3.937 \(\int (a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}} x^{10} \, dx\)

Optimal. Leaf size=150 \[ \frac{8 d^2 x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac{16 d^3 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac{x^9 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}-\frac{2 d x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac{a x^{11} \left (c+\frac{d}{x^2}\right )^{3/2}}{11 c} \]

[Out]

(-16*d^3*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^3)/(3465*c^5) + (8*d^2*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^5)/(
1155*c^4) - (2*d*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^7)/(231*c^3) + ((11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^9)/
(99*c^2) + (a*(c + d/x^2)^(3/2)*x^11)/(11*c)

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Rubi [A]  time = 0.0725086, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac{8 d^2 x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac{16 d^3 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac{x^9 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}-\frac{2 d x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac{a x^{11} \left (c+\frac{d}{x^2}\right )^{3/2}}{11 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^10,x]

[Out]

(-16*d^3*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^3)/(3465*c^5) + (8*d^2*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^5)/(
1155*c^4) - (2*d*(11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^7)/(231*c^3) + ((11*b*c - 8*a*d)*(c + d/x^2)^(3/2)*x^9)/
(99*c^2) + (a*(c + d/x^2)^(3/2)*x^11)/(11*c)

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^{10} \, dx &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^{11}}{11 c}+\frac{(11 b c-8 a d) \int \sqrt{c+\frac{d}{x^2}} x^8 \, dx}{11 c}\\ &=\frac{(11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^{11}}{11 c}-\frac{(2 d (11 b c-8 a d)) \int \sqrt{c+\frac{d}{x^2}} x^6 \, dx}{33 c^2}\\ &=-\frac{2 d (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac{(11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^{11}}{11 c}+\frac{\left (8 d^2 (11 b c-8 a d)\right ) \int \sqrt{c+\frac{d}{x^2}} x^4 \, dx}{231 c^3}\\ &=\frac{8 d^2 (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^5}{1155 c^4}-\frac{2 d (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac{(11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^{11}}{11 c}-\frac{\left (16 d^3 (11 b c-8 a d)\right ) \int \sqrt{c+\frac{d}{x^2}} x^2 \, dx}{1155 c^4}\\ &=-\frac{16 d^3 (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^3}{3465 c^5}+\frac{8 d^2 (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^5}{1155 c^4}-\frac{2 d (11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac{(11 b c-8 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^{11}}{11 c}\\ \end{align*}

Mathematica [A]  time = 0.0708872, size = 108, normalized size = 0.72 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (a \left (240 c^2 d^2 x^4-280 c^3 d x^6+315 c^4 x^8-192 c d^3 x^2+128 d^4\right )+11 b c \left (-30 c^2 d x^4+35 c^3 x^6+24 c d^2 x^2-16 d^3\right )\right )}{3465 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^10,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)*(11*b*c*(-16*d^3 + 24*c*d^2*x^2 - 30*c^2*d*x^4 + 35*c^3*x^6) + a*(128*d^4 - 192
*c*d^3*x^2 + 240*c^2*d^2*x^4 - 280*c^3*d*x^6 + 315*c^4*x^8)))/(3465*c^5)

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Maple [A]  time = 0.006, size = 113, normalized size = 0.8 \begin{align*}{\frac{x \left ( 315\,a{x}^{8}{c}^{4}-280\,a{c}^{3}d{x}^{6}+385\,b{c}^{4}{x}^{6}+240\,a{c}^{2}{d}^{2}{x}^{4}-330\,b{c}^{3}d{x}^{4}-192\,ac{d}^{3}{x}^{2}+264\,b{c}^{2}{d}^{2}{x}^{2}+128\,a{d}^{4}-176\,bc{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{c}^{5}}\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)*x^10*(c+d/x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.949773, size = 213, normalized size = 1.42 \begin{align*} \frac{{\left (35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} x^{9} - 135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d x^{7} + 189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{2} x^{5} - 105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{3} x^{3}\right )} b}{315 \, c^{4}} + \frac{{\left (315 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}} x^{11} - 1540 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} d x^{9} + 2970 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d^{2} x^{7} - 2772 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{3} x^{5} + 1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{4} x^{3}\right )} a}{3465 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^10*(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35243, size = 294, normalized size = 1.96 \begin{align*} \frac{{\left (315 \, a c^{5} x^{11} + 35 \,{\left (11 \, b c^{5} + a c^{4} d\right )} x^{9} + 5 \,{\left (11 \, b c^{4} d - 8 \, a c^{3} d^{2}\right )} x^{7} - 6 \,{\left (11 \, b c^{3} d^{2} - 8 \, a c^{2} d^{3}\right )} x^{5} + 8 \,{\left (11 \, b c^{2} d^{3} - 8 \, a c d^{4}\right )} x^{3} - 16 \,{\left (11 \, b c d^{4} - 8 \, a d^{5}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^10*(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

