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

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

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Rubi [A]  time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]))))

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]]

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*}

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Mathematica [A]  time = 0.08, size = 90, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (x^2 \left (\frac {c x^2}{d}+1\right ) \left (-16 c^3 x^6+24 c^2 d x^4-30 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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 138, normalized size = 1.03 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (176 a c^4 d x^{10}-88 a c^3 d^2 x^8+66 a c^2 d^3 x^6-55 a c d^4 x^4-385 a d^5 x^2-128 b c^5 x^{10}+64 b c^4 d x^8-48 b c^3 d^2 x^6+40 b c^2 d^3 x^4-35 b c d^4 x^2-315 b d^5\right )}{3465 d^5 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-315*b*d^5 - 35*b*c*d^4*x^2 - 385*a*d^5*x^2 + 40*b*c^2*d^3*x^4 - 55*a*c*d^4*x^4 - 48*b
*c^3*d^2*x^6 + 66*a*c^2*d^3*x^6 + 64*b*c^4*d*x^8 - 88*a*c^3*d^2*x^8 - 128*b*c^5*x^10 + 176*a*c^4*d*x^10))/(346
5*d^5*x^10)

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fricas [A]  time = 0.53, size = 133, normalized size = 0.99 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 2.98, size = 430, normalized size = 3.21 \begin {gather*} \frac {32 \, {\left (3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {9}{2}} d \mathrm {sgn}\relax (x) + 7392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {11}{2}} d \mathrm {sgn}\relax (x) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {9}{2}} d^{2} \mathrm {sgn}\relax (x) + 2640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {11}{2}} d^{2} \mathrm {sgn}\relax (x) - 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {9}{2}} d^{3} \mathrm {sgn}\relax (x) - 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {11}{2}} d^{3} \mathrm {sgn}\relax (x) + 1815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {9}{2}} d^{4} \mathrm {sgn}\relax (x) + 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {11}{2}} d^{4} \mathrm {sgn}\relax (x) - 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {9}{2}} d^{5} \mathrm {sgn}\relax (x) - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {11}{2}} d^{5} \mathrm {sgn}\relax (x) + 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {9}{2}} d^{6} \mathrm {sgn}\relax (x) + 8 \, b c^{\frac {11}{2}} d^{6} \mathrm {sgn}\relax (x) - 11 \, a c^{\frac {9}{2}} d^{7} \mathrm {sgn}\relax (x)\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{11}} \end {gather*}

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

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maple [A]  time = 0.05, size = 118, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \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 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}\right ) \left (c \,x^{2}+d \right )}{3465 d^{5} x^{10}} \end {gather*}

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 = 0.47, size = 152, normalized size = 1.13 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 5.61, size = 210, normalized size = 1.57 \begin {gather*} \frac {16\,a\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^4}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{11\,x^{10}}-\frac {a\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {128\,b\,c^5\,\sqrt {c+\frac {d}{x^2}}}{3465\,d^5}-\frac {a\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,d\,x^6}-\frac {b\,c\,\sqrt {c+\frac {d}{x^2}}}{99\,d\,x^8}+\frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d^2\,x^4}-\frac {8\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3\,x^2}+\frac {8\,b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{693\,d^2\,x^6}-\frac {16\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^3\,x^4}+\frac {64\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{3465\,d^4\,x^2} \end {gather*}

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]

(16*a*c^4*(c + d/x^2)^(1/2))/(315*d^4) - (b*(c + d/x^2)^(1/2))/(11*x^10) - (a*(c + d/x^2)^(1/2))/(9*x^8) - (12
8*b*c^5*(c + d/x^2)^(1/2))/(3465*d^5) - (a*c*(c + d/x^2)^(1/2))/(63*d*x^6) - (b*c*(c + d/x^2)^(1/2))/(99*d*x^8
) + (2*a*c^2*(c + d/x^2)^(1/2))/(105*d^2*x^4) - (8*a*c^3*(c + d/x^2)^(1/2))/(315*d^3*x^2) + (8*b*c^2*(c + d/x^
2)^(1/2))/(693*d^2*x^6) - (16*b*c^3*(c + d/x^2)^(1/2))/(1155*d^3*x^4) + (64*b*c^4*(c + d/x^2)^(1/2))/(3465*d^4
*x^2)

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sympy [A]  time = 5.86, size = 146, normalized size = 1.09 \begin {gather*} - \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 {gather*}

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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