3.792 \(\int \frac{\left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=159 \[ -\frac{c^3 (3 b c-8 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{128 d^{5/2}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{128 d^2 x}+\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{64 d x^3}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-8 a d)}{48 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{8 d x^3} \]

[Out]

(c*(3*b*c - 8*a*d)*Sqrt[c + d/x^2])/(64*d*x^3) + ((3*b*c - 8*a*d)*(c + d/x^2)^(3
/2))/(48*d*x^3) - (b*(c + d/x^2)^(5/2))/(8*d*x^3) + (c^2*(3*b*c - 8*a*d)*Sqrt[c
+ d/x^2])/(128*d^2*x) - (c^3*(3*b*c - 8*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)
])/(128*d^(5/2))

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Rubi [A]  time = 0.282074, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{c^3 (3 b c-8 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{128 d^{5/2}}+\frac{c^2 \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{128 d^2 x}+\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-8 a d)}{64 d x^3}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-8 a d)}{48 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{8 d x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b/x^2)*(c + d/x^2)^(3/2))/x^4,x]

[Out]

(c*(3*b*c - 8*a*d)*Sqrt[c + d/x^2])/(64*d*x^3) + ((3*b*c - 8*a*d)*(c + d/x^2)^(3
/2))/(48*d*x^3) - (b*(c + d/x^2)^(5/2))/(8*d*x^3) + (c^2*(3*b*c - 8*a*d)*Sqrt[c
+ d/x^2])/(128*d^2*x) - (c^3*(3*b*c - 8*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)
])/(128*d^(5/2))

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Rubi in Sympy [A]  time = 23.5431, size = 143, normalized size = 0.9 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{8 d x^{3}} + \frac{c^{3} \left (8 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{128 d^{\frac{5}{2}}} - \frac{c^{2} \sqrt{c + \frac{d}{x^{2}}} \left (8 a d - 3 b c\right )}{128 d^{2} x} - \frac{c \sqrt{c + \frac{d}{x^{2}}} \left (8 a d - 3 b c\right )}{64 d x^{3}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (8 a d - 3 b c\right )}{48 d x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**4,x)

[Out]

-b*(c + d/x**2)**(5/2)/(8*d*x**3) + c**3*(8*a*d - 3*b*c)*atanh(sqrt(d)/(x*sqrt(c
+ d/x**2)))/(128*d**(5/2)) - c**2*sqrt(c + d/x**2)*(8*a*d - 3*b*c)/(128*d**2*x)
- c*sqrt(c + d/x**2)*(8*a*d - 3*b*c)/(64*d*x**3) - (c + d/x**2)**(3/2)*(8*a*d -
3*b*c)/(48*d*x**3)

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Mathematica [A]  time = 0.430904, size = 171, normalized size = 1.08 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (3 c^3 x^8 \log (x) (8 a d-3 b c)+3 c^3 x^8 (3 b c-8 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )+\sqrt{d} \sqrt{c x^2+d} \left (8 a d x^2 \left (3 c^2 x^4+14 c d x^2+8 d^2\right )+b \left (-9 c^3 x^6+6 c^2 d x^4+72 c d^2 x^2+48 d^3\right )\right )\right )}{384 d^{5/2} x^7 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b/x^2)*(c + d/x^2)^(3/2))/x^4,x]

[Out]

-(Sqrt[c + d/x^2]*(Sqrt[d]*Sqrt[d + c*x^2]*(8*a*d*x^2*(8*d^2 + 14*c*d*x^2 + 3*c^
2*x^4) + b*(48*d^3 + 72*c*d^2*x^2 + 6*c^2*d*x^4 - 9*c^3*x^6)) + 3*c^3*(-3*b*c +
8*a*d)*x^8*Log[x] + 3*c^3*(3*b*c - 8*a*d)*x^8*Log[d + Sqrt[d]*Sqrt[d + c*x^2]]))
/(384*d^(5/2)*x^7*Sqrt[d + c*x^2])

