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3.2016 \(\int \frac{(a+\frac{b}{x^3})^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4}+\frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{11/2}}{33 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{9/2}}{9 b^4} \]

[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)

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Rubi [A]  time = 0.0417916, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{7/2}}{7 b^4}+\frac{2 a^3 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^4}-\frac{2 \left (a+\frac{b}{x^3}\right )^{11/2}}{33 b^4}+\frac{2 a \left (a+\frac{b}{x^3}\right )^{9/2}}{9 b^4} \]

Antiderivative was successfully verified.

[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 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{\left (a+\frac{b}{x^3}\right )^{3/2}}{x^{13}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int 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 (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]  time = 0.0151555, size = 62, normalized size = 0.78 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right )^2 \left (-40 a^2 b x^6+16 a^3 x^9+70 a b^2 x^3-105 b^3\right )}{3465 b^4 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)^(3/2)/x^13,x]

[Out]

(2*Sqrt[a + b/x^3]*(b + a*x^3)^2*(-105*b^3 + 70*a*b^2*x^3 - 40*a^2*b*x^6 + 16*a^3*x^9))/(3465*b^4*x^15)

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Maple [A]  time = 0.005, size = 61, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 16\,{a}^{3}{x}^{9}-40\,{a}^{2}b{x}^{6}+70\,{x}^{3}a{b}^{2}-105\,{b}^{3} \right ) }{3465\,{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((a+b/x^3)^(3/2)/x^13,x)

[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

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Maxima [A]  time = 1.00128, size = 86, normalized size = 1.08 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.47918, size = 173, normalized size = 2.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 6.11066, size = 2317, normalized size = 28.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [B]  time = 1.29104, size = 181, normalized size = 2.26 \begin{align*} -\frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} - 135 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a + 189 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{315 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{11}{2}} - 1540 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}} a + 2970 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{4}}{b^{3}}\right )}}{10395 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/10395*(11*(35*(a + b/x^3)^(9/2) - 135*(a + b/x^3)^(7/2)*a + 189*(a + b/x^3)^(5/2)*a^2 - 105*(a + b/x^3)^(3/
2)*a^3)*a/b^3 + (315*(a + b/x^3)^(11/2) - 1540*(a + b/x^3)^(9/2)*a + 2970*(a + b/x^3)^(7/2)*a^2 - 2772*(a + b/
x^3)^(5/2)*a^3 + 1155*(a + b/x^3)^(3/2)*a^4)/b^3)/b

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