3.2015 \(\int \frac{(a+\frac{b}{x^3})^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3} \]

[Out]

(-2*a^2*(a + b/x^3)^(5/2))/(15*b^3) + (4*a*(a + b/x^3)^(7/2))/(21*b^3) - (2*(a + b/x^3)^(9/2))/(27*b^3)

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Rubi [A]  time = 0.0316208, antiderivative size = 59, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^3)^(3/2)/x^10,x]

[Out]

(-2*a^2*(a + b/x^3)^(5/2))/(15*b^3) + (4*a*(a + b/x^3)^(7/2))/(21*b^3) - (2*(a + b/x^3)^(9/2))/(27*b^3)

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^{10}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{3/2}}{b^2}-\frac{2 a (a+b x)^{5/2}}{b^2}+\frac{(a+b x)^{7/2}}{b^2}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{9/2}}{27 b^3}\\ \end{align*}

Mathematica [A]  time = 0.013689, size = 51, normalized size = 0.86 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right )^2 \left (8 a^2 x^6-20 a b x^3+35 b^2\right )}{945 b^3 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)^(3/2)/x^10,x]

[Out]

(-2*Sqrt[a + b/x^3]*(b + a*x^3)^2*(35*b^2 - 20*a*b*x^3 + 8*a^2*x^6))/(945*b^3*x^12)

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Maple [A]  time = 0.006, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-20\,{x}^{3}ab+35\,{b}^{2} \right ) }{945\,{b}^{3}{x}^{9}} \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^10,x)

[Out]

-2/945*(a*x^3+b)*(8*a^2*x^6-20*a*b*x^3+35*b^2)*((a*x^3+b)/x^3)^(3/2)/b^3/x^9

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Maxima [A]  time = 0.990275, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{9}{2}}}{27 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} a}{21 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a^{2}}{15 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^10,x, algorithm="maxima")

[Out]

-2/27*(a + b/x^3)^(9/2)/b^3 + 4/21*(a + b/x^3)^(7/2)*a/b^3 - 2/15*(a + b/x^3)^(5/2)*a^2/b^3

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Fricas [A]  time = 1.46949, size = 146, normalized size = 2.47 \begin{align*} -\frac{2 \,{\left (8 \, a^{4} x^{12} - 4 \, a^{3} b x^{9} + 3 \, a^{2} b^{2} x^{6} + 50 \, a b^{3} x^{3} + 35 \, b^{4}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{945 \, b^{3} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^10,x, algorithm="fricas")

[Out]

-2/945*(8*a^4*x^12 - 4*a^3*b*x^9 + 3*a^2*b^2*x^6 + 50*a*b^3*x^3 + 35*b^4)*sqrt((a*x^3 + b)/x^3)/(b^3*x^12)

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Sympy [B]  time = 3.86603, size = 1001, normalized size = 16.97 \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**10,x)

[Out]

-16*a**(23/2)*b**(9/2)*x**21*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2)
+ 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 40*a**(21/2)*b**(11/2)*x**18*sqrt(a*x**3/b +
1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/
2)*b**10*x**(27/2)) - 30*a**(19/2)*b**(13/2)*x**15*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**
(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 110*a**(17/2)*b**(15/2
)*x**12*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9
*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 380*a**(15/2)*b**(17/2)*x**9*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b*
*7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) -
516*a**(13/2)*b**(19/2)*x**6*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2)
+ 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) - 310*a**(11/2)*b**(21/2)*x**3*sqrt(a*x**3/b
+ 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9
/2)*b**10*x**(27/2)) - 70*a**(9/2)*b**(23/2)*sqrt(a*x**3/b + 1)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)
*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) + 16*a**12*b**4*x**(45/2)/(945
*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10
*x**(27/2)) + 48*a**11*b**5*x**(39/2)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**
(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) + 48*a**10*b**6*x**(33/2)/(945*a**(15/2)*b**7*x**(45/2)
+ 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945*a**(9/2)*b**10*x**(27/2)) + 16*a**9*b**7
*x**(27/2)/(945*a**(15/2)*b**7*x**(45/2) + 2835*a**(13/2)*b**8*x**(39/2) + 2835*a**(11/2)*b**9*x**(33/2) + 945
*a**(9/2)*b**10*x**(27/2))

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Giac [B]  time = 1.18971, size = 143, normalized size = 2.42 \begin{align*} -\frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} - 42 \,{\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 )} a}{b^{2}} + \frac{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}}{b^{2}}\right )}}{945 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^10,x, algorithm="giac")

[Out]

-2/945*(3*(15*(a + b/x^3)^(7/2) - 42*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*a^2)*a/b^2 + (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)/b^2)/b