3.979 \(\int \frac{a+\frac{b}{x^2}}{(c+\frac{d}{x^2})^{3/2} x^9} \, dx\)

Optimal. Leaf size=126 \[ \frac{c^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{d^5}+\frac{c^3 (b c-a d)}{d^5 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5} \]

[Out]

(c^3*(b*c - a*d))/(d^5*Sqrt[c + d/x^2]) + (c^2*(4*b*c - 3*a*d)*Sqrt[c + d/x^2])/d^5 - (c*(2*b*c - a*d)*(c + d/
x^2)^(3/2))/d^5 + ((4*b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^5) - (b*(c + d/x^2)^(7/2))/(7*d^5)

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Rubi [A]  time = 0.0889519, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac{c^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{d^5}+\frac{c^3 (b c-a d)}{d^5 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-a d)}{5 d^5}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]

[Out]

(c^3*(b*c - a*d))/(d^5*Sqrt[c + d/x^2]) + (c^2*(4*b*c - 3*a*d)*Sqrt[c + d/x^2])/d^5 - (c*(2*b*c - a*d)*(c + d/
x^2)^(3/2))/d^5 + ((4*b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^5) - (b*(c + d/x^2)^(7/2))/(7*d^5)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0296777, size = 104, normalized size = 0.83 \[ \frac{b \left (-16 c^2 d^2 x^4+64 c^3 d x^6+128 c^4 x^8+8 c d^3 x^2-5 d^4\right )-7 a d x^2 \left (8 c^2 d x^4+16 c^3 x^6-2 c d^2 x^2+d^3\right )}{35 d^5 x^8 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^9),x]

[Out]

(-7*a*d*x^2*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6) + b*(-5*d^4 + 8*c*d^3*x^2 - 16*c^2*d^2*x^4 + 64*c^3
*d*x^6 + 128*c^4*x^8))/(35*d^5*Sqrt[c + d/x^2]*x^8)

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Maple [A]  time = 0.007, size = 118, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 112\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}+56\,a{c}^{2}{d}^{2}{x}^{6}-64\,b{c}^{3}d{x}^{6}-14\,ac{d}^{3}{x}^{4}+16\,b{c}^{2}{d}^{2}{x}^{4}+7\,a{d}^{4}{x}^{2}-8\,bc{d}^{3}{x}^{2}+5\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{35\,{d}^{5}{x}^{10}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x)

[Out]

-1/35*(112*a*c^3*d*x^8-128*b*c^4*x^8+56*a*c^2*d^2*x^6-64*b*c^3*d*x^6-14*a*c*d^3*x^4+16*b*c^2*d^2*x^4+7*a*d^4*x
^2-8*b*c*d^3*x^2+5*b*d^4)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^5/x^10

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Maxima [A]  time = 0.955883, size = 204, normalized size = 1.62 \begin{align*} -\frac{1}{35} \, b{\left (\frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{5}} - \frac{28 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{5}} + \frac{70 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{5}} - \frac{140 \, \sqrt{c + \frac{d}{x^{2}}} c^{3}}{d^{5}} - \frac{35 \, c^{4}}{\sqrt{c + \frac{d}{x^{2}}} d^{5}}\right )} - \frac{1}{5} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{4}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{4}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{4}} + \frac{5 \, c^{3}}{\sqrt{c + \frac{d}{x^{2}}} d^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x, algorithm="maxima")

[Out]

-1/35*b*(5*(c + d/x^2)^(7/2)/d^5 - 28*(c + d/x^2)^(5/2)*c/d^5 + 70*(c + d/x^2)^(3/2)*c^2/d^5 - 140*sqrt(c + d/
x^2)*c^3/d^5 - 35*c^4/(sqrt(c + d/x^2)*d^5)) - 1/5*a*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*s
qrt(c + d/x^2)*c^2/d^4 + 5*c^3/(sqrt(c + d/x^2)*d^4))

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Fricas [A]  time = 1.48715, size = 252, normalized size = 2. \begin{align*} \frac{{\left (16 \,{\left (8 \, b c^{4} - 7 \, a c^{3} d\right )} x^{8} + 8 \,{\left (8 \, b c^{3} d - 7 \, a c^{2} d^{2}\right )} x^{6} - 5 \, b d^{4} - 2 \,{\left (8 \, b c^{2} d^{2} - 7 \, a c d^{3}\right )} x^{4} +{\left (8 \, b c d^{3} - 7 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \,{\left (c d^{5} x^{8} + d^{6} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x, algorithm="fricas")

[Out]

1/35*(16*(8*b*c^4 - 7*a*c^3*d)*x^8 + 8*(8*b*c^3*d - 7*a*c^2*d^2)*x^6 - 5*b*d^4 - 2*(8*b*c^2*d^2 - 7*a*c*d^3)*x
^4 + (8*b*c*d^3 - 7*a*d^4)*x^2)*sqrt((c*x^2 + d)/x^2)/(c*d^5*x^8 + d^6*x^6)

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Sympy [A]  time = 12.6597, size = 122, normalized size = 0.97 \begin{align*} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7 d^{5}} - \frac{c^{3} \left (a d - b c\right )}{d^{5} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (a d - 4 b c\right )}{5 d^{5}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (- 3 a c d + 6 b c^{2}\right )}{3 d^{5}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (3 a c^{2} d - 4 b c^{3}\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**9,x)

[Out]

-b*(c + d/x**2)**(7/2)/(7*d**5) - c**3*(a*d - b*c)/(d**5*sqrt(c + d/x**2)) - (c + d/x**2)**(5/2)*(a*d - 4*b*c)
/(5*d**5) - (c + d/x**2)**(3/2)*(-3*a*c*d + 6*b*c**2)/(3*d**5) - sqrt(c + d/x**2)*(3*a*c**2*d - 4*b*c**3)/d**5

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^9), x)