b*x)^(1/2)*b^3*x^2-3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x
+b)/a^(1/2))*a*b^4*x^2-12*(a*x^2+b*x)^(3/2)*a^(1/2)*b^3)/(x*(a*x+b))^(1/2)/b^4/(a*x+b)^3/a^(1/2)

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Maxima [A]
time = 0.29, size = 47, normalized size = 0.85 \begin {gather*} -\frac {2 \, \sqrt {a + \frac {b}{x}}}{b^{3}} - \frac {4 \, a}{\sqrt {a + \frac {b}{x}} b^{3}} + \frac {2 \, a^{2}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 55, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (8 \, a^{2} x^{2} + 12 \, a b x + 3 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^4,x, algorithm="fricas")

[Out]

-2/3*(8*a^2*x^2 + 12*a*b*x + 3*b^2)*sqrt((a*x + b)/x)/(a^2*b^3*x^2 + 2*a*b^4*x + b^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (46) = 92\).
time = 0.73, size = 136, normalized size = 2.47 \begin {gather*} \begin {cases} - \frac {16 a^{2} x^{2}}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} - \frac {24 a b x}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} - \frac {6 b^{2}}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{3 a^{\frac {5}{2}} x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**4,x)

[Out]

Piecewise((-16*a**2*x**2/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)) - 24*a*b*x/(3*a*b**3*x**2*sqrt
(a + b/x) + 3*b**4*x*sqrt(a + b/x)) - 6*b**2/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)), Ne(b, 0))
, (-1/(3*a**(5/2)*x**3), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^4,x, algorithm="giac")

[Out]

integrate(1/((a + b/x)^(5/2)*x^4), x)

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Mupad [B]
time = 1.35, size = 40, normalized size = 0.73 \begin {gather*} -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (8\,a^2\,x^2+12\,a\,b\,x+3\,b^2\right )}{3\,b^3\,{\left (b+a\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x)^(5/2)),x)

[Out]

-(2*(a + b/x)^(1/2)*(3*b^2 + 8*a^2*x^2 + 12*a*b*x))/(3*b^3*(b + a*x)^2)

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