\(\int \frac {(a+\frac {b}{x^3})^{3/2}}{x^{13}} \, dx\) [2016]

Optimal result

Integrand size = 15, antiderivative size = 80 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=\frac {2 a^3 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{9/2}}{9 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{11/2}}{33 b^4} \]

[Out]

2/15*a^3*(a+b/x^3)^(5/2)/b^4-2/7*a^2*(a+b/x^3)^(7/2)/b^4+2/9*a*(a+b/x^3)^(9/2)/b^4-2/33*(a+b/x^3)^(11/2)/b^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=\frac {2 a^3 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{11/2}}{33 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{9/2}}{9 b^4} \]

[In]

Int[(a + b/x^3)^(3/2)/x^13,x]

[Out]

(2*a^3*(a + b/x^3)^(5/2))/(15*b^4) - (2*a^2*(a + b/x^3)^(7/2))/(7*b^4) + (2*a*(a + b/x^3)^(9/2))/(9*b^4) - (2*
(a + b/x^3)^(11/2))/(33*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*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int x^3 (a+b x)^{3/2} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{3/2}}{b^3}+\frac {3 a^2 (a+b x)^{5/2}}{b^3}-\frac {3 a (a+b x)^{7/2}}{b^3}+\frac {(a+b x)^{9/2}}{b^3}\right ) \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = \frac {2 a^3 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^4}-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^4}+\frac {2 a \left (a+\frac {b}{x^3}\right )^{9/2}}{9 b^4}-\frac {2 \left (a+\frac {b}{x^3}\right )^{11/2}}{33 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2} \left (b+a x^3\right ) \left (105 b^3-70 a b^2 x^3+40 a^2 b x^6-16 a^3 x^9\right )}{3465 b^4 x^{12}} \]

[In]

Integrate[(a + b/x^3)^(3/2)/x^13,x]

[Out]

(-2*(a + b/x^3)^(3/2)*(b + a*x^3)*(105*b^3 - 70*a*b^2*x^3 + 40*a^2*b*x^6 - 16*a^3*x^9))/(3465*b^4*x^12)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {2 \left (a \,x^{3}+b \right ) \left (16 a^{3} x^{9}-40 a^{2} b \,x^{6}+70 a \,b^{2} x^{3}-105 b^{3}\right ) \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}{3465 x^{12} b^{4}}\)	\(61\)
risch	\(\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \left (16 a^{5} x^{15}-8 a^{4} b \,x^{12}+6 a^{3} b^{2} x^{9}-5 a^{2} b^{3} x^{6}-140 a \,b^{4} x^{3}-105 b^{5}\right )}{3465 x^{15} b^{4}}\)	\(76\)
trager	\(\frac {2 \left (16 a^{5} x^{15}-8 a^{4} b \,x^{12}+6 a^{3} b^{2} x^{9}-5 a^{2} b^{3} x^{6}-140 a \,b^{4} x^{3}-105 b^{5}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{3465 x^{15} b^{4}}\)	\(80\)
default	\(\frac {2 \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}} \left (16 a^{4} x^{12}-24 a^{3} b \,x^{9}+30 a^{2} b^{2} x^{6}-35 a \,b^{3} x^{3}-105 b^{4}\right ) \sqrt {a \,x^{4}+b x}}{3465 x^{12} b^{4} \sqrt {x \left (a \,x^{3}+b \right )}}\)	\(87\)

[In]

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

[Out]

2/3465*(a*x^3+b)*(16*a^3*x^9-40*a^2*b*x^6+70*a*b^2*x^3-105*b^3)*((a*x^3+b)/x^3)^(3/2)/x^12/b^4

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=\frac {2 \, {\left (16 \, a^{5} x^{15} - 8 \, a^{4} b x^{12} + 6 \, a^{3} b^{2} x^{9} - 5 \, a^{2} b^{3} x^{6} - 140 \, a b^{4} x^{3} - 105 \, b^{5}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{3465 \, b^{4} x^{15}} \]

[In]

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

[Out]

2/3465*(16*a^5*x^15 - 8*a^4*b*x^12 + 6*a^3*b^2*x^9 - 5*a^2*b^3*x^6 - 140*a*b^4*x^3 - 105*b^5)*sqrt((a*x^3 + b)
/x^3)/(b^4*x^15)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2317 vs. \(2 (75) = 150\).

