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3.1997 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^{13}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}+\frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4} \]

[Out]

(2*a^3*(a + b/x^3)^(3/2))/(9*b^4) - (2*a^2*(a + b/x^3)^(5/2))/(5*b^4) + (2*a*(a + b/x^3)^(7/2))/(7*b^4) - (2*(
a + b/x^3)^(9/2))/(27*b^4)

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Rubi [A]  time = 0.0413452, antiderivative size = 80, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}+\frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^3]/x^13,x]

[Out]

(2*a^3*(a + b/x^3)^(3/2))/(9*b^4) - (2*a^2*(a + b/x^3)^(5/2))/(5*b^4) + (2*a*(a + b/x^3)^(7/2))/(7*b^4) - (2*(
a + b/x^3)^(9/2))/(27*b^4)

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{\sqrt{a+\frac{b}{x^3}}}{x^{13}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int x^3 \sqrt{a+b x} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{3/2}}{b^3}-\frac{3 a (a+b x)^{5/2}}{b^3}+\frac{(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^4}\\ \end{align*}

Mathematica [A]  time = 0.013227, size = 60, normalized size = 0.75 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right ) \left (-24 a^2 b x^6+16 a^3 x^9+30 a b^2 x^3-35 b^3\right )}{945 b^4 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^3]/x^13,x]

[Out]

(2*Sqrt[a + b/x^3]*(b + a*x^3)*(-35*b^3 + 30*a*b^2*x^3 - 24*a^2*b*x^6 + 16*a^3*x^9))/(945*b^4*x^12)

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Maple [A]  time = 0.006, size = 61, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}-24\,{a}^{2}b{x}^{6}+30\,{x}^{3}a{b}^{2}-35\,{b}^{3} \right ) }{945\,{x}^{12}{b}^{4}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^3)^(1/2)/x^13,x)

[Out]

2/945*(a*x^3+b)*(16*a^3*x^9-24*a^2*b*x^6+30*a*b^2*x^3-35*b^3)*((a*x^3+b)/x^3)^(1/2)/x^12/b^4

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Maxima [A]  time = 0.965404, size = 86, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}}}{27 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a}{7 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}}{9 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^13,x, algorithm="maxima")

[Out]

-2/27*(a + b/x^3)^(9/2)/b^4 + 2/7*(a + b/x^3)^(7/2)*a/b^4 - 2/5*(a + b/x^3)^(5/2)*a^2/b^4 + 2/9*(a + b/x^3)^(3
/2)*a^3/b^4

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Fricas [A]  time = 1.51645, size = 144, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (16 \, a^{4} x^{12} - 8 \, a^{3} b x^{9} + 6 \, a^{2} b^{2} x^{6} - 5 \, a b^{3} x^{3} - 35 \, b^{4}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{945 \, b^{4} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^13,x, algorithm="fricas")

[Out]

2/945*(16*a^4*x^12 - 8*a^3*b*x^9 + 6*a^2*b^2*x^6 - 5*a*b^3*x^3 - 35*b^4)*sqrt((a*x^3 + b)/x^3)/(b^4*x^12)

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Sympy [B]  time = 5.04386, size = 2317, normalized size = 28.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**3)**(1/2)/x**13,x)

[Out]

32*a**(29/2)*b**(23/2)*x**30*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2
) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670
*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) + 176*a**(27/2)*b**(25/2)*x**27*sqrt(a*x**3/b + 1)/
(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(
15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*
x**(27/2)) + 396*a**(25/2)*b**(27/2)*x**24*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*
b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x*
*(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) + 462*a**(23/2)*b**(29/2)*x**21*sqrt(
a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2
) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a
**(9/2)*b**21*x**(27/2)) + 210*a**(21/2)*b**(31/2)*x**18*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5
670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(
13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 378*a**(19/2)*b**(33/
2)*x**15*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*
b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**
(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 1134*a**(17/2)*b**(35/2)*x**12*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**1
5*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/
2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 1494*a
**(15/2)*b**(37/2)*x**9*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 1
4175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(
11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 1098*a**(13/2)*b**(39/2)*x**6*sqrt(a*x**3/b + 1)/(945*
a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)
*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(2
7/2)) - 430*a**(11/2)*b**(41/2)*x**3*sqrt(a*x**3/b + 1)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*
x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2
) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 70*a**(9/2)*b**(43/2)*sqrt(a*x**3/b + 1)/
(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(
15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*
x**(27/2)) - 32*a**15*b**11*x**(63/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*
a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)
*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 192*a**14*b**12*x**(57/2)/(945*a**(21/2)*b**15*x**(63/2) +
5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**
(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 480*a**13*b**13*x**(
51/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 1890
0*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*
b**21*x**(27/2)) - 640*a**12*b**14*x**(45/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) +
 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a*
*(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 480*a**11*b**15*x**(39/2)/(945*a**(21/2)*b**15*x**(6
3/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14
175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2)) - 192*a**10*b**
16*x**(33/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57/2) + 14175*a**(17/2)*b**17*x**(51/2)
 + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 5670*a**(11/2)*b**20*x**(33/2) + 945*a*
*(9/2)*b**21*x**(27/2)) - 32*a**9*b**17*x**(27/2)/(945*a**(21/2)*b**15*x**(63/2) + 5670*a**(19/2)*b**16*x**(57
/2) + 14175*a**(17/2)*b**17*x**(51/2) + 18900*a**(15/2)*b**18*x**(45/2) + 14175*a**(13/2)*b**19*x**(39/2) + 56
70*a**(11/2)*b**20*x**(33/2) + 945*a**(9/2)*b**21*x**(27/2))

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Giac [A]  time = 1.2002, size = 77, normalized size = 0.96 \begin{align*} -\frac{2 \,{\left (35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a + 189 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}\right )}}{945 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/x^13,x, algorithm="giac")

[Out]

-2/945*(35*(a + b/x^3)^(9/2) - 135*(a + b/x^3)^(7/2)*a + 189*(a + b/x^3)^(5/2)*a^2 - 105*(a + b/x^3)^(3/2)*a^3
)/b^4

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