3.780 \(\int \frac{\left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{\left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^2} \]

[Out]

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

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Rubi [A]  time = 0.122247, antiderivative size = 46, 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 \[ \frac{\left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.3842, size = 39, normalized size = 0.85 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7 d^{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0777933, size = 49, normalized size = 1.07 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (7 a d x^2-2 b c x^2+5 b d\right )}{35 d^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d/x^2]*(d + c*x^2)^2*(5*b*d - 2*b*c*x^2 + 7*a*d*x^2))/(35*d^2*x^6)

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Maple [A]  time = 0.01, size = 48, normalized size = 1. \[ -{\frac{ \left ( 7\,ad{x}^{2}-2\,bc{x}^{2}+5\,bd \right ) \left ( c{x}^{2}+d \right ) }{35\,{d}^{2}{x}^{4}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*((c*x^2+d)/x^2)^(3/2)*(7*a*d*x^2-2*b*c*x^2+5*b*d)*(c*x^2+d)/d^2/x^4

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Maxima [A]  time = 1.37175, size = 66, normalized size = 1.43 \[ -\frac{a{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, d} - \frac{1}{35} \,{\left (\frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{2}} - \frac{7 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{2}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*a*(c + d/x^2)^(5/2)/d - 1/35*(5*(c + d/x^2)^(7/2)/d^2 - 7*(c + d/x^2)^(5/2)
*c/d^2)*b

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Fricas [A]  time = 0.284685, size = 113, normalized size = 2.46 \[ \frac{{\left ({\left (2 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} -{\left (b c^{2} d + 14 \, a c d^{2}\right )} x^{4} - 5 \, b d^{3} -{\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \, d^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*((2*b*c^3 - 7*a*c^2*d)*x^6 - (b*c^2*d + 14*a*c*d^2)*x^4 - 5*b*d^3 - (8*b*c*
d^2 + 7*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^2*x^6)

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Sympy [A]  time = 7.37081, size = 138, normalized size = 3. \[ - \frac{a c \left (\begin{cases} \frac{\sqrt{c}}{x^{2}} & \text{for}\: d = 0 \\\frac{2 \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right )}{2} - \frac{a \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d} - \frac{b c \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} - \frac{b \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**3,x)

[Out]

-a*c*Piecewise((sqrt(c)/x**2, Eq(d, 0)), (2*(c + d/x**2)**(3/2)/(3*d), True))/2
- a*(-c*(c + d/x**2)**(3/2)/3 + (c + d/x**2)**(5/2)/5)/d - b*c*(-c*(c + d/x**2)*
*(3/2)/3 + (c + d/x**2)**(5/2)/5)/d**2 - b*(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**2

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GIAC/XCAS [A]  time = 0.759235, size = 500, normalized size = 10.87 \[ \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{7}{2}}{\rm sign}\left (x\right ) - 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{5}{2}} d{\rm sign}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{7}{2}} d{\rm sign}\left (x\right ) + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{5}{2}} d^{2}{\rm sign}\left (x\right ) + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{7}{2}} d^{2}{\rm sign}\left (x\right ) - 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{5}{2}} d^{3}{\rm sign}\left (x\right ) + 28 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{7}{2}} d^{3}{\rm sign}\left (x\right ) + 77 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{5}{2}} d^{4}{\rm sign}\left (x\right ) + 14 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{7}{2}} d^{4}{\rm sign}\left (x\right ) - 14 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{5}{2}} d^{5}{\rm sign}\left (x\right ) - 2 \, b c^{\frac{7}{2}} d^{5}{\rm sign}\left (x\right ) + 7 \, a c^{\frac{5}{2}} d^{6}{\rm sign}\left (x\right )\right )}}{35 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(35*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(5/2)*sign(x) + 70*(sqrt(c)*x - sq
rt(c*x^2 + d))^10*b*c^(7/2)*sign(x) - 70*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(5
/2)*d*sign(x) + 70*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(7/2)*d*sign(x) + 105*(sq
rt(c)*x - sqrt(c*x^2 + d))^8*a*c^(5/2)*d^2*sign(x) + 140*(sqrt(c)*x - sqrt(c*x^2
+ d))^6*b*c^(7/2)*d^2*sign(x) - 140*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(5/2)*d
^3*sign(x) + 28*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(7/2)*d^3*sign(x) + 77*(sqrt
(c)*x - sqrt(c*x^2 + d))^4*a*c^(5/2)*d^4*sign(x) + 14*(sqrt(c)*x - sqrt(c*x^2 +
d))^2*b*c^(7/2)*d^4*sign(x) - 14*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(5/2)*d^5*s
ign(x) - 2*b*c^(7/2)*d^5*sign(x) + 7*a*c^(5/2)*d^6*sign(x))/((sqrt(c)*x - sqrt(c
*x^2 + d))^2 - d)^7