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Maple [B]  time = 0.029, size = 316, normalized size = 2. \[{\frac{1}{384\,{x}^{5}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( -8\,a{c}^{3} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{8}{d}^{7/2}+3\,b{c}^{4} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{8}{d}^{5/2}+8\,a{c}^{2} \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{6}{d}^{7/2}-3\,b{c}^{3} \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{6}{d}^{5/2}-24\,a{c}^{3}\sqrt{c{x}^{2}+d}{x}^{8}{d}^{9/2}+9\,b{c}^{4}\sqrt{c{x}^{2}+d}{x}^{8}{d}^{7/2}+24\,a{c}^{3}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{8}{d}^{5}-9\,b{c}^{4}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{8}{d}^{4}+16\,ac \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{4}{d}^{9/2}-6\,b{c}^{2} \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{4}{d}^{7/2}-64\,a \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}{d}^{11/2}+24\,bc \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}{d}^{9/2}-48\,b \left ( c{x}^{2}+d \right ) ^{5/2}{d}^{11/2} \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(3/2)/x^4,x)

[Out]

1/384*((c*x^2+d)/x^2)^(3/2)/x^5*(-8*a*c^3*(c*x^2+d)^(3/2)*x^8*d^(7/2)+3*b*c^4*(c
*x^2+d)^(3/2)*x^8*d^(5/2)+8*a*c^2*(c*x^2+d)^(5/2)*x^6*d^(7/2)-3*b*c^3*(c*x^2+d)^
(5/2)*x^6*d^(5/2)-24*a*c^3*(c*x^2+d)^(1/2)*x^8*d^(9/2)+9*b*c^4*(c*x^2+d)^(1/2)*x
^8*d^(7/2)+24*a*c^3*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^8*d^5-9*b*c^4*ln(2*(d^
(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^8*d^4+16*a*c*(c*x^2+d)^(5/2)*x^4*d^(9/2)-6*b*c^2*(
c*x^2+d)^(5/2)*x^4*d^(7/2)-64*a*(c*x^2+d)^(5/2)*x^2*d^(11/2)+24*b*c*(c*x^2+d)^(5
/2)*x^2*d^(9/2)-48*b*(c*x^2+d)^(5/2)*d^(11/2))/(c*x^2+d)^(3/2)/d^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305749, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (3 \, b c^{4} - 8 \, a c^{3} d\right )} \sqrt{d} x^{7} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (3 \, b c^{3} d - 8 \, a c^{2} d^{2}\right )} x^{6} - 48 \, b d^{4} - 2 \,{\left (3 \, b c^{2} d^{2} + 56 \, a c d^{3}\right )} x^{4} - 8 \,{\left (9 \, b c d^{3} + 8 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{768 \, d^{3} x^{7}}, \frac{3 \,{\left (3 \, b c^{4} - 8 \, a c^{3} d\right )} \sqrt{-d} x^{7} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \,{\left (3 \, b c^{3} d - 8 \, a c^{2} d^{2}\right )} x^{6} - 48 \, b d^{4} - 2 \,{\left (3 \, b c^{2} d^{2} + 56 \, a c d^{3}\right )} x^{4} - 8 \,{\left (9 \, b c d^{3} + 8 \, a d^{4}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{384 \, d^{3} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[-1/768*(3*(3*b*c^4 - 8*a*c^3*d)*sqrt(d)*x^7*log(-(2*d*x*sqrt((c*x^2 + d)/x^2) +
(c*x^2 + 2*d)*sqrt(d))/x^2) - 2*(3*(3*b*c^3*d - 8*a*c^2*d^2)*x^6 - 48*b*d^4 - 2
*(3*b*c^2*d^2 + 56*a*c*d^3)*x^4 - 8*(9*b*c*d^3 + 8*a*d^4)*x^2)*sqrt((c*x^2 + d)/
x^2))/(d^3*x^7), 1/384*(3*(3*b*c^4 - 8*a*c^3*d)*sqrt(-d)*x^7*arctan(sqrt(-d)/(x*
sqrt((c*x^2 + d)/x^2))) + (3*(3*b*c^3*d - 8*a*c^2*d^2)*x^6 - 48*b*d^4 - 2*(3*b*c
^2*d^2 + 56*a*c*d^3)*x^4 - 8*(9*b*c*d^3 + 8*a*d^4)*x^2)*sqrt((c*x^2 + d)/x^2))/(
d^3*x^7)]

