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

Optimal. Leaf size=123 \[ \frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{3/2}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-6 a d)}{24 d x}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-6 a d)}{16 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{6 d x} \]

[Out]

(c*(b*c - 6*a*d)*Sqrt[c + d/x^2])/(16*d*x) + ((b*c - 6*a*d)*(c + d/x^2)^(3/2))/(
24*d*x) - (b*(c + d/x^2)^(5/2))/(6*d*x) + (c^2*(b*c - 6*a*d)*ArcTanh[Sqrt[d]/(Sq
rt[c + d/x^2]*x)])/(16*d^(3/2))

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Rubi [A]  time = 0.19716, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{3/2}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-6 a d)}{24 d x}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-6 a d)}{16 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{6 d x} \]

Antiderivative was successfully verified.

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

[Out]

(c*(b*c - 6*a*d)*Sqrt[c + d/x^2])/(16*d*x) + ((b*c - 6*a*d)*(c + d/x^2)^(3/2))/(
24*d*x) - (b*(c + d/x^2)^(5/2))/(6*d*x) + (c^2*(b*c - 6*a*d)*ArcTanh[Sqrt[d]/(Sq
rt[c + d/x^2]*x)])/(16*d^(3/2))

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Rubi in Sympy [A]  time = 15.5839, size = 104, normalized size = 0.85 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{6 d x} - \frac{c^{2} \left (6 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{16 d^{\frac{3}{2}}} - \frac{c \sqrt{c + \frac{d}{x^{2}}} \left (6 a d - b c\right )}{16 d x} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (6 a d - b c\right )}{24 d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(c + d/x**2)**(5/2)/(6*d*x) - c**2*(6*a*d - b*c)*atanh(sqrt(d)/(x*sqrt(c + d/
x**2)))/(16*d**(3/2)) - c*sqrt(c + d/x**2)*(6*a*d - b*c)/(16*d*x) - (c + d/x**2)
**(3/2)*(6*a*d - b*c)/(24*d*x)

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Mathematica [A]  time = 0.227799, size = 147, normalized size = 1.2 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (\sqrt{d} \sqrt{c x^2+d} \left (6 a d x^2 \left (5 c x^2+2 d\right )+b \left (3 c^2 x^4+14 c d x^2+8 d^2\right )\right )+3 c^2 x^6 \log (x) (b c-6 a d)-3 c^2 x^6 (b c-6 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{48 d^{3/2} x^5 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 273, normalized size = 2.2 \[ -{\frac{1}{48\,{x}^{3}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( -6\,a{c}^{2} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{6}{d}^{5/2}+b{c}^{3} \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}{x}^{6}{d}^{{\frac{3}{2}}}+18\,a{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}{d}^{4}+6\,ac \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{4}{d}^{5/2}-b{c}^{2} \left ( c{x}^{2}+d \right ) ^{{\frac{5}{2}}}{x}^{4}{d}^{{\frac{3}{2}}}-18\,a{c}^{2}\sqrt{c{x}^{2}+d}{x}^{6}{d}^{7/2}+3\,b{c}^{3}\sqrt{c{x}^{2}+d}{x}^{6}{d}^{5/2}-3\,b{c}^{3}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}{d}^{3}+12\,a \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}{d}^{7/2}-2\,bc \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}{d}^{5/2}+8\,b \left ( c{x}^{2}+d \right ) ^{5/2}{d}^{7/2} \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*((c*x^2+d)/x^2)^(3/2)/x^3*(-6*a*c^2*(c*x^2+d)^(3/2)*x^6*d^(5/2)+b*c^3*(c*x
^2+d)^(3/2)*x^6*d^(3/2)+18*a*c^2*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^6*d^4+6*a
*c*(c*x^2+d)^(5/2)*x^4*d^(5/2)-b*c^2*(c*x^2+d)^(5/2)*x^4*d^(3/2)-18*a*c^2*(c*x^2
+d)^(1/2)*x^6*d^(7/2)+3*b*c^3*(c*x^2+d)^(1/2)*x^6*d^(5/2)-3*b*c^3*ln(2*(d^(1/2)*
(c*x^2+d)^(1/2)+d)/x)*x^6*d^3+12*a*(c*x^2+d)^(5/2)*x^2*d^(7/2)-2*b*c*(c*x^2+d)^(
5/2)*x^2*d^(5/2)+8*b*(c*x^2+d)^(5/2)*d^(7/2))/(c*x^2+d)^(3/2)/d^(9/2)

