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

Optimal. Leaf size=117 \[ \frac {8 d^2 x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{3465 c^4}-\frac {4 d x^7 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{693 c^3}+\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{99 c^2}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c} \]

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Rubi [A]  time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \begin {gather*} \frac {8 d^2 x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{3465 c^4}+\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{99 c^2}-\frac {4 d x^7 \left (c+\frac {d}{x^2}\right )^{5/2} (11 b c-6 a d)}{693 c^3}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]

[Out]

(8*d^2*(11*b*c - 6*a*d)*(c + d/x^2)^(5/2)*x^5)/(3465*c^4) - (4*d*(11*b*c - 6*a*d)*(c + d/x^2)^(5/2)*x^7)/(693*
c^3) + ((11*b*c - 6*a*d)*(c + d/x^2)^(5/2)*x^9)/(99*c^2) + (a*(c + d/x^2)^(5/2)*x^11)/(11*c)

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]

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 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]

Rubi steps

\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{11 c}+\frac {(11 b c-6 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx}{11 c}\\ &=\frac {(11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{11 c}-\frac {(4 d (11 b c-6 a d)) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6 \, dx}{99 c^2}\\ &=-\frac {4 d (11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{693 c^3}+\frac {(11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{11 c}+\frac {\left (8 d^2 (11 b c-6 a d)\right ) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4 \, dx}{693 c^3}\\ &=\frac {8 d^2 (11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{3465 c^4}-\frac {4 d (11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{693 c^3}+\frac {(11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{11 c}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 89, normalized size = 0.76 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (3 a \left (105 c^3 x^6-70 c^2 d x^4+40 c d^2 x^2-16 d^3\right )+11 b c \left (35 c^2 x^4-20 c d x^2+8 d^2\right )\right )}{3465 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(11*b*c*(8*d^2 - 20*c*d*x^2 + 35*c^2*x^4) + 3*a*(-16*d^3 + 40*c*d^2*x^2 - 70*
c^2*d*x^4 + 105*c^3*x^6)))/(3465*c^4)

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IntegrateAlgebraic [A]  time = 0.10, size = 90, normalized size = 0.77 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (315 a c^3 x^6-210 a c^2 d x^4+120 a c d^2 x^2-48 a d^3+385 b c^3 x^4-220 b c^2 d x^2+88 b c d^2\right )}{3465 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(88*b*c*d^2 - 48*a*d^3 - 220*b*c^2*d*x^2 + 120*a*c*d^2*x^2 + 385*b*c^3*x^4 -
210*a*c^2*d*x^4 + 315*a*c^3*x^6))/(3465*c^4)

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fricas [A]  time = 0.44, size = 132, normalized size = 1.13 \begin {gather*} \frac {{\left (315 \, a c^{5} x^{11} + 35 \, {\left (11 \, b c^{5} + 12 \, a c^{4} d\right )} x^{9} + 5 \, {\left (110 \, b c^{4} d + 3 \, a c^{3} d^{2}\right )} x^{7} + 3 \, {\left (11 \, b c^{3} d^{2} - 6 \, a c^{2} d^{3}\right )} x^{5} - 4 \, {\left (11 \, b c^{2} d^{3} - 6 \, a c d^{4}\right )} x^{3} + 8 \, {\left (11 \, b c d^{4} - 6 \, a d^{5}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3465*(315*a*c^5*x^11 + 35*(11*b*c^5 + 12*a*c^4*d)*x^9 + 5*(110*b*c^4*d + 3*a*c^3*d^2)*x^7 + 3*(11*b*c^3*d^2
- 6*a*c^2*d^3)*x^5 - 4*(11*b*c^2*d^3 - 6*a*c*d^4)*x^3 + 8*(11*b*c*d^4 - 6*a*d^5)*x)*sqrt((c*x^2 + d)/x^2)/c^4

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giac [A]  time = 0.18, size = 140, normalized size = 1.20 \begin {gather*} -\frac {8 \, {\left (11 \, b c d^{\frac {9}{2}} - 6 \, a d^{\frac {11}{2}}\right )} \mathrm {sgn}\relax (x)}{3465 \, c^{4}} + \frac {315 \, {\left (c x^{2} + d\right )}^{\frac {11}{2}} a \mathrm {sgn}\relax (x) + 385 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} b c \mathrm {sgn}\relax (x) - 1155 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a d \mathrm {sgn}\relax (x) - 990 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c d \mathrm {sgn}\relax (x) + 1485 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d^{2} \mathrm {sgn}\relax (x) + 693 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d^{2} \mathrm {sgn}\relax (x) - 693 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{3} \mathrm {sgn}\relax (x)}{3465 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3465*(11*b*c*d^(9/2) - 6*a*d^(11/2))*sgn(x)/c^4 + 1/3465*(315*(c*x^2 + d)^(11/2)*a*sgn(x) + 385*(c*x^2 + d)
^(9/2)*b*c*sgn(x) - 1155*(c*x^2 + d)^(9/2)*a*d*sgn(x) - 990*(c*x^2 + d)^(7/2)*b*c*d*sgn(x) + 1485*(c*x^2 + d)^
(7/2)*a*d^2*sgn(x) + 693*(c*x^2 + d)^(5/2)*b*c*d^2*sgn(x) - 693*(c*x^2 + d)^(5/2)*a*d^3*sgn(x))/c^4