Time = 2.16 (sec) , antiderivative size = 2317, normalized size of antiderivative = 28.96 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

32*a**(33/2)*b**(23/2)*x**33*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63
/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20
790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) + 176*a**(31/2)*b**(25/2)*x**30*sqrt(a*x**3/b
+ 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 693
00*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11
/2)*b**21*x**(33/2)) + 396*a**(29/2)*b**(27/2)*x**27*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 2079
0*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15
/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) + 462*a**(27/2)*b**(29
/2)*x**24*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/
2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20
*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 1848*a**(23/2)*b**(33/2)*x**18*sqrt(a*x**3/b + 1)/(3465*a**(23/
2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18
*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2
)) - 5544*a**(21/2)*b**(35/2)*x**15*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16
*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/
2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 8844*a**(19/2)*b**(37/2)*x**12*sqrt(a
*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/
2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 346
5*a**(11/2)*b**21*x**(33/2)) - 8448*a**(17/2)*b**(39/2)*x**9*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2
) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 5197
5*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 4840*a**(15/
2)*b**(41/2)*x**6*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975
*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/
2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 1540*a**(13/2)*b**(43/2)*x**3*sqrt(a*x**3/b + 1)/(3465*
a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2
)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x
**(33/2)) - 210*a**(11/2)*b**(45/2)*sqrt(a*x**3/b + 1)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16
*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/
2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 32*a**17*b**11*x**(69/2)/(3465*a**(23
/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**1
8*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/
2)) - 192*a**16*b**12*x**(63/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(
19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b*
*20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 480*a**15*b**13*x**(57/2)/(3465*a**(23/2)*b**15*x**(69/2) +
20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a*
*(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 640*a**14*b**14*
x**(51/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2)
+ 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a
**(11/2)*b**21*x**(33/2)) - 480*a**13*b**15*x**(45/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*
x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2
) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2)) - 192*a**12*b**16*x**(39/2)/(3465*a**(23
/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(19/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**1
8*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/
2)) - 32*a**11*b**17*x**(33/2)/(3465*a**(23/2)*b**15*x**(69/2) + 20790*a**(21/2)*b**16*x**(63/2) + 51975*a**(1
9/2)*b**17*x**(57/2) + 69300*a**(17/2)*b**18*x**(51/2) + 51975*a**(15/2)*b**19*x**(45/2) + 20790*a**(13/2)*b**
20*x**(39/2) + 3465*a**(11/2)*b**21*x**(33/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {11}{2}}}{33 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}} a}{9 \, b^{4}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{3}}{15 \, b^{4}} \]

[In]

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

[Out]

-2/33*(a + b/x^3)^(11/2)/b^4 + 2/9*(a + b/x^3)^(9/2)*a/b^4 - 2/7*(a + b/x^3)^(7/2)*a^2/b^4 + 2/15*(a + b/x^3)^
(5/2)*a^3/b^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=-\frac {2 \, {\left (105 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {11}{2}} - 385 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {9}{2}} a + 495 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} a^{2} - 231 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{3}\right )}}{3465 \, b^{4}} \]

[In]

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

[Out]

-2/3465*(105*(a + b/x^3)^(11/2) - 385*(a + b/x^3)^(9/2)*a + 495*(a + b/x^3)^(7/2)*a^2 - 231*(a + b/x^3)^(5/2)*
a^3)/b^4

Mupad [B] (verification not implemented)

Time = 8.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^{13}} \, dx=\frac {32\,a^5\,\sqrt {a+\frac {b}{x^3}}}{3465\,b^4}-\frac {2\,b\,\sqrt {a+\frac {b}{x^3}}}{33\,x^{15}}-\frac {8\,a\,\sqrt {a+\frac {b}{x^3}}}{99\,x^{12}}-\frac {16\,a^4\,\sqrt {a+\frac {b}{x^3}}}{3465\,b^3\,x^3}+\frac {4\,a^3\,\sqrt {a+\frac {b}{x^3}}}{1155\,b^2\,x^6}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x^3}}}{693\,b\,x^9} \]

[In]

int((a + b/x^3)^(3/2)/x^13,x)

[Out]

(32*a^5*(a + b/x^3)^(1/2))/(3465*b^4) - (2*b*(a + b/x^3)^(1/2))/(33*x^15) - (8*a*(a + b/x^3)^(1/2))/(99*x^12)
- (16*a^4*(a + b/x^3)^(1/2))/(3465*b^3*x^3) + (4*a^3*(a + b/x^3)^(1/2))/(1155*b^2*x^6) - (2*a^2*(a + b/x^3)^(1
/2))/(693*b*x^9)