3.21.23 \(\int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{13}} \, dx\) [2023]

Optimal. Leaf size=80 \[ \frac {2 a^3 \sqrt {a+\frac {b}{x^3}}}{3 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{5/2}}{5 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^4} \]

[Out]

-2/3*a^2*(a+b/x^3)^(3/2)/b^4+2/5*a*(a+b/x^3)^(5/2)/b^4-2/21*(a+b/x^3)^(7/2)/b^4+2/3*a^3*(a+b/x^3)^(1/2)/b^4

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Rubi [A]
time = 0.03, antiderivative size = 80, 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 = {272, 45} \begin {gather*} \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 \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} \end {gather*}

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 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*} \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x^{13}} \, dx &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\left (\frac {1}{3} \text {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*}

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Mathematica [A]
time = 1.33, size = 53, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x^3}} \left (-5 b^3+6 a b^2 x^3-8 a^2 b x^6+16 a^3 x^9\right )}{105 b^4 x^9} \end {gather*}

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 = 0.06, size = 83, normalized size = 1.04

method result size
trager \(\frac {2 \left (16 a^{3} x^{9}-8 a^{2} b \,x^{6}+6 a \,b^{2} x^{3}-5 b^{3}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{105 x^{9} b^{4}}\) \(58\)
gosper \(\frac {2 \left (a \,x^{3}+b \right ) \left (16 a^{3} x^{9}-8 a^{2} b \,x^{6}+6 a \,b^{2} x^{3}-5 b^{3}\right )}{105 x^{12} b^{4} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) \(61\)
risch \(\frac {2 \left (a \,x^{3}+b \right ) \left (16 a^{3} x^{9}-8 a^{2} b \,x^{6}+6 a \,b^{2} x^{3}-5 b^{3}\right )}{105 x^{12} b^{4} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\) \(61\)
default \(\frac {2 \left (16 a^{3} x^{9}-8 a^{2} b \,x^{6}+6 a \,b^{2} x^{3}-5 b^{3}\right ) \sqrt {a \,x^{4}+b x}\, \left (a \,x^{3}+b \right )}{105 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x^{12} b^{4} \sqrt {x \left (a \,x^{3}+b \right )}}\) \(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105/((a*x^3+b)/x^3)^(1/2)/x^12*(16*a^3*x^9-8*a^2*b*x^6+6*a*b^2*x^3-5*b^3)*(a*x^4+b*x)^(1/2)*(a*x^3+b)/b^4/(x
*(a*x^3+b))^(1/2)

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Maxima [A]
time = 0.31, size = 64, normalized size = 0.80 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 53, normalized size = 0.66 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 2183 vs. \(2 (75) = 150\).
time = 1.94, size = 2183, normalized size = 27.29 \begin {gather*} \text {Too large to display} \end {gather*}

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 [A]
time = 1.05, size = 61, normalized size = 0.76 \begin {gather*} \frac {2 \, \sqrt {a + \frac {b}{x^{3}}} a^{3}}{3 \, b^{4}} - \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 21 \, {\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 )}}{105 \, b^{4}} \end {gather*}

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]

2/3*sqrt(a + b/x^3)*a^3/b^4 - 2/105*(5*(a + b/x^3)^(7/2) - 21*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*a^2)/
b^4

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Mupad [B]
time = 1.44, size = 73, normalized size = 0.91 \begin {gather*} \frac {32\,a^3\,\sqrt {a+\frac {b}{x^3}}}{105\,b^4}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{21\,b\,x^9}+\frac {4\,a\,\sqrt {a+\frac {b}{x^3}}}{35\,b^2\,x^6}-\frac {16\,a^2\,\sqrt {a+\frac {b}{x^3}}}{105\,b^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^13*(a + b/x^3)^(1/2)),x)

[Out]

(32*a^3*(a + b/x^3)^(1/2))/(105*b^4) - (2*(a + b/x^3)^(1/2))/(21*b*x^9) + (4*a*(a + b/x^3)^(1/2))/(35*b^2*x^6)
 - (16*a^2*(a + b/x^3)^(1/2))/(105*b^3*x^3)

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