\(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx\) [1920]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{b^3}+\frac {2 a \left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^3} \]

[Out]

2/3*a*(a+b/x^2)^(3/2)/b^3-1/5*(a+b/x^2)^(5/2)/b^3-a^2*(a+b/x^2)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {a^2 \sqrt {a+\frac {b}{x^2}}}{b^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{5 b^3}+\frac {2 a \left (a+\frac {b}{x^2}\right )^{3/2}}{3 b^3} \]

[In]

Int[1/(Sqrt[a + b/x^2]*x^7),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \left (3 b^2-4 a b x^2+8 a^2 x^4\right )}{15 b^3 x^4} \]

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^7),x]

[Out]

-1/15*(Sqrt[a + b/x^2]*(3*b^2 - 4*a*b*x^2 + 8*a^2*x^4))/(b^3*x^4)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82

method	result	size
trager	\(-\frac {\left (8 a^{2} x^{4}-4 a b \,x^{2}+3 b^{2}\right ) \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{15 x^{4} b^{3}}\)	\(47\)
gosper	\(-\frac {\left (a \,x^{2}+b \right ) \left (8 a^{2} x^{4}-4 a b \,x^{2}+3 b^{2}\right )}{15 x^{6} b^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)
default	\(-\frac {\left (a \,x^{2}+b \right ) \left (8 a^{2} x^{4}-4 a b \,x^{2}+3 b^{2}\right )}{15 x^{6} b^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)
risch	\(-\frac {\left (a \,x^{2}+b \right ) \left (8 a^{2} x^{4}-4 a b \,x^{2}+3 b^{2}\right )}{15 x^{6} b^{3} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(50\)

[In]

int(1/(a+b/x^2)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/15/x^4*(8*a^2*x^4-4*a*b*x^2+3*b^2)/b^3*(-(-a*x^2-b)/x^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {{\left (8 \, a^{2} x^{4} - 4 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{15 \, b^{3} x^{4}} \]

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-1/15*(8*a^2*x^4 - 4*a*b*x^2 + 3*b^2)*sqrt((a*x^2 + b)/x^2)/(b^3*x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (49) = 98\).

Time = 1.18 (sec) , antiderivative size = 750, normalized size of antiderivative = 13.16 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=- \frac {8 a^{\frac {15}{2}} b^{\frac {9}{2}} x^{10} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} - \frac {20 a^{\frac {13}{2}} b^{\frac {11}{2}} x^{8} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} - \frac {15 a^{\frac {11}{2}} b^{\frac {13}{2}} x^{6} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} - \frac {5 a^{\frac {9}{2}} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} - \frac {5 a^{\frac {7}{2}} b^{\frac {17}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} - \frac {3 a^{\frac {5}{2}} b^{\frac {19}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} + \frac {8 a^{8} b^{4} x^{11}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} + \frac {24 a^{7} b^{5} x^{9}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} + \frac {24 a^{6} b^{6} x^{7}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} + \frac {8 a^{5} b^{7} x^{5}}{15 a^{\frac {11}{2}} b^{7} x^{11} + 45 a^{\frac {9}{2}} b^{8} x^{9} + 45 a^{\frac {7}{2}} b^{9} x^{7} + 15 a^{\frac {5}{2}} b^{10} x^{5}} \]

[In]

integrate(1/(a+b/x**2)**(1/2)/x**7,x)

[Out]

-8*a**(15/2)*b**(9/2)*x**10*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*
b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 20*a**(13/2)*b**(11/2)*x**8*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11
+ 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 15*a**(11/2)*b**(13/2)*x**6*sqrt(a
*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5
) - 5*a**(9/2)*b**(15/2)*x**4*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2
)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 5*a**(7/2)*b**(17/2)*x**2*sqrt(a*x**2/b + 1)/(15*a**(11/2)*b**7*x**11
+ 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) - 3*a**(5/2)*b**(19/2)*sqrt(a*x**2/b
+ 1)/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x**5) + 8*a
**8*b**4*x**11/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)*b**10*x*
*5) + 24*a**7*b**5*x**9/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15*a**(5/2)
*b**10*x**5) + 24*a**6*b**6*x**7/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x**7 + 15
*a**(5/2)*b**10*x**5) + 8*a**5*b**7*x**5/(15*a**(11/2)*b**7*x**11 + 45*a**(9/2)*b**8*x**9 + 45*a**(7/2)*b**9*x
**7 + 15*a**(5/2)*b**10*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, b^{3}} + \frac {2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a}{3 \, b^{3}} - \frac {\sqrt {a + \frac {b}{x^{2}}} a^{2}}{b^{3}} \]

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/5*(a + b/x^2)^(5/2)/b^3 + 2/3*(a + b/x^2)^(3/2)*a/b^3 - sqrt(a + b/x^2)*a^2/b^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=\frac {16 \, {\left (10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} - 5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} b + b^{2}\right )} a^{\frac {5}{2}}}{15 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{5} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(a+b/x^2)^(1/2)/x^7,x, algorithm="giac")

[Out]

16/15*(10*(sqrt(a)*x - sqrt(a*x^2 + b))^4 - 5*(sqrt(a)*x - sqrt(a*x^2 + b))^2*b + b^2)*a^(5/2)/(((sqrt(a)*x -
sqrt(a*x^2 + b))^2 - b)^5*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^7} \, dx=-\frac {3\,b^2\,\sqrt {a+\frac {b}{x^2}}+8\,a^2\,x^4\,\sqrt {a+\frac {b}{x^2}}-4\,a\,b\,x^2\,\sqrt {a+\frac {b}{x^2}}}{15\,b^3\,x^4} \]

[In]

int(1/(x^7*(a + b/x^2)^(1/2)),x)

[Out]

-(3*b^2*(a + b/x^2)^(1/2) + 8*a^2*x^4*(a + b/x^2)^(1/2) - 4*a*b*x^2*(a + b/x^2)^(1/2))/(15*b^3*x^4)