3.2021 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{13}} \, dx\)
Optimal. Leaf size=80 \[ \frac{2 a^3 \sqrt{a+\frac{b}{x^3}}}{3 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4} \]
[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 = 0.114078, 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
\[ \frac{2 a^3 \sqrt{a+\frac{b}{x^3}}}{3 b^4}-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{5 b^4} \]
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)
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Rubi in Sympy [A] time = 14.1965, size = 75, normalized size = 0.94 \[ \frac{2 a^{3} \sqrt{a + \frac{b}{x^{3}}}}{3 b^{4}} - \frac{2 a^{2} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{3 b^{4}} + \frac{2 a \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{7}{2}}}{21 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**13/(a+b/x**3)**(1/2),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)
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Mathematica [A] time = 0.0558844, size = 53, normalized size = 0.66 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (16 a^3 x^9-8 a^2 b x^6+6 a b^2 x^3-5 b^3\right )}{105 b^4 x^9} \]
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 = 0.009, size = 61, normalized size = 0.8 \[{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}-8\,{a}^{2}b{x}^{6}+6\,a{b}^{2}{x}^{3}-5\,{b}^{3} \right ) }{105\,{x}^{12}{b}^{4}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \]
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 = 1.43814, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}}}{21 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a}{5 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{4}} + \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^3)*x^13),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 = 0.244643, size = 72, normalized size = 0.9 \[ \frac{2 \,{\left (16 \, a^{3} x^{9} - 8 \, a^{2} b x^{6} + 6 \, a b^{2} x^{3} - 5 \, b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{105 \, b^{4} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^3)*x^13),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 [A] time = 28.733, size = 2183, normalized size = 27.29 \[ \text{result too large to display} \]
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**(2
1/2)*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**1
7*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**(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)) + 42*a**(15/2)*b**(33/2)*x**12*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)) - 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**(2
1/2)) - 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**13*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**1
5*x**(57/2) + 630*a**(17/2)*b**16*x**(51/2) + 1575*a**(15/2)*b**17*x**(45/2) + 2
100*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**(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) + 63
0*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**(1
1/2)*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**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/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^3)*x^13),x, algorithm="giac")
[Out]
integrate(1/(sqrt(a + b/x^3)*x^13), x)