3.766 \(\int \frac{\left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}}}{x^7} \, dx\)

Optimal. Leaf size=104 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]

[Out]

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

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Rubi [A]  time = 0.234944, antiderivative size = 104, 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 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-a d)}{7 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-2 a d)}{5 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{9/2}}{9 d^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.8243, size = 92, normalized size = 0.88 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9 d^{4}} - \frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{4}} + \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (2 a d - 3 b c\right )}{5 d^{4}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}} \left (a d - 3 b c\right )}{7 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.104687, size = 91, normalized size = 0.88 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (3 a d x^2 \left (8 c^2 x^4-12 c d x^2+15 d^2\right )+b \left (-16 c^3 x^6+24 c^2 d x^4-30 c d^2 x^2+35 d^3\right )\right )}{315 d^4 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^7,x]

[Out]

-(Sqrt[c + d/x^2]*(d + c*x^2)*(3*a*d*x^2*(15*d^2 - 12*c*d*x^2 + 8*c^2*x^4) + b*(
35*d^3 - 30*c*d^2*x^2 + 24*c^2*d*x^4 - 16*c^3*x^6)))/(315*d^4*x^8)

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Maple [A]  time = 0.011, size = 94, normalized size = 0.9 \[ -{\frac{ \left ( 24\,a{c}^{2}d{x}^{6}-16\,b{c}^{3}{x}^{6}-36\,ac{d}^{2}{x}^{4}+24\,b{c}^{2}d{x}^{4}+45\,a{d}^{3}{x}^{2}-30\,bc{d}^{2}{x}^{2}+35\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{315\,{d}^{4}{x}^{8}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(1/2)/x^7,x)

[Out]

-1/315*((c*x^2+d)/x^2)^(1/2)*(24*a*c^2*d*x^6-16*b*c^3*x^6-36*a*c*d^2*x^4+24*b*c^
2*d*x^4+45*a*d^3*x^2-30*b*c*d^2*x^2+35*b*d^3)*(c*x^2+d)/d^4/x^8

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Maxima [A]  time = 1.39348, size = 159, normalized size = 1.53 \[ -\frac{1}{315} \, b{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{4}} - \frac{135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{4}} + \frac{189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{4}} - \frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{3}}{d^{4}}\right )} - \frac{1}{105} \, a{\left (\frac{15 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{3}} - \frac{42 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{3}} + \frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="maxima")

[Out]

-1/315*b*(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) - 1/105*a*(15*(c + d/x^2)^(7/2
)/d^3 - 42*(c + d/x^2)^(5/2)*c/d^3 + 35*(c + d/x^2)^(3/2)*c^2/d^3)

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Fricas [A]  time = 0.330539, size = 147, normalized size = 1.41 \[ \frac{{\left (8 \,{\left (2 \, b c^{4} - 3 \, a c^{3} d\right )} x^{8} - 4 \,{\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x^{6} - 35 \, b d^{4} + 3 \,{\left (2 \, b c^{2} d^{2} - 3 \, a c d^{3}\right )} x^{4} - 5 \,{\left (b c d^{3} + 9 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, d^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.75093, size = 112, normalized size = 1.08 \[ - \frac{a \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} - \frac{b \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.819908, size = 500, normalized size = 4.81 \[ \frac{16 \,{\left (210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{7}{2}} d{\rm sign}\left (x\right ) + 378 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{9}{2}} d{\rm sign}\left (x\right ) + 63 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{7}{2}} d^{2}{\rm sign}\left (x\right ) + 168 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{9}{2}} d^{2}{\rm sign}\left (x\right ) - 42 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{7}{2}} d^{3}{\rm sign}\left (x\right ) - 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{9}{2}} d^{3}{\rm sign}\left (x\right ) + 108 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{7}{2}} d^{4}{\rm sign}\left (x\right ) + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{9}{2}} d^{4}{\rm sign}\left (x\right ) - 27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{7}{2}} d^{5}{\rm sign}\left (x\right ) - 2 \, b c^{\frac{9}{2}} d^{5}{\rm sign}\left (x\right ) + 3 \, a c^{\frac{7}{2}} d^{6}{\rm sign}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)/x^7,x, algorithm="giac")

[Out]

16/315*(210*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(7/2)*sign(x) + 630*(sqrt(c)*x
- sqrt(c*x^2 + d))^10*b*c^(9/2)*sign(x) - 315*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a
*c^(7/2)*d*sign(x) + 378*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(9/2)*d*sign(x) + 6
3*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(7/2)*d^2*sign(x) + 168*(sqrt(c)*x - sqrt(
c*x^2 + d))^6*b*c^(9/2)*d^2*sign(x) - 42*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(7/
2)*d^3*sign(x) - 72*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(9/2)*d^3*sign(x) + 108*
(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(7/2)*d^4*sign(x) + 18*(sqrt(c)*x - sqrt(c*x
^2 + d))^2*b*c^(9/2)*d^4*sign(x) - 27*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(7/2)*
d^5*sign(x) - 2*b*c^(9/2)*d^5*sign(x) + 3*a*c^(7/2)*d^6*sign(x))/((sqrt(c)*x - s
qrt(c*x^2 + d))^2 - d)^9