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257574, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c^{3} - 6 \, a c^{2} d\right )} \sqrt{d} x^{5} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (b c^{2} d + 10 \, a c d^{2}\right )} x^{4} + 8 \, b d^{3} + 2 \,{\left (7 \, b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, d^{2} x^{5}}, -\frac{3 \,{\left (b c^{3} - 6 \, a c^{2} d\right )} \sqrt{-d} x^{5} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \,{\left (b c^{2} d + 10 \, a c d^{2}\right )} x^{4} + 8 \, b d^{3} + 2 \,{\left (7 \, b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, d^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(3*(b*c^3 - 6*a*c^2*d)*sqrt(d)*x^5*log((2*d*x*sqrt((c*x^2 + d)/x^2) - (c*
x^2 + 2*d)*sqrt(d))/x^2) + 2*(3*(b*c^2*d + 10*a*c*d^2)*x^4 + 8*b*d^3 + 2*(7*b*c*
d^2 + 6*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*x^5), -1/48*(3*(b*c^3 - 6*a*c^2*
d)*sqrt(-d)*x^5*arctan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))) + (3*(b*c^2*d + 10*a*
c*d^2)*x^4 + 8*b*d^3 + 2*(7*b*c*d^2 + 6*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*
x^5)]

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Sympy [A]  time = 26.2621, size = 253, normalized size = 2.06 \[ - \frac{a c^{\frac{3}{2}} \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} - \frac{a c^{\frac{3}{2}}}{8 x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 a \sqrt{c} d}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 a c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 \sqrt{d}} - \frac{a d^{2}}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{b c^{\frac{5}{2}}}{16 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{17 b c^{\frac{3}{2}}}{48 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{11 b \sqrt{c} d}{24 x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{16 d^{\frac{3}{2}}} - \frac{b d^{2}}{6 \sqrt{c} x^{7} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260858, size = 234, normalized size = 1.9 \[ -\frac{\frac{3 \,{\left (b c^{4}{\rm sign}\left (x\right ) - 6 \, a c^{3} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d} + \frac{3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} b c^{4}{\rm sign}\left (x\right ) + 30 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} a c^{3} d{\rm sign}\left (x\right ) + 8 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{4} d{\rm sign}\left (x\right ) - 48 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{3} d^{2}{\rm sign}\left (x\right ) - 3 \, \sqrt{c x^{2} + d} b c^{4} d^{2}{\rm sign}\left (x\right ) + 18 \, \sqrt{c x^{2} + d} a c^{3} d^{3}{\rm sign}\left (x\right )}{c^{3} d x^{6}}}{48 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(3*(b*c^4*sign(x) - 6*a*c^3*d*sign(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/(s
qrt(-d)*d) + (3*(c*x^2 + d)^(5/2)*b*c^4*sign(x) + 30*(c*x^2 + d)^(5/2)*a*c^3*d*s
ign(x) + 8*(c*x^2 + d)^(3/2)*b*c^4*d*sign(x) - 48*(c*x^2 + d)^(3/2)*a*c^3*d^2*si
gn(x) - 3*sqrt(c*x^2 + d)*b*c^4*d^2*sign(x) + 18*sqrt(c*x^2 + d)*a*c^3*d^3*sign(
x))/(c^3*d*x^6))/c