3.2023 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{13}} \, dx\)
Optimal. Leaf size=80 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}+\frac{2 a^3 \sqrt{a+\frac{b}{x^3}}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4} \]
[Out]
(2*a^3*Sqrt[a + b/x^3])/(3*b^4) - (2*a^2*(a + b/x^3)^(3/2))/(3*b^4) + (2*a*(a + b/x^3)^(5/2))/(5*b^4) - (2*(a
+ b/x^3)^(7/2))/(21*b^4)
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Rubi [A] time = 0.039535, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}+\frac{2 a^3 \sqrt{a+\frac{b}{x^3}}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[a + b/x^3]*x^13),x]
[Out]
(2*a^3*Sqrt[a + b/x^3])/(3*b^4) - (2*a^2*(a + b/x^3)^(3/2))/(3*b^4) + (2*a*(a + b/x^3)^(5/2))/(5*b^4) - (2*(a
+ b/x^3)^(7/2))/(21*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{1}{\sqrt{a+\frac{b}{x^3}} x^{13}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{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}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a^3 \sqrt{a+\frac{b}{x^3}}}{3 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^4}\\ \end{align*}
Mathematica [A] time = 0.0189044, size = 53, normalized size = 0.66 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (-8 a^2 b x^6+16 a^3 x^9+6 a b^2 x^3-5 b^3\right )}{105 b^4 x^9} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[a + b/x^3]*x^13),x]
[Out]
(2*Sqrt[a + b/x^3]*(-5*b^3 + 6*a*b^2*x^3 - 8*a^2*b*x^6 + 16*a^3*x^9))/(105*b^4*x^9)
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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}-8\,{a}^{2}b{x}^{6}+6\,{x}^{3}a{b}^{2}-5\,{b}^{3} \right ) }{105\,{x}^{12}{b}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^13/(a+b/x^3)^(1/2),x)
[Out]
2/105*(a*x^3+b)*(16*a^3*x^9-8*a^2*b*x^6+6*a*b^2*x^3-5*b^3)/x^12/b^4/((a*x^3+b)/x^3)^(1/2)
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Maxima [A] time = 0.962616, size = 86, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}}}{21 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a}{5 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{4}} + \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{3}}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^13/(a+b/x^3)^(1/2),x, algorithm="maxima")
[Out]
-2/21*(a + b/x^3)^(7/2)/b^4 + 2/5*(a + b/x^3)^(5/2)*a/b^4 - 2/3*(a + b/x^3)^(3/2)*a^2/b^4 + 2/3*sqrt(a + b/x^3
)*a^3/b^4
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Fricas [A] time = 1.49936, size = 119, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (16 \, a^{3} x^{9} - 8 \, a^{2} b x^{6} + 6 \, a b^{2} x^{3} - 5 \, b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{105 \, b^{4} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^13/(a+b/x^3)^(1/2),x, algorithm="fricas")
[Out]
2/105*(16*a^3*x^9 - 8*a^2*b*x^6 + 6*a*b^2*x^3 - 5*b^3)*sqrt((a*x^3 + b)/x^3)/(b^4*x^9)
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Sympy [B] time = 7.78618, size = 2183, normalized size = 27.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**13/(a+b/x**3)**(1/2),x)
[Out]
32*a**(25/2)*b**(23/2)*x**27*sqrt(a*x**3/b + 1)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2)
+ 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(
9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) + 176*a**(23/2)*b**(25/2)*x**24*sqrt(a*x**3/b + 1)/(105*a
**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**
18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) +
396*a**(21/2)*b**(27/2)*x**21*sqrt(a*x**3/b + 1)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/
2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a*
*(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) + 462*a**(19/2)*b**(29/2)*x**18*sqrt(a*x**3/b + 1)/(105
*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b
**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2))
+ 280*a**(17/2)*b**(31/2)*x**15*sqrt(a*x**3/b + 1)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(5
1/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*
a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) + 42*a**(15/2)*b**(33/2)*x**12*sqrt(a*x**3/b + 1)/(10
5*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*
b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)
) - 84*a**(13/2)*b**(35/2)*x**9*sqrt(a*x**3/b + 1)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51
/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a
**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 94*a**(11/2)*b**(37/2)*x**6*sqrt(a*x**3/b + 1)/(105*
a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b*
*18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2))
- 48*a**(9/2)*b**(39/2)*x**3*sqrt(a*x**3/b + 1)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2)
+ 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(
9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 10*a**(7/2)*b**(41/2)*sqrt(a*x**3/b + 1)/(105*a**(19/2)
*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(3
9/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 32*a**1
3*b**11*x**(57/2)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45
/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**
(7/2)*b**21*x**(21/2)) - 192*a**12*b**12*x**(51/2)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51
/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a
**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 480*a**11*b**13*x**(45/2)/(105*a**(19/2)*b**15*x**(5
7/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*
a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 640*a**10*b**14*x**
(39/2)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*
a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21
*x**(21/2)) - 480*a**9*b**15*x**(33/2)/(105*a**(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a
**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**2
0*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 192*a**8*b**16*x**(27/2)/(105*a**(19/2)*b**15*x**(57/2) + 630*a*
*(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18*x**(39/2) + 1575*a**(11/2)*b**
19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2)) - 32*a**7*b**17*x**(21/2)/(105*a**
(19/2)*b**15*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2100*a**(13/2)*b**18
*x**(39/2) + 1575*a**(11/2)*b**19*x**(33/2) + 630*a**(9/2)*b**20*x**(27/2) + 105*a**(7/2)*b**21*x**(21/2))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{13}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^13/(a+b/x^3)^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(a + b/x^3)*x^13), x)