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

Optimal. Leaf size=121 \[ \frac{x \left (c+\frac{d}{x^2}\right )^{3/2} (2 a d+3 b c)}{3 c}-\frac{d \sqrt{c+\frac{d}{x^2}} (2 a d+3 b c)}{2 c x}-\frac{1}{2} \sqrt{d} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )+\frac{a x^3 \left (c+\frac{d}{x^2}\right )^{5/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.178709, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{x \left (c+\frac{d}{x^2}\right )^{3/2} (2 a d+3 b c)}{3 c}-\frac{d \sqrt{c+\frac{d}{x^2}} (2 a d+3 b c)}{2 c x}-\frac{1}{2} \sqrt{d} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )+\frac{a x^3 \left (c+\frac{d}{x^2}\right )^{5/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8302, size = 105, normalized size = 0.87 \[ \frac{a x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{3 c} - \frac{\sqrt{d} \left (2 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{2} - \frac{d \sqrt{c + \frac{d}{x^{2}}} \left (2 a d + 3 b c\right )}{2 c x} + \frac{x \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (2 a d + 3 b c\right )}{3 c} \]

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]

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

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Mathematica [A]  time = 0.173112, size = 128, normalized size = 1.06 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (3 \sqrt{d} x^2 \log (x) (2 a d+3 b c)-3 \sqrt{d} x^2 (2 a d+3 b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )+\sqrt{c x^2+d} \left (2 a c x^4+8 a d x^2+6 b c x^2-3 b d\right )\right )}{6 x \sqrt{c x^2+d}} \]

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

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Maple [A]  time = 0.018, size = 170, normalized size = 1.4 \[ -{\frac{x}{6\,d} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 6\,a{d}^{5/2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{2}+9\,{d}^{3/2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) bc{x}^{2}-2\,a \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}d-3\,bc \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}+3\,b \left ( c{x}^{2}+d \right ) ^{5/2}-6\,a\sqrt{c{x}^{2}+d}{d}^{2}{x}^{2}-9\,bc\sqrt{c{x}^{2}+d}{x}^{2}d \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*((c*x^2+d)/x^2)^(3/2)*x*(6*a*d^(5/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^
2+9*d^(3/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*b*c*x^2-2*a*(c*x^2+d)^(3/2)*x^2*
d-3*b*c*(c*x^2+d)^(3/2)*x^2+3*b*(c*x^2+d)^(5/2)-6*a*(c*x^2+d)^(1/2)*d^2*x^2-9*b*
c*(c*x^2+d)^(1/2)*x^2*d)/(c*x^2+d)^(3/2)/d

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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 = 0.23976, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{d} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \,{\left (2 \, a c x^{4} + 2 \,{\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{12 \, x}, -\frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{-d} x \arctan \left (\frac{d}{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (2 \, a c x^{4} + 2 \,{\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, x}\right ] \]

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]

[1/12*(3*(3*b*c + 2*a*d)*sqrt(d)*x*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^
2) + 2*d)/x^2) + 2*(2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3*b*d)*sqrt((c*x^2 + d)/
x^2))/x, -1/6*(3*(3*b*c + 2*a*d)*sqrt(-d)*x*arctan(d/(sqrt(-d)*x*sqrt((c*x^2 + d
)/x^2))) - (2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3*b*d)*sqrt((c*x^2 + d)/x^2))/x]

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Sympy [A]  time = 10.4421, size = 202, normalized size = 1.67 \[ \frac{a \sqrt{c} d x}{\sqrt{1 + \frac{d}{c x^{2}}}} + \frac{a c \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3} + \frac{a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3} - a d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )} + \frac{a d^{2}}{\sqrt{c} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{3}{2}} x}{\sqrt{1 + \frac{d}{c x^{2}}}} - \frac{b \sqrt{c} d \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} + \frac{b \sqrt{c} d}{x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b c \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2} \]

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*sqrt(c)*d*x/sqrt(1 + d/(c*x**2)) + a*c*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/3 + a*d
**(3/2)*sqrt(c*x**2/d + 1)/3 - a*d**(3/2)*asinh(sqrt(d)/(sqrt(c)*x)) + a*d**2/(s
qrt(c)*x*sqrt(1 + d/(c*x**2))) + b*c**(3/2)*x/sqrt(1 + d/(c*x**2)) - b*sqrt(c)*d
*sqrt(1 + d/(c*x**2))/(2*x) + b*sqrt(c)*d/(x*sqrt(1 + d/(c*x**2))) - 3*b*c*sqrt(
d)*asinh(sqrt(d)/(sqrt(c)*x))/2

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GIAC/XCAS [A]  time = 0.246426, size = 155, normalized size = 1.28 \[ \frac{2 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c{\rm sign}\left (x\right ) + 6 \, \sqrt{c x^{2} + d} b c^{2}{\rm sign}\left (x\right ) + 6 \, \sqrt{c x^{2} + d} a c d{\rm sign}\left (x\right ) - \frac{3 \, \sqrt{c x^{2} + d} b c d{\rm sign}\left (x\right )}{x^{2}} + \frac{3 \,{\left (3 \, b c^{2} d{\rm sign}\left (x\right ) + 2 \, a c d^{2}{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d}}}{6 \, c} \]

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]

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