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

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

[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 = 0.284083, 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 \[ -\frac{c^3 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac{3 c \left (c+\frac{d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^5} \]

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)

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Rubi in Sympy [A]  time = 28.5536, size = 121, normalized size = 0.9 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11 d^{5}} + \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{5}} - \frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (3 a d - 4 b c\right )}{5 d^{5}} + \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}} \left (a d - 2 b c\right )}{7 d^{5}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}} \left (a d - 4 b c\right )}{9 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.124379, size = 113, normalized size = 0.84 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (11 a d x^2 \left (-16 c^3 x^6+24 c^2 d x^4-30 c d^2 x^2+35 d^3\right )+b \left (128 c^4 x^8-192 c^3 d x^6+240 c^2 d^2 x^4-280 c d^3 x^2+315 d^4\right )\right )}{3465 d^5 x^{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d/x^2]*(d + c*x^2)*(11*a*d*x^2*(35*d^3 - 30*c*d^2*x^2 + 24*c^2*d*x^4
- 16*c^3*x^6) + b*(315*d^4 - 280*c*d^3*x^2 + 240*c^2*d^2*x^4 - 192*c^3*d*x^6 + 1
28*c^4*x^8)))/(3465*d^5*x^10)

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Maple [A]  time = 0.011, size = 118, normalized size = 0.9 \[{\frac{ \left ( 176\,a{c}^{3}d{x}^{8}-128\,b{c}^{4}{x}^{8}-264\,a{c}^{2}{d}^{2}{x}^{6}+192\,b{c}^{3}d{x}^{6}+330\,ac{d}^{3}{x}^{4}-240\,b{c}^{2}{d}^{2}{x}^{4}-385\,a{d}^{4}{x}^{2}+280\,bc{d}^{3}{x}^{2}-315\,b{d}^{4} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{d}^{5}{x}^{10}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

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+19
2*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^2-31
5*b*d^4)*(c*x^2+d)/d^5/x^10

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Maxima [A]  time = 1.38267, size = 205, normalized size = 1.53 \[ -\frac{1}{3465} \, b{\left (\frac{315 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}}}{d^{5}} - \frac{1540 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c}{d^{5}} + \frac{2970 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{2}}{d^{5}} - \frac{2772 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{3}}{d^{5}} + \frac{1155 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{4}}{d^{5}}\right )} - \frac{1}{315} \, a{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{d^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^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 = 0.487867, size = 180, normalized size = 1.34 \[ -\frac{{\left (16 \,{\left (8 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 8 \,{\left (8 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 6 \,{\left (8 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} + 315 \, b d^{5} - 5 \,{\left (8 \, b c^{2} d^{3} - 11 \, a c d^{4}\right )} x^{4} + 35 \,{\left (b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, d^{5} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^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^10)

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Sympy [A]  time = 5.07913, size = 146, normalized size = 1.09 \[ - \frac{a \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} - \frac{b \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\right )}{d^{5}} \]

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/XCAS [A]  time = 1.23267, size = 581, normalized size = 4.34 \[ \frac{32 \,{\left (3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{11}{2}}{\rm sign}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{9}{2}} d{\rm sign}\left (x\right ) + 7392 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{11}{2}} d{\rm sign}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{9}{2}} d^{2}{\rm sign}\left (x\right ) + 2640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{11}{2}} d^{2}{\rm sign}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{9}{2}} d^{3}{\rm sign}\left (x\right ) - 1320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{11}{2}} d^{3}{\rm sign}\left (x\right ) + 1815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{9}{2}} d^{4}{\rm sign}\left (x\right ) + 440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{11}{2}} d^{4}{\rm sign}\left (x\right ) - 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{9}{2}} d^{5}{\rm sign}\left (x\right ) - 88 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{11}{2}} d^{5}{\rm sign}\left (x\right ) + 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{9}{2}} d^{6}{\rm sign}\left (x\right ) + 8 \, b c^{\frac{11}{2}} d^{6}{\rm sign}\left (x\right ) - 11 \, a c^{\frac{9}{2}} d^{7}{\rm sign}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)/x^9,x, algorithm="giac")

[Out]

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