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

Optimal. Leaf size=115 \[ \frac{x^2 \left (c+\frac{d}{x^2}\right )^{3/2} (a d+4 b c)}{8 c}-\frac{3 d \sqrt{c+\frac{d}{x^2}} (a d+4 b c)}{8 c}+\frac{3 d (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 \sqrt{c}}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{5/2}}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.255156, antiderivative size = 115, 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 \[ \frac{x^2 \left (c+\frac{d}{x^2}\right )^{3/2} (a d+4 b c)}{8 c}-\frac{3 d \sqrt{c+\frac{d}{x^2}} (a d+4 b c)}{8 c}+\frac{3 d (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 \sqrt{c}}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{5/2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.7374, size = 104, normalized size = 0.9 \[ \frac{a x^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{4 c} - \frac{3 d \sqrt{c + \frac{d}{x^{2}}} \left (a d + 4 b c\right )}{8 c} + \frac{x^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d + 4 b c\right )}{8 c} + \frac{3 d \left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{8 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4*(c + d/x**2)**(5/2)/(4*c) - 3*d*sqrt(c + d/x**2)*(a*d + 4*b*c)/(8*c) + x*
*2*(c + d/x**2)**(3/2)*(a*d + 4*b*c)/(8*c) + 3*d*(a*d + 4*b*c)*atanh(sqrt(c + d/
x**2)/sqrt(c))/(8*sqrt(c))

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Mathematica [A]  time = 0.166814, size = 91, normalized size = 0.79 \[ \frac{1}{8} \sqrt{c+\frac{d}{x^2}} \left (\frac{3 d x (a d+4 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{\sqrt{c} \sqrt{c x^2+d}}+2 a c x^4+5 a d x^2+4 b c x^2-8 b d\right ) \]

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

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Maple [A]  time = 0.017, size = 174, normalized size = 1.5 \[{\frac{{x}^{2}}{8\,d} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 2\,a{x}^{2} \left ( c{x}^{2}+d \right ) ^{3/2}\sqrt{c}d+8\,b{c}^{3/2}{x}^{2} \left ( c{x}^{2}+d \right ) ^{3/2}+12\,bc{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) x-8\,b \left ( c{x}^{2}+d \right ) ^{5/2}\sqrt{c}+3\,a{d}^{2}{x}^{2}\sqrt{c{x}^{2}+d}\sqrt{c}+12\,b{c}^{3/2}{x}^{2}\sqrt{c{x}^{2}+d}d+3\,a{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) x \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \]

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]

1/8*((c*x^2+d)/x^2)^(3/2)*x^2*(2*a*x^2*(c*x^2+d)^(3/2)*c^(1/2)*d+8*b*c^(3/2)*x^2
*(c*x^2+d)^(3/2)+12*b*c*d^2*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x-8*b*(c*x^2+d)^(5/2)*
c^(1/2)+3*a*d^2*x^2*(c*x^2+d)^(1/2)*c^(1/2)+12*b*c^(3/2)*x^2*(c*x^2+d)^(1/2)*d+3
*a*d^3*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x)/(c*x^2+d)^(3/2)/c^(1/2)/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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24319, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, b c d + a d^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, a c^{2} x^{4} - 8 \, b c d +{\left (4 \, b c^{2} + 5 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c}, -\frac{3 \,{\left (4 \, b c d + a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (2 \, a c^{2} x^{4} - 8 \, b c d +{\left (4 \, b c^{2} + 5 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c}\right ] \]

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]

[1/16*(3*(4*b*c*d + a*d^2)*sqrt(c)*log(-2*c*x^2*sqrt((c*x^2 + d)/x^2) - (2*c*x^2
+ d)*sqrt(c)) + 2*(2*a*c^2*x^4 - 8*b*c*d + (4*b*c^2 + 5*a*c*d)*x^2)*sqrt((c*x^2
+ d)/x^2))/c, -1/8*(3*(4*b*c*d + a*d^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt((c*x^2 +
d)/x^2)) - (2*a*c^2*x^4 - 8*b*c*d + (4*b*c^2 + 5*a*c*d)*x^2)*sqrt((c*x^2 + d)/x^
2))/c]

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Sympy [A]  time = 31.9244, size = 216, normalized size = 1.88 \[ \frac{a c^{2} x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a c \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{3}{2}} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{a d^{\frac{3}{2}} x}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 \sqrt{c}} + \frac{3 b \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2} + \frac{b c \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} - \frac{b c \sqrt{d} x}{\sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b d^{\frac{3}{2}}}{x \sqrt{\frac{c x^{2}}{d} + 1}} \]

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**2*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*a*c*sqrt(d)*x**3/(8*sqrt(c*x**2/d
+ 1)) + a*d**(3/2)*x*sqrt(c*x**2/d + 1)/2 + a*d**(3/2)*x/(8*sqrt(c*x**2/d + 1))
+ 3*a*d**2*asinh(sqrt(c)*x/sqrt(d))/(8*sqrt(c)) + 3*b*sqrt(c)*d*asinh(sqrt(c)*x
/sqrt(d))/2 + b*c*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 - b*c*sqrt(d)*x/sqrt(c*x**2/d +
1) - b*d**(3/2)/(x*sqrt(c*x**2/d + 1))

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GIAC/XCAS [A]  time = 0.303322, size = 170, normalized size = 1.48 \[ \frac{2 \, b \sqrt{c} d^{2}{\rm sign}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d} + \frac{1}{8} \,{\left (2 \, a c x^{2}{\rm sign}\left (x\right ) + \frac{4 \, b c^{3}{\rm sign}\left (x\right ) + 5 \, a c^{2} d{\rm sign}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + d} x - \frac{3 \,{\left (4 \, b c^{\frac{3}{2}} d{\rm sign}\left (x\right ) + a \sqrt{c} d^{2}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right )}{16 \, c} \]

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*b*sqrt(c)*d^2*sign(x)/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d) + 1/8*(2*a*c*x^2*s
ign(x) + (4*b*c^3*sign(x) + 5*a*c^2*d*sign(x))/c^2)*sqrt(c*x^2 + d)*x - 3/16*(4*
b*c^(3/2)*d*sign(x) + a*sqrt(c)*d^2*sign(x))*ln((sqrt(c)*x - sqrt(c*x^2 + d))^2)
/c