3.808 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^5} \, dx\)

Optimal. Leaf size=68 \[ \frac{\sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{d^3}+\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^3} \]

[Out]

(c*(b*c - a*d))/(d^3*Sqrt[c + d/x^2]) + ((2*b*c - a*d)*Sqrt[c + d/x^2])/d^3 - (b
*(c + d/x^2)^(3/2))/(3*d^3)

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Rubi [A]  time = 0.172851, antiderivative size = 68, 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 \[ \frac{\sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{d^3}+\frac{c (b c-a d)}{d^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x]

[Out]

(c*(b*c - a*d))/(d^3*Sqrt[c + d/x^2]) + ((2*b*c - a*d)*Sqrt[c + d/x^2])/d^3 - (b
*(c + d/x^2)^(3/2))/(3*d^3)

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Rubi in Sympy [A]  time = 16.955, size = 61, normalized size = 0.9 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d^{3}} - \frac{c \left (a d - b c\right )}{d^{3} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (a d - 2 b c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.074885, size = 60, normalized size = 0.88 \[ \frac{b \left (8 c^2 x^4+4 c d x^2-d^2\right )-3 a d x^2 \left (2 c x^2+d\right )}{3 d^3 x^4 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x]

[Out]

(-3*a*d*x^2*(d + 2*c*x^2) + b*(-d^2 + 4*c*d*x^2 + 8*c^2*x^4))/(3*d^3*Sqrt[c + d/
x^2]*x^4)

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Maple [A]  time = 0.011, size = 69, normalized size = 1. \[ -{\frac{ \left ( 6\,acd{x}^{4}-8\,b{c}^{2}{x}^{4}+3\,a{d}^{2}{x}^{2}-4\,bcd{x}^{2}+b{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{3\,{d}^{3}{x}^{6}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^5,x)

[Out]

-1/3*(6*a*c*d*x^4-8*b*c^2*x^4+3*a*d^2*x^2-4*b*c*d*x^2+b*d^2)*(c*x^2+d)/((c*x^2+d
)/x^2)^(3/2)/d^3/x^6

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Maxima [A]  time = 1.46481, size = 109, normalized size = 1.6 \[ -\frac{1}{3} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{3}} - \frac{6 \, \sqrt{c + \frac{d}{x^{2}}} c}{d^{3}} - \frac{3 \, c^{2}}{\sqrt{c + \frac{d}{x^{2}}} d^{3}}\right )} - a{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{c}{\sqrt{c + \frac{d}{x^{2}}} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x, algorithm="maxima")

[Out]

-1/3*b*((c + d/x^2)^(3/2)/d^3 - 6*sqrt(c + d/x^2)*c/d^3 - 3*c^2/(sqrt(c + d/x^2)
*d^3)) - a*(sqrt(c + d/x^2)/d^2 + c/(sqrt(c + d/x^2)*d^2))

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Fricas [A]  time = 0.225291, size = 99, normalized size = 1.46 \[ \frac{{\left (2 \,{\left (4 \, b c^{2} - 3 \, a c d\right )} x^{4} - b d^{2} +{\left (4 \, b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \,{\left (c d^{3} x^{4} + d^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 37.0914, size = 476, normalized size = 7. \[ a \left (\begin{cases} - \frac{2 c}{d^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{1}{d x^{2} \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\- \frac{1}{4 c^{\frac{3}{2}} x^{4}} & \text{otherwise} \end{cases}\right ) + b \left (\frac{8 c^{\frac{9}{2}} d^{\frac{7}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} + \frac{12 c^{\frac{7}{2}} d^{\frac{9}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} + \frac{3 c^{\frac{5}{2}} d^{\frac{11}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} - \frac{c^{\frac{3}{2}} d^{\frac{13}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} - \frac{8 c^{5} d^{3} x^{7}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} - \frac{16 c^{4} d^{4} x^{5}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}} - \frac{8 c^{3} d^{5} x^{3}}{3 c^{\frac{7}{2}} d^{6} x^{7} + 6 c^{\frac{5}{2}} d^{7} x^{5} + 3 c^{\frac{3}{2}} d^{8} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**5,x)

[Out]

a*Piecewise((-2*c/(d**2*sqrt(c + d/x**2)) - 1/(d*x**2*sqrt(c + d/x**2)), Ne(d, 0
)), (-1/(4*c**(3/2)*x**4), True)) + b*(8*c**(9/2)*d**(7/2)*x**6*sqrt(c*x**2/d +
1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) + 12*c**
(7/2)*d**(9/2)*x**4*sqrt(c*x**2/d + 1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x
**5 + 3*c**(3/2)*d**8*x**3) + 3*c**(5/2)*d**(11/2)*x**2*sqrt(c*x**2/d + 1)/(3*c*
*(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) - c**(3/2)*d**(1
3/2)*sqrt(c*x**2/d + 1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2
)*d**8*x**3) - 8*c**5*d**3*x**7/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3
*c**(3/2)*d**8*x**3) - 16*c**4*d**4*x**5/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7
*x**5 + 3*c**(3/2)*d**8*x**3) - 8*c**3*d**5*x**3/(3*c**(7/2)*d**6*x**7 + 6*c**(5
/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^5),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^5), x)