3.1750 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^6} \, dx\)
Optimal. Leaf size=97 \[ \frac{2 a^4}{3 b^5 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{8 a^3}{b^5 \sqrt{a+\frac{b}{x}}}-\frac{12 a^2 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5} \]
[Out]
(2*a^4)/(3*b^5*(a + b/x)^(3/2)) - (8*a^3)/(b^5*Sqrt[a + b/x]) - (12*a^2*Sqrt[a + b/x])/b^5 + (8*a*(a + b/x)^(3
/2))/(3*b^5) - (2*(a + b/x)^(5/2))/(5*b^5)
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Rubi [A] time = 0.0393913, antiderivative size = 97, normalized size of antiderivative = 1.,
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} \[ \frac{2 a^4}{3 b^5 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{8 a^3}{b^5 \sqrt{a+\frac{b}{x}}}-\frac{12 a^2 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b/x)^(5/2)*x^6),x]
[Out]
(2*a^4)/(3*b^5*(a + b/x)^(3/2)) - (8*a^3)/(b^5*Sqrt[a + b/x]) - (12*a^2*Sqrt[a + b/x])/b^5 + (8*a*(a + b/x)^(3
/2))/(3*b^5) - (2*(a + b/x)^(5/2))/(5*b^5)
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]]
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])
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^6} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{5/2}}-\frac{4 a^3}{b^4 (a+b x)^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+b x}}-\frac{4 a \sqrt{a+b x}}{b^4}+\frac{(a+b x)^{3/2}}{b^4}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^4}{3 b^5 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{8 a^3}{b^5 \sqrt{a+\frac{b}{x}}}-\frac{12 a^2 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}\\ \end{align*}
Mathematica [A] time = 0.035599, size = 69, normalized size = 0.71 \[ -\frac{2 \left (48 a^2 b^2 x^2+192 a^3 b x^3+128 a^4 x^4-8 a b^3 x+3 b^4\right )}{15 b^5 x^3 \sqrt{a+\frac{b}{x}} (a x+b)} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b/x)^(5/2)*x^6),x]
[Out]
(-2*(3*b^4 - 8*a*b^3*x + 48*a^2*b^2*x^2 + 192*a^3*b*x^3 + 128*a^4*x^4))/(15*b^5*Sqrt[a + b/x]*x^3*(b + a*x))
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Maple [A] time = 0.006, size = 66, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}+192\,{a}^{3}{x}^{3}b+48\,{a}^{2}{x}^{2}{b}^{2}-8\,ax{b}^{3}+3\,{b}^{4} \right ) }{15\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x)^(5/2)/x^6,x)
[Out]
-2/15*(a*x+b)*(128*a^4*x^4+192*a^3*b*x^3+48*a^2*b^2*x^2-8*a*b^3*x+3*b^4)/x^5/b^5/((a*x+b)/x)^(5/2)
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Maxima [A] time = 1.02301, size = 109, normalized size = 1.12 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{5}} - \frac{12 \, \sqrt{a + \frac{b}{x}} a^{2}}{b^{5}} - \frac{8 \, a^{3}}{\sqrt{a + \frac{b}{x}} b^{5}} + \frac{2 \, a^{4}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="maxima")
[Out]
-2/5*(a + b/x)^(5/2)/b^5 + 8/3*(a + b/x)^(3/2)*a/b^5 - 12*sqrt(a + b/x)*a^2/b^5 - 8*a^3/(sqrt(a + b/x)*b^5) +
2/3*a^4/((a + b/x)^(3/2)*b^5)
________________________________________________________________________________________
Fricas [A] time = 1.503, size = 176, normalized size = 1.81 \begin{align*} -\frac{2 \,{\left (128 \, a^{4} x^{4} + 192 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 8 \, a b^{3} x + 3 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{15 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="fricas")
[Out]
-2/15*(128*a^4*x^4 + 192*a^3*b*x^3 + 48*a^2*b^2*x^2 - 8*a*b^3*x + 3*b^4)*sqrt((a*x + b)/x)/(a^2*b^5*x^4 + 2*a*
b^6*x^3 + b^7*x^2)
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Sympy [B] time = 5.28847, size = 2032, normalized size = 20.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)**(5/2)/x**6,x)
[Out]
-256*a**(21/2)*b**(33/2)*x**8*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 2
25*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26
*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 1408*a**(19/2)*b**(35/2)*x**7*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x*
*(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a
**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 3168*a**(17/2)*b**(37/2)*x
**6*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/
2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b*
*27*x**(5/2)) - 3696*a**(15/2)*b**(39/2)*x**5*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**
22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 9
0*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 2310*a**(13/2)*b**(41/2)*x**4*sqrt(a*x/b + 1)/(15*a*
*(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x
**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 696*a**(11
/2)*b**(43/2)*x**3*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2
)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) +
15*a**(5/2)*b**27*x**(5/2)) - 68*a**(9/2)*b**(45/2)*x**2*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a
**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*
x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 8*a**(7/2)*b**(47/2)*x*sqrt(a*x/b + 1)/(
15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b*
*24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 6*a**
(5/2)*b**(49/2)*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b
**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15
*a**(5/2)*b**27*x**(5/2)) + 256*a**11*b**16*x**(17/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(1
5/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/
2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) + 1536*a**10*b**17*x**(15/2)/(15*a**(17/2)*b**21*x**(17/2) + 9
0*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**
25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) + 3840*a**9*b**18*x**(13/2)/(15*a**(17/
2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11
/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) + 5120*a**8*b**19
*x**(11/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*
a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5
/2)) + 3840*a**7*b**20*x**(9/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b
**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15
*a**(5/2)*b**27*x**(5/2)) + 1536*a**6*b**21*x**(7/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15
/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2
)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) + 256*a**5*b**22*x**(5/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a*
*(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x
**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2))
________________________________________________________________________________________
Giac [A] time = 1.23564, size = 163, normalized size = 1.68 \begin{align*} \frac{2}{15} \, b{\left (\frac{5 \,{\left (a^{4} - \frac{12 \,{\left (a x + b\right )} a^{3}}{x}\right )} x}{{\left (a x + b\right )} b^{6} \sqrt{\frac{a x + b}{x}}} - \frac{90 \, a^{2} b^{24} \sqrt{\frac{a x + b}{x}} - \frac{20 \,{\left (a x + b\right )} a b^{24} \sqrt{\frac{a x + b}{x}}}{x} + \frac{3 \,{\left (a x + b\right )}^{2} b^{24} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{b^{30}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="giac")
[Out]
2/15*b*(5*(a^4 - 12*(a*x + b)*a^3/x)*x/((a*x + b)*b^6*sqrt((a*x + b)/x)) - (90*a^2*b^24*sqrt((a*x + b)/x) - 20
*(a*x + b)*a*b^24*sqrt((a*x + b)/x)/x + 3*(a*x + b)^2*b^24*sqrt((a*x + b)/x)/x^2)/b^30)