3.2043 \(\int \frac{1}{(a+\frac{b}{x^3})^{3/2} x^{13}} \, dx\)
Optimal. Leaf size=78 \[ -\frac{2 a^3}{3 b^4 \sqrt{a+\frac{b}{x^3}}}-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4} \]
[Out]
(-2*a^3)/(3*b^4*Sqrt[a + b/x^3]) - (2*a^2*Sqrt[a + b/x^3])/b^4 + (2*a*(a + b/x^3)^(3/2))/(3*b^4) - (2*(a + b/x
^3)^(5/2))/(15*b^4)
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Rubi [A] time = 0.0400413, antiderivative size = 78, 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} \[ -\frac{2 a^3}{3 b^4 \sqrt{a+\frac{b}{x^3}}}-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b/x^3)^(3/2)*x^13),x]
[Out]
(-2*a^3)/(3*b^4*Sqrt[a + b/x^3]) - (2*a^2*Sqrt[a + b/x^3])/b^4 + (2*a*(a + b/x^3)^(3/2))/(3*b^4) - (2*(a + b/x
^3)^(5/2))/(15*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}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{13}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{3/2}}+\frac{3 a^2}{b^3 \sqrt{a+b x}}-\frac{3 a \sqrt{a+b x}}{b^3}+\frac{(a+b x)^{3/2}}{b^3}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 a^3}{3 b^4 \sqrt{a+\frac{b}{x^3}}}-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4}\\ \end{align*}
Mathematica [A] time = 0.0138454, size = 51, normalized size = 0.65 \[ -\frac{2 \left (8 a^2 b x^6+16 a^3 x^9-2 a b^2 x^3+b^3\right )}{15 b^4 x^9 \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b/x^3)^(3/2)*x^13),x]
[Out]
(-2*(b^3 - 2*a*b^2*x^3 + 8*a^2*b*x^6 + 16*a^3*x^9))/(15*b^4*Sqrt[a + b/x^3]*x^9)
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Maple [A] time = 0.006, size = 59, 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}-2\,{x}^{3}a{b}^{2}+{b}^{3} \right ) }{15\,{x}^{12}{b}^{4}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x^3)^(3/2)/x^13,x)
[Out]
-2/15*(a*x^3+b)*(16*a^3*x^9+8*a^2*b*x^6-2*a*b^2*x^3+b^3)/x^12/b^4/((a*x^3+b)/x^3)^(3/2)
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Maxima [A] time = 0.970286, size = 86, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{3 \, b^{4}} - \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{2}}{b^{4}} - \frac{2 \, a^{3}}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="maxima")
[Out]
-2/15*(a + b/x^3)^(5/2)/b^4 + 2/3*(a + b/x^3)^(3/2)*a/b^4 - 2*sqrt(a + b/x^3)*a^2/b^4 - 2/3*a^3/(sqrt(a + b/x^
3)*b^4)
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Fricas [A] time = 1.85248, size = 132, normalized size = 1.69 \begin{align*} -\frac{2 \,{\left (16 \, a^{3} x^{9} + 8 \, a^{2} b x^{6} - 2 \, a b^{2} x^{3} + b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{15 \,{\left (a b^{4} x^{9} + b^{5} x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="fricas")
[Out]
-2/15*(16*a^3*x^9 + 8*a^2*b*x^6 - 2*a*b^2*x^3 + b^3)*sqrt((a*x^3 + b)/x^3)/(a*b^4*x^9 + b^5*x^6)
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Sympy [B] time = 7.98018, size = 2048, normalized size = 26.26 \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**3)**(3/2)/x**13,x)
[Out]
-32*a**(21/2)*b**(23/2)*x**24*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2)
+ 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b
**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 176*a**(19/2)*b**(25/2)*x**21*sqrt(a*x**3/b + 1)/(15*a**(17/2)
*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2
) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 396*a**(17/2)*
b**(27/2)*x**18*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2
)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2)
+ 15*a**(5/2)*b**21*x**(15/2)) - 462*a**(15/2)*b**(29/2)*x**15*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/
2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/
2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 290*a**(13/2)*b**(31/2)*x**1
2*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39
/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)
*b**21*x**(15/2)) - 92*a**(11/2)*b**(33/2)*x**9*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2
)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/
2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 16*a**(9/2)*b**(35/2)*x**6*sqrt(a*x**3/b + 1
)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)
*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) -
6*a**(7/2)*b**(37/2)*x**3*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 2
25*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**2
0*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) - 2*a**(5/2)*b**(39/2)*sqrt(a*x**3/b + 1)/(15*a**(17/2)*b**15*x**(5
1/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(
9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 32*a**11*b**11*x**(51/2)/(
15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b*
*18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 19
2*a**10*b**12*x**(45/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**
(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5
/2)*b**21*x**(15/2)) + 480*a**9*b**13*x**(39/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) +
225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b*
*20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 640*a**8*b**14*x**(33/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**
(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x*
*(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 480*a**7*b**15*x**(27/2)/(15*a**(17/2)*
b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2)
+ 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(15/2)) + 192*a**6*b**16*
x**(21/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*b**17*x**(39/2) + 300*a
**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) + 15*a**(5/2)*b**21*x**(
15/2)) + 32*a**5*b**17*x**(15/2)/(15*a**(17/2)*b**15*x**(51/2) + 90*a**(15/2)*b**16*x**(45/2) + 225*a**(13/2)*
b**17*x**(39/2) + 300*a**(11/2)*b**18*x**(33/2) + 225*a**(9/2)*b**19*x**(27/2) + 90*a**(7/2)*b**20*x**(21/2) +
15*a**(5/2)*b**21*x**(15/2))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{13}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x^3)^(3/2)/x^13,x, algorithm="giac")
[Out]
integrate(1/((a + b/x^3)^(3/2)*x^13), x)