3.1996 \(\int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3} \]

[Out]

-2/9*a^2*(a+b/x^3)^(3/2)/b^3+4/15*a*(a+b/x^3)^(5/2)/b^3-2/21*(a+b/x^3)^(7/2)/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} \[ -\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*a^2*(a + b/x^3)^(3/2))/(9*b^3) + (4*a*(a + b/x^3)^(5/2))/(15*b^3) - (2*(a + b/x^3)^(7/2))/(21*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 {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 0.83 \[ -\frac {2 \sqrt {a+\frac {b}{x^3}} \left (a x^3+b\right ) \left (8 a^2 x^6-12 a b x^3+15 b^2\right )}{315 b^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*Sqrt[a + b/x^3]*(b + a*x^3)*(15*b^2 - 12*a*b*x^3 + 8*a^2*x^6))/(315*b^3*x^9)

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fricas [A]  time = 0.96, size = 53, normalized size = 0.90 \[ -\frac {2 \, {\left (8 \, a^{3} x^{9} - 4 \, a^{2} b x^{6} + 3 \, a b^{2} x^{3} + 15 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{315 \, b^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="fricas")

[Out]

-2/315*(8*a^3*x^9 - 4*a^2*b*x^6 + 3*a*b^2*x^3 + 15*b^3)*sqrt((a*x^3 + b)/x^3)/(b^3*x^9)

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giac [A]  time = 0.22, size = 43, normalized size = 0.73 \[ -\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 42 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a + 35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}\right )}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="giac")

[Out]

-2/315*(15*(a + b/x^3)^(7/2) - 42*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*a^2)/b^3

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maple [A]  time = 0.01, size = 50, normalized size = 0.85 \[ -\frac {2 \left (a \,x^{3}+b \right ) \left (8 a^{2} x^{6}-12 a b \,x^{3}+15 b^{2}\right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{315 b^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^3)^(1/2)/x^10,x)

[Out]

-2/315*(a*x^3+b)*(8*a^2*x^6-12*a*b*x^3+15*b^2)*((a*x^3+b)/x^3)^(1/2)/b^3/x^9

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maxima [A]  time = 0.89, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}}}{21 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a}{15 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^10,x, algorithm="maxima")

[Out]

-2/21*(a + b/x^3)^(7/2)/b^3 + 4/15*(a + b/x^3)^(5/2)*a/b^3 - 2/9*(a + b/x^3)^(3/2)*a^2/b^3

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mupad [B]  time = 1.94, size = 70, normalized size = 1.19 \[ \frac {8\,a^2\,\sqrt {a+\frac {b}{x^3}}}{315\,b^2\,x^3}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x^3}}}{315\,b^3}-\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{105\,b\,x^6}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{21\,x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^3)^(1/2)/x^10,x)

[Out]

(8*a^2*(a + b/x^3)^(1/2))/(315*b^2*x^3) - (16*a^3*(a + b/x^3)^(1/2))/(315*b^3) - (2*a*(a + b/x^3)^(1/2))/(105*
b*x^6) - (2*(a + b/x^3)^(1/2))/(21*x^9)

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sympy [B]  time = 2.91, size = 913, normalized size = 15.47 \[ - \frac {16 a^{\frac {19}{2}} b^{\frac {9}{2}} x^{18} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {40 a^{\frac {17}{2}} b^{\frac {11}{2}} x^{15} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {30 a^{\frac {15}{2}} b^{\frac {13}{2}} x^{12} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {15}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {100 a^{\frac {11}{2}} b^{\frac {17}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {96 a^{\frac {9}{2}} b^{\frac {19}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {30 a^{\frac {7}{2}} b^{\frac {21}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {16 a^{10} b^{4} x^{\frac {39}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {48 a^{9} b^{5} x^{\frac {33}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {48 a^{8} b^{6} x^{\frac {27}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {16 a^{7} b^{7} x^{\frac {21}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**3)**(1/2)/x**10,x)

[Out]

-16*a**(19/2)*b**(9/2)*x**18*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) +
945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(17/2)*b**(11/2)*x**15*sqrt(a*x**3/b + 1)
/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**
10*x**(21/2)) - 30*a**(15/2)*b**(13/2)*x**12*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*
b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(13/2)*b**(15/2)*x**9*sqr
t(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 3
15*a**(7/2)*b**10*x**(21/2)) - 100*a**(11/2)*b**(17/2)*x**6*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) +
945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 96*a**(9/2)*b**(
19/2)*x**3*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9
*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 30*a**(7/2)*b**(21/2)*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(
39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 16*a**10*
b**4*x**(39/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 31
5*a**(7/2)*b**10*x**(21/2)) + 48*a**9*b**5*x**(33/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33
/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 48*a**8*b**6*x**(27/2)/(315*a**(13/2)*b**7
*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 16*a
**7*b**7*x**(21/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2)
+ 315*a**(7/2)*b**10*x**(21/2))

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