3.1708 \(\int \frac {(a+\frac {b}{x})^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3} \]

[Out]

-2/5*a^2*(a+b/x)^(5/2)/b^3+4/7*a*(a+b/x)^(7/2)/b^3-2/9*(a+b/x)^(9/2)/b^3

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Rubi [A]  time = 0.03, antiderivative size = 59, 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} \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(5/2))/(5*b^3) + (4*a*(a + b/x)^(7/2))/(7*b^3) - (2*(a + b/x)^(9/2))/(9*b^3)

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^4} \, dx &=-\operatorname {Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 47, normalized size = 0.80 \[ -\frac {2 \sqrt {a+\frac {b}{x}} (a x+b)^2 \left (8 a^2 x^2-20 a b x+35 b^2\right )}{315 b^3 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)/x^4,x]

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^2*(35*b^2 - 20*a*b*x + 8*a^2*x^2))/(315*b^3*x^4)

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fricas [A]  time = 1.38, size = 60, normalized size = 1.02 \[ -\frac {2 \, {\left (8 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 50 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{315 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(8*a^4*x^4 - 4*a^3*b*x^3 + 3*a^2*b^2*x^2 + 50*a*b^3*x + 35*b^4)*sqrt((a*x + b)/x)/(b^3*x^4)

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giac [B]  time = 0.32, size = 208, normalized size = 3.53 \[ \frac {2 \, {\left (420 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} \mathrm {sgn}\relax (x) + 1575 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b \mathrm {sgn}\relax (x) + 2583 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm {sgn}\relax (x) + 2310 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{3} \mathrm {sgn}\relax (x) + 1170 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} \mathrm {sgn}\relax (x) + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} \mathrm {sgn}\relax (x) + 35 \, b^{6} \mathrm {sgn}\relax (x)\right )}}{315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(420*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sgn(x) + 1575*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b*sgn
(x) + 2583*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^2*sgn(x) + 2310*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b
^3*sgn(x) + 1170*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sgn(x) + 315*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*
b^5*sgn(x) + 35*b^6*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^9

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maple [A]  time = 0.01, size = 44, normalized size = 0.75 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-20 a b x +35 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{315 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(a*x+b)*(8*a^2*x^2-20*a*b*x+35*b^2)*((a*x+b)/x)^(3/2)/b^3/x^3

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maxima [A]  time = 0.99, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(a + b/x)^(9/2)/b^3 + 4/7*(a + b/x)^(7/2)*a/b^3 - 2/5*(a + b/x)^(5/2)*a^2/b^3

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mupad [B]  time = 1.89, size = 88, normalized size = 1.49 \[ \frac {8\,a^3\,\sqrt {a+\frac {b}{x}}}{315\,b^2\,x}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{9\,x^4}-\frac {16\,a^4\,\sqrt {a+\frac {b}{x}}}{315\,b^3}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b\,x^2}-\frac {20\,a\,\sqrt {a+\frac {b}{x}}}{63\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(3/2)/x^4,x)

[Out]

(8*a^3*(a + b/x)^(1/2))/(315*b^2*x) - (2*b*(a + b/x)^(1/2))/(9*x^4) - (16*a^4*(a + b/x)^(1/2))/(315*b^3) - (2*
a^2*(a + b/x)^(1/2))/(105*b*x^2) - (20*a*(a + b/x)^(1/2))/(63*x^3)

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sympy [B]  time = 2.40, size = 986, normalized size = 16.71 \[ - \frac {16 a^{\frac {23}{2}} b^{\frac {9}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {40 a^{\frac {21}{2}} b^{\frac {11}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {30 a^{\frac {19}{2}} b^{\frac {13}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {110 a^{\frac {17}{2}} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {380 a^{\frac {15}{2}} b^{\frac {17}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {516 a^{\frac {13}{2}} b^{\frac {19}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {310 a^{\frac {11}{2}} b^{\frac {21}{2}} x \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} - \frac {70 a^{\frac {9}{2}} b^{\frac {23}{2}} \sqrt {\frac {a x}{b} + 1}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {16 a^{12} b^{4} x^{\frac {15}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {48 a^{11} b^{5} x^{\frac {13}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {48 a^{10} b^{6} x^{\frac {11}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} + \frac {16 a^{9} b^{7} x^{\frac {9}{2}}}{315 a^{\frac {15}{2}} b^{7} x^{\frac {15}{2}} + 945 a^{\frac {13}{2}} b^{8} x^{\frac {13}{2}} + 945 a^{\frac {11}{2}} b^{9} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{10} x^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**(23/2)*b**(9/2)*x**7*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945
*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 40*a**(21/2)*b**(11/2)*x**6*sqrt(a*x/b + 1)/(315*a*
*(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(
9/2)) - 30*a**(19/2)*b**(13/2)*x**5*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/
2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 110*a**(17/2)*b**(15/2)*x**4*sqrt(a*x/b + 1
)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b
**10*x**(9/2)) - 380*a**(15/2)*b**(17/2)*x**3*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b*
*8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 516*a**(13/2)*b**(19/2)*x**2*sqrt
(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*
a**(9/2)*b**10*x**(9/2)) - 310*a**(11/2)*b**(21/2)*x*sqrt(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(1
3/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) - 70*a**(9/2)*b**(23/2)*sqrt
(a*x/b + 1)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*
a**(9/2)*b**10*x**(9/2)) + 16*a**12*b**4*x**(15/2)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2
) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) + 48*a**11*b**5*x**(13/2)/(315*a**(15/2)*b**7*
x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2)) + 48*a*
*10*b**6*x**(11/2)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**(13/2) + 945*a**(11/2)*b**9*x**(11/2)
+ 315*a**(9/2)*b**10*x**(9/2)) + 16*a**9*b**7*x**(9/2)/(315*a**(15/2)*b**7*x**(15/2) + 945*a**(13/2)*b**8*x**
(13/2) + 945*a**(11/2)*b**9*x**(11/2) + 315*a**(9/2)*b**10*x**(9/2))

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