3.936 \(\int \frac{(a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}}}{x^9} \, dx\)

Optimal. Leaf size=134 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5} \]

[Out]

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

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Rubi [A]  time = 0.101759, antiderivative size = 134, 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} \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}-\frac{c^3 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0633148, size = 90, normalized size = 0.67 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (x^2 \left (\frac{c x^2}{d}+1\right ) \left (24 c^2 d x^4-16 c^3 x^6-30 c d^2 x^2+35 d^3\right ) (8 b c-11 a d)-315 b d^3 \left (c x^2+d\right )\right )}{3465 d^4 x^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*(-315*b*d^3*(d + c*x^2) + (8*b*c - 11*a*d)*x^2*(1 + (c*x^2)/d)*(35*d^3 - 30*c*d^2*x^2 + 24*c^
2*d*x^4 - 16*c^3*x^6)))/(3465*d^4*x^10)

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Maple [A]  time = 0.007, size = 118, normalized size = 0.9 \begin{align*}{\frac{ \left ( 176\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}-264\,a{c}^{2}{d}^{2}{x}^{6}+192\,b{c}^{3}d{x}^{6}+330\,ac{d}^{3}{x}^{4}-240\,b{c}^{2}{d}^{2}{x}^{4}-385\,a{d}^{4}{x}^{2}+280\,bc{d}^{3}{x}^{2}-315\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{d}^{5}{x}^{10}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)*(c+d/x^2)^(1/2)/x^9,x)

[Out]

1/3465*((c*x^2+d)/x^2)^(1/2)*(176*a*c^3*d*x^8-128*b*c^4*x^8-264*a*c^2*d^2*x^6+192*b*c^3*d*x^6+330*a*c*d^3*x^4-
240*b*c^2*d^2*x^4-385*a*d^4*x^2+280*b*c*d^3*x^2-315*b*d^4)*(c*x^2+d)/d^5/x^10

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Maxima [A]  time = 0.957736, size = 205, normalized size = 1.53 \begin{align*} -\frac{1}{3465} \, b{\left (\frac{315 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}}}{d^{5}} - \frac{1540 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c}{d^{5}} + \frac{2970 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{2}}{d^{5}} - \frac{2772 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{3}}{d^{5}} + \frac{1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{4}}{d^{5}}\right )} - \frac{1}{315} \, a{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{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)^(1/2)/x^9,x, algorithm="maxima")

[Out]

-1/3465*b*(315*(c + d/x^2)^(11/2)/d^5 - 1540*(c + d/x^2)^(9/2)*c/d^5 + 2970*(c + d/x^2)^(7/2)*c^2/d^5 - 2772*(
c + d/x^2)^(5/2)*c^3/d^5 + 1155*(c + d/x^2)^(3/2)*c^4/d^5) - 1/315*a*(35*(c + d/x^2)^(9/2)/d^4 - 135*(c + d/x^
2)^(7/2)*c/d^4 + 189*(c + d/x^2)^(5/2)*c^2/d^4 - 105*(c + d/x^2)^(3/2)*c^3/d^4)

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Fricas [A]  time = 1.79989, size = 302, normalized size = 2.25 \begin{align*} -\frac{{\left (16 \,{\left (8 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 8 \,{\left (8 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 6 \,{\left (8 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} + 315 \, b d^{5} - 5 \,{\left (8 \, b c^{2} d^{3} - 11 \, a c d^{4}\right )} x^{4} + 35 \,{\left (b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, d^{5} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^9,x, algorithm="fricas")

[Out]

-1/3465*(16*(8*b*c^5 - 11*a*c^4*d)*x^10 - 8*(8*b*c^4*d - 11*a*c^3*d^2)*x^8 + 6*(8*b*c^3*d^2 - 11*a*c^2*d^3)*x^
6 + 315*b*d^5 - 5*(8*b*c^2*d^3 - 11*a*c*d^4)*x^4 + 35*(b*c*d^4 + 11*a*d^5)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^5*x^1
0)

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Sympy [A]  time = 5.68921, size = 146, normalized size = 1.09 \begin{align*} - \frac{a \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} - \frac{b \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\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)**(1/2)/x**9,x)

[Out]

-a*(-c**3*(c + d/x**2)**(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + d/x**2)**(9/
2)/9)/d**4 - b*(c**4*(c + d/x**2)**(3/2)/3 - 4*c**3*(c + d/x**2)**(5/2)/5 + 6*c**2*(c + d/x**2)**(7/2)/7 - 4*c
*(c + d/x**2)**(9/2)/9 + (c + d/x**2)**(11/2)/11)/d**5

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Giac [B]  time = 7.0966, size = 581, normalized size = 4.34 \begin{align*} \frac{32 \,{\left (3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{9}{2}} d \mathrm{sgn}\left (x\right ) + 7392 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{11}{2}} d \mathrm{sgn}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{9}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 2640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{11}{2}} d^{2} \mathrm{sgn}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{9}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 1320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{11}{2}} d^{3} \mathrm{sgn}\left (x\right ) + 1815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{9}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{11}{2}} d^{4} \mathrm{sgn}\left (x\right ) - 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{9}{2}} d^{5} \mathrm{sgn}\left (x\right ) - 88 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{11}{2}} d^{5} \mathrm{sgn}\left (x\right ) + 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{9}{2}} d^{6} \mathrm{sgn}\left (x\right ) + 8 \, b c^{\frac{11}{2}} d^{6} \mathrm{sgn}\left (x\right ) - 11 \, a c^{\frac{9}{2}} d^{7} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(1/2)/x^9,x, algorithm="giac")

[Out]

32/3465*(3465*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(9/2)*sgn(x) + 11088*(sqrt(c)*x - sqrt(c*x^2 + d))^12*b*c^(
11/2)*sgn(x) - 4851*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(9/2)*d*sgn(x) + 7392*(sqrt(c)*x - sqrt(c*x^2 + d))^1
0*b*c^(11/2)*d*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(9/2)*d^2*sgn(x) + 2640*(sqrt(c)*x - sqrt(c*x
^2 + d))^8*b*c^(11/2)*d^2*sgn(x) - 165*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(9/2)*d^3*sgn(x) - 1320*(sqrt(c)*x
- sqrt(c*x^2 + d))^6*b*c^(11/2)*d^3*sgn(x) + 1815*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(9/2)*d^4*sgn(x) + 440*(
sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(11/2)*d^4*sgn(x) - 605*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(9/2)*d^5*sgn(x
) - 88*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(11/2)*d^5*sgn(x) + 121*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(9/2)*d
^6*sgn(x) + 8*b*c^(11/2)*d^6*sgn(x) - 11*a*c^(9/2)*d^7*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^11