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

Optimal. Leaf size=90 \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.209742, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8482, size = 76, normalized size = 0.84 \[ \frac{a x^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{4 c} - \frac{x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (a d - 4 b c\right )}{8 c} - \frac{d \left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{8 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x**3*(c+d/x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0969584, size = 99, normalized size = 1.1 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{c x^2+d} \left (a \left (2 c x^2+d\right )+4 b c\right )+d (4 b c-a d) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )\right )}{8 c^{3/2} \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 125, normalized size = 1.4 \[{\frac{x}{8}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,ax \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{3/2}-adx\sqrt{c{x}^{2}+d}{c}^{{\frac{3}{2}}}+4\,bx\sqrt{c{x}^{2}+d}{c}^{5/2}+4\,bd\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-a{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) c \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^3*(c+d/x^2)^(1/2),x)

[Out]

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

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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)*sqrt(c + d/x^2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237586, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, b c d - a d^{2}\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c^{2}}, -\frac{{\left (4 \, b c d - a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 22.2215, size = 144, normalized size = 1.6 \[ \frac{a c x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{3}{2}} x}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{3}{2}}} + \frac{b \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**3*(c+d/x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222341, size = 144, normalized size = 1.6 \[ \frac{1}{8} \,{\left (2 \, a x^{2}{\rm sign}\left (x\right ) + \frac{4 \, b c^{2}{\rm sign}\left (x\right ) + a c d{\rm sign}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + d} x - \frac{{\left (4 \, b c d{\rm sign}\left (x\right ) - a d^{2}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{8 \, c^{\frac{3}{2}}} + \frac{{\left (4 \, b c d{\rm ln}\left (\sqrt{d}\right ) - a d^{2}{\rm ln}\left (\sqrt{d}\right )\right )}{\rm sign}\left (x\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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