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]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, 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 = {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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 38, normalized size = 1.00 \[ \frac {2 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^2} \]

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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fricas [A]  time = 0.70, size = 49, normalized size = 1.29 \[ \frac {2 \, {\left (2 \, a^{3} x^{3} - a^{2} b x^{2} - 8 \, a b^{2} x - 5 \, b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{35 \, b^{2} x^{3}} \]

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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giac [B]  time = 0.25, size = 177, normalized size = 4.66 \[ \frac {2 \, {\left (35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 105 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b \mathrm {sgn}\relax (x) + 140 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{2} \mathrm {sgn}\relax (x) + 98 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{3} \mathrm {sgn}\relax (x) + 35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{4} \mathrm {sgn}\relax (x) + 5 \, b^{5} \mathrm {sgn}\relax (x)\right )}}{35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7}} \]

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

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maple [A]  time = 0.00, size = 33, normalized size = 0.87 \[ \frac {2 \left (a x +b \right ) \left (2 a x -5 b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{35 b^{2} x^{2}} \]

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 = 1.05, size = 30, normalized size = 0.79 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b^{2}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{5 \, b^{2}} \]

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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mupad [B]  time = 1.73, size = 68, normalized size = 1.79 \[ \frac {4\,a^3\,\sqrt {a+\frac {b}{x}}}{35\,b^2}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{7\,x^3}-\frac {16\,a\,\sqrt {a+\frac {b}{x}}}{35\,x^2}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{35\,b\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(3/2)/x^3,x)

[Out]

(4*a^3*(a + b/x)^(1/2))/(35*b^2) - (2*b*(a + b/x)^(1/2))/(7*x^3) - (16*a*(a + b/x)^(1/2))/(35*x^2) - (2*a^2*(a
+ b/x)^(1/2))/(35*b*x)

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sympy [B]  time = 1.47, size = 360, normalized size = 9.47 \[ \frac {4 a^{\frac {15}{2}} b^{\frac {3}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} + \frac {2 a^{\frac {13}{2}} b^{\frac {5}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} - \frac {18 a^{\frac {11}{2}} b^{\frac {7}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} - \frac {26 a^{\frac {9}{2}} b^{\frac {9}{2}} x \sqrt {\frac {a x}{b} + 1}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} - \frac {10 a^{\frac {7}{2}} b^{\frac {11}{2}} \sqrt {\frac {a x}{b} + 1}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} - \frac {4 a^{8} b x^{\frac {9}{2}}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} - \frac {4 a^{7} b^{2} x^{\frac {7}{2}}}{35 a^{\frac {9}{2}} b^{3} x^{\frac {9}{2}} + 35 a^{\frac {7}{2}} b^{4} x^{\frac {7}{2}}} \]

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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