1/3465*(315*a*c^5*x^11 + 35*(11*b*c^5 + a*c^4*d)*x^9 + 5*(11*b*c^4*d - 8*a*c^3*d^2)*x^7 - 6*(11*b*c^3*d^2 - 8*
a*c^2*d^3)*x^5 + 8*(11*b*c^2*d^3 - 8*a*c*d^4)*x^3 - 16*(11*b*c*d^4 - 8*a*d^5)*x)*sqrt((c*x^2 + d)/x^2)/c^5

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Sympy [B]  time = 9.08156, size = 1386, normalized size = 9.24 \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**2)*x**10*(c+d/x**2)**(1/2),x)

[Out]

315*a*c**9*d**(33/2)*x**18*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18
*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1295*a*c**8*d**(35/2)*x**16*sqrt(c*x**2/d + 1)/(3465*c**9*d
**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1990*a*
c**7*d**(37/2)*x**14*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4
+ 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1358*a*c**6*d**(39/2)*x**12*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x
**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 343*a*c**5*d*
*(41/2)*x**10*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860
*c**6*d**19*x**2 + 3465*c**5*d**20) + 35*a*c**4*d**(43/2)*x**8*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 1386
0*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 280*a*c**3*d**(45/2)*x*
*6*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19
*x**2 + 3465*c**5*d**20) + 560*a*c**2*d**(47/2)*x**4*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**
17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 448*a*c*d**(49/2)*x**2*sqrt(c*x**
2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*
c**5*d**20) + 128*a*d**(51/2)*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d*
*18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 35*b*c**7*d**(19/2)*x**14*sqrt(c*x**2/d + 1)/(315*c**7*d
**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 110*b*c**6*d**(21/2)*x**12*sqrt(c*x**
2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 114*b*c**5*d**(23
/2)*x**10*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)
+ 40*b*c**4*d**(25/2)*x**8*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2
+ 315*c**4*d**12) - 5*b*c**3*d**(27/2)*x**6*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 94
5*c**5*d**11*x**2 + 315*c**4*d**12) - 30*b*c**2*d**(29/2)*x**4*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c*
*6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) - 40*b*c*d**(31/2)*x**2*sqrt(c*x**2/d + 1)/(315*c**7*d**
9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) - 16*b*d**(33/2)*sqrt(c*x**2/d + 1)/(315*
c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)

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Giac [A]  time = 1.13441, size = 217, normalized size = 1.45 \begin{align*} \frac{\frac{11 \,{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} b \mathrm{sgn}\left (x\right )}{c^{3}} + \frac{{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} a \mathrm{sgn}\left (x\right )}{c^{4}}}{3465 \, c} + \frac{16 \,{\left (11 \, b c d^{\frac{9}{2}} - 8 \, a d^{\frac{11}{2}}\right )} \mathrm{sgn}\left (x\right )}{3465 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^10*(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

1/3465*(11*(35*(c*x^2 + d)^(9/2) - 135*(c*x^2 + d)^(7/2)*d + 189*(c*x^2 + d)^(5/2)*d^2 - 105*(c*x^2 + d)^(3/2)
*d^3)*b*sgn(x)/c^3 + (315*(c*x^2 + d)^(11/2) - 1540*(c*x^2 + d)^(9/2)*d + 2970*(c*x^2 + d)^(7/2)*d^2 - 2772*(c
*x^2 + d)^(5/2)*d^3 + 1155*(c*x^2 + d)^(3/2)*d^4)*a*sgn(x)/c^4)/c + 16/3465*(11*b*c*d^(9/2) - 8*a*d^(11/2))*sg
n(x)/c^5