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

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

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Rubi [A]  time = 0.07, antiderivative size = 150, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 ) \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*}

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Mathematica [A]  time = 0.08, size = 108, normalized size = 0.72 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (a \left (315 c^4 x^8-280 c^3 d x^6+240 c^2 d^2 x^4-192 c d^3 x^2+128 d^4\right )+11 b c \left (35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right )\right )}{3465 c^5} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.10, size = 112, normalized size = 0.75 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (315 a c^4 x^8-280 a c^3 d x^6+240 a c^2 d^2 x^4-192 a c d^3 x^2+128 a d^4+385 b c^4 x^6-330 b c^3 d x^4+264 b c^2 d^2 x^2-176 b c d^3\right )}{3465 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 131, normalized size = 0.87 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.22, size = 175, normalized size = 1.17 \begin {gather*} \frac {16 \, {\left (11 \, b c d^{\frac {9}{2}} - 8 \, a d^{\frac {11}{2}}\right )} \mathrm {sgn}\relax (x)}{3465 \, c^{5}} + \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) - 1540 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a d \mathrm {sgn}\relax (x) - 1485 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c d \mathrm {sgn}\relax (x) + 2970 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d^{2} \mathrm {sgn}\relax (x) + 2079 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d^{2} \mathrm {sgn}\relax (x) - 2772 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{3} \mathrm {sgn}\relax (x) - 1155 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c d^{3} \mathrm {sgn}\relax (x) + 1155 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d^{4} \mathrm {sgn}\relax (x)}{3465 \, c^{5}} \end {gather*}

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]

16/3465*(11*b*c*d^(9/2) - 8*a*d^(11/2))*sgn(x)/c^5 + 1/3465*(315*(c*x^2 + d)^(11/2)*a*sgn(x) + 385*(c*x^2 + d)
^(9/2)*b*c*sgn(x) - 1540*(c*x^2 + d)^(9/2)*a*d*sgn(x) - 1485*(c*x^2 + d)^(7/2)*b*c*d*sgn(x) + 2970*(c*x^2 + d)
^(7/2)*a*d^2*sgn(x) + 2079*(c*x^2 + d)^(5/2)*b*c*d^2*sgn(x) - 2772*(c*x^2 + d)^(5/2)*a*d^3*sgn(x) - 1155*(c*x^
2 + d)^(3/2)*b*c*d^3*sgn(x) + 1155*(c*x^2 + d)^(3/2)*a*d^4*sgn(x))/c^5

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maple [A]  time = 0.05, size = 113, normalized size = 0.75 \begin {gather*} \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \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 a c \,d^{3} x^{2}+264 b \,c^{2} d^{2} x^{2}+128 a \,d^{4}-176 b c \,d^{3}\right ) \left (c \,x^{2}+d \right ) x}{3465 c^{5}} \end {gather*}

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 = 0.64, size = 158, normalized size = 1.05 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 4.59, size = 117, normalized size = 0.78 \begin {gather*} \sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,x^{11}}{11}+\frac {x\,\left (128\,a\,d^5-176\,b\,c\,d^4\right )}{3465\,c^5}+\frac {x^9\,\left (385\,b\,c^5+35\,a\,d\,c^4\right )}{3465\,c^5}-\frac {d\,x^7\,\left (8\,a\,d-11\,b\,c\right )}{693\,c^2}+\frac {2\,d^2\,x^5\,\left (8\,a\,d-11\,b\,c\right )}{1155\,c^3}-\frac {8\,d^3\,x^3\,\left (8\,a\,d-11\,b\,c\right )}{3465\,c^4}\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)^(1/2),x)

[Out]

(c + d/x^2)^(1/2)*((a*x^11)/11 + (x*(128*a*d^5 - 176*b*c*d^4))/(3465*c^5) + (x^9*(385*b*c^5 + 35*a*c^4*d))/(34
65*c^5) - (d*x^7*(8*a*d - 11*b*c))/(693*c^2) + (2*d^2*x^5*(8*a*d - 11*b*c))/(1155*c^3) - (8*d^3*x^3*(8*a*d - 1
1*b*c))/(3465*c^4))

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sympy [B]  time = 6.44, size = 1386, normalized size = 9.24

result too large to display

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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