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Sympy [A]  time = 42.2857, size = 287, normalized size = 1.81 \[ - \frac{a c^{\frac{5}{2}}}{16 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{17 a c^{\frac{3}{2}}}{48 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{11 a \sqrt{c} d}{24 x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{a c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{16 d^{\frac{3}{2}}} - \frac{a d^{2}}{6 \sqrt{c} x^{7} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{3 b c^{\frac{7}{2}}}{128 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{5}{2}}}{128 d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{13 b c^{\frac{3}{2}}}{64 x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{5 b \sqrt{c} d}{16 x^{7} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{128 d^{\frac{5}{2}}} - \frac{b d^{2}}{8 \sqrt{c} x^{9} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**4,x)

[Out]

-a*c**(5/2)/(16*d*x*sqrt(1 + d/(c*x**2))) - 17*a*c**(3/2)/(48*x**3*sqrt(1 + d/(c
*x**2))) - 11*a*sqrt(c)*d/(24*x**5*sqrt(1 + d/(c*x**2))) + a*c**3*asinh(sqrt(d)/
(sqrt(c)*x))/(16*d**(3/2)) - a*d**2/(6*sqrt(c)*x**7*sqrt(1 + d/(c*x**2))) + 3*b*
c**(7/2)/(128*d**2*x*sqrt(1 + d/(c*x**2))) + b*c**(5/2)/(128*d*x**3*sqrt(1 + d/(
c*x**2))) - 13*b*c**(3/2)/(64*x**5*sqrt(1 + d/(c*x**2))) - 5*b*sqrt(c)*d/(16*x**
7*sqrt(1 + d/(c*x**2))) - 3*b*c**4*asinh(sqrt(d)/(sqrt(c)*x))/(128*d**(5/2)) - b
*d**2/(8*sqrt(c)*x**9*sqrt(1 + d/(c*x**2)))

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GIAC/XCAS [A]  time = 0.27015, size = 289, normalized size = 1.82 \[ \frac{\frac{3 \,{\left (3 \, b c^{5}{\rm sign}\left (x\right ) - 8 \, a c^{4} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d^{2}} + \frac{9 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} b c^{5}{\rm sign}\left (x\right ) - 24 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} a c^{4} d{\rm sign}\left (x\right ) - 33 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} b c^{5} d{\rm sign}\left (x\right ) - 40 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} a c^{4} d^{2}{\rm sign}\left (x\right ) - 33 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{5} d^{2}{\rm sign}\left (x\right ) + 88 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{4} d^{3}{\rm sign}\left (x\right ) + 9 \, \sqrt{c x^{2} + d} b c^{5} d^{3}{\rm sign}\left (x\right ) - 24 \, \sqrt{c x^{2} + d} a c^{4} d^{4}{\rm sign}\left (x\right )}{c^{4} d^{2} x^{8}}}{384 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/384*(3*(3*b*c^5*sign(x) - 8*a*c^4*d*sign(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/
(sqrt(-d)*d^2) + (9*(c*x^2 + d)^(7/2)*b*c^5*sign(x) - 24*(c*x^2 + d)^(7/2)*a*c^4
*d*sign(x) - 33*(c*x^2 + d)^(5/2)*b*c^5*d*sign(x) - 40*(c*x^2 + d)^(5/2)*a*c^4*d
^2*sign(x) - 33*(c*x^2 + d)^(3/2)*b*c^5*d^2*sign(x) + 88*(c*x^2 + d)^(3/2)*a*c^4
*d^3*sign(x) + 9*sqrt(c*x^2 + d)*b*c^5*d^3*sign(x) - 24*sqrt(c*x^2 + d)*a*c^4*d^
4*sign(x))/(c^4*d^2*x^8))/c