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maple [A]  time = 0.05, size = 91, normalized size = 0.78 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (315 a \,x^{6} c^{3}-210 a \,c^{2} d \,x^{4}+385 b \,c^{3} x^{4}+120 a c \,d^{2} x^{2}-220 b \,c^{2} d \,x^{2}-48 a \,d^{3}+88 b c \,d^{2}\right ) \left (c \,x^{2}+d \right ) x^{3}}{3465 c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x)

[Out]

1/3465*((c*x^2+d)/x^2)^(3/2)*x^3*(315*a*c^3*x^6-210*a*c^2*d*x^4+385*b*c^3*x^4+120*a*c*d^2*x^2-220*b*c^2*d*x^2-
48*a*d^3+88*b*c*d^2)*(c*x^2+d)/c^4

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maxima [A]  time = 0.49, size = 124, normalized size = 1.06 \begin {gather*} \frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5}\right )} b}{315 \, c^{3}} + \frac {{\left (105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} x^{11} - 385 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} d x^{9} + 495 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{2} x^{7} - 231 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} x^{5}\right )} a}{1155 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*(c + d/x^2)^(9/2)*x^9 - 90*(c + d/x^2)^(7/2)*d*x^7 + 63*(c + d/x^2)^(5/2)*d^2*x^5)*b/c^3 + 1/1155*(1
05*(c + d/x^2)^(11/2)*x^11 - 385*(c + d/x^2)^(9/2)*d*x^9 + 495*(c + d/x^2)^(7/2)*d^2*x^7 - 231*(c + d/x^2)^(5/
2)*d^3*x^5)*a/c^4

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mupad [B]  time = 4.57, size = 118, normalized size = 1.01 \begin {gather*} \sqrt {c+\frac {d}{x^2}}\,\left (\frac {x^9\,\left (385\,b\,c^5+420\,a\,d\,c^4\right )}{3465\,c^4}-\frac {x\,\left (48\,a\,d^5-88\,b\,c\,d^4\right )}{3465\,c^4}+\frac {a\,c\,x^{11}}{11}+\frac {d\,x^7\,\left (3\,a\,d+110\,b\,c\right )}{693\,c}-\frac {d^2\,x^5\,\left (6\,a\,d-11\,b\,c\right )}{1155\,c^2}+\frac {4\,d^3\,x^3\,\left (6\,a\,d-11\,b\,c\right )}{3465\,c^3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(a + b/x^2)*(c + d/x^2)^(3/2),x)

[Out]

(c + d/x^2)^(1/2)*((x^9*(385*b*c^5 + 420*a*c^4*d))/(3465*c^4) - (x*(48*a*d^5 - 88*b*c*d^4))/(3465*c^4) + (a*c*
x^11)/11 + (d*x^7*(3*a*d + 110*b*c))/(693*c) - (d^2*x^5*(6*a*d - 11*b*c))/(1155*c^2) + (4*d^3*x^3*(6*a*d - 11*
b*c))/(3465*c^3))

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sympy [B]  time = 9.65, size = 2304, normalized size = 19.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**10,x)

[Out]

315*a*c**10*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**1
8*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1295*a*c**9*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**8*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**7*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) + 35*a*c**7*d*
*(21/2)*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) + 343*a*c**6*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) + 110*a*c**6*d**(23/2)*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) + 35*a*c**5*d**(43/2)*x**8*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 + 34
65*c**5*d**20) + 114*a*c**5*d**(25/2)*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) + 280*a*c**4*d**(45/2)*x**6*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) + 40*a*c**4*d**(27/2)*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) + 560*a
*c**3*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) - 5*a*c**3*d**(29/2)*x**6*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) + 448*a*c**2*d**(49/2)*x**2*sqrt(c*x**2/d + 1)/(34
65*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)
- 30*a*c**2*d**(31/2)*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) + 128*a*c*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) - 40*a*c*d**(33/2)*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*a*d**(35/2)*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) + 35*b*c**8*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**7*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**6*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**5*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) + 15*b*c**5*d**(11/2)*x**10*sqrt(c*x**2/d + 1)/(105*c**5*d**
4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 5*b*c**4*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 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 33*b*c**4*d**(13/2)*x**8*sqrt(c*x**2/d + 1)/(1
05*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 30*b*c**3*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) + 17*b*c**3*d**(15/2)*x**6*sqrt(c*x*
*2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 40*b*c**2*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) + 3*b*c**2*d**(17/2)*x**
4*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 16*b*c*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) + 12*b*c*d**(19/2)*x*
*2*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 8*b*d**(21/2)*sqrt(c*x**2/d
+ 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6)

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