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

Optimal. Leaf size=86 \[ \frac{a x^5 \left (c+\frac{d}{x^2}\right )^{5/2}}{5 c}-b d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )+b d x \sqrt{c+\frac{d}{x^2}}+\frac{1}{3} b x^3 \left (c+\frac{d}{x^2}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.177461, antiderivative size = 86, 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 \[ \frac{a x^5 \left (c+\frac{d}{x^2}\right )^{5/2}}{5 c}-b d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )+b d x \sqrt{c+\frac{d}{x^2}}+\frac{1}{3} b x^3 \left (c+\frac{d}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7138, size = 75, normalized size = 0.87 \[ \frac{a x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 c} - b d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )} + b d x \sqrt{c + \frac{d}{x^{2}}} + \frac{b x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} \]

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]

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

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Mathematica [A]  time = 0.202274, size = 109, normalized size = 1.27 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c x^2+d} \left (3 a \left (c x^2+d\right )^2+5 b c \left (c x^2+4 d\right )\right )-15 b c d^{3/2} \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )+15 b c d^{3/2} \log (x)\right )}{15 c \sqrt{c x^2+d}} \]

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]*x*(Sqrt[d + c*x^2]*(3*a*(d + c*x^2)^2 + 5*b*c*(4*d + c*x^2)) +
15*b*c*d^(3/2)*Log[x] - 15*b*c*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + c*x^2]]))/(15*c*
Sqrt[d + c*x^2])

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Maple [A]  time = 0.015, size = 99, normalized size = 1.2 \[ -{\frac{{x}^{3}}{15\,c} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 15\,b{d}^{3/2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) c-3\,a \left ( c{x}^{2}+d \right ) ^{5/2}-5\,b \left ( c{x}^{2}+d \right ) ^{3/2}c-15\,b\sqrt{c{x}^{2}+d}dc \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*((c*x^2+d)/x^2)^(3/2)*x^3*(15*b*d^(3/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x
)*c-3*a*(c*x^2+d)^(5/2)-5*b*(c*x^2+d)^(3/2)*c-15*b*(c*x^2+d)^(1/2)*d*c)/(c*x^2+d
)^(3/2)/c

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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 = 0.236264, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b c d^{\frac{3}{2}} \log \left (-\frac{c x^{2} - 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \,{\left (3 \, a c^{2} x^{5} +{\left (5 \, b c^{2} + 6 \, a c d\right )} x^{3} +{\left (20 \, b c d + 3 \, a d^{2}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{30 \, c}, -\frac{15 \, b c \sqrt{-d} d \arctan \left (\frac{d}{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (3 \, a c^{2} x^{5} +{\left (5 \, b c^{2} + 6 \, a c d\right )} x^{3} +{\left (20 \, b c d + 3 \, a d^{2}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*b*c*d^(3/2)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2
) + 2*(3*a*c^2*x^5 + (5*b*c^2 + 6*a*c*d)*x^3 + (20*b*c*d + 3*a*d^2)*x)*sqrt((c*x
^2 + d)/x^2))/c, -1/15*(15*b*c*sqrt(-d)*d*arctan(d/(sqrt(-d)*x*sqrt((c*x^2 + d)/
x^2))) - (3*a*c^2*x^5 + (5*b*c^2 + 6*a*c*d)*x^3 + (20*b*c*d + 3*a*d^2)*x)*sqrt((
c*x^2 + d)/x^2))/c]

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Sympy [A]  time = 8.6311, size = 184, normalized size = 2.14 \[ \frac{a c \sqrt{d} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{2 a d^{\frac{3}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{a d^{\frac{5}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c} + \frac{b \sqrt{c} d x}{\sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3} + \frac{b d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3} - b d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )} + \frac{b d^{2}}{\sqrt{c} x \sqrt{1 + \frac{d}{c x^{2}}}} \]

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

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GIAC/XCAS [A]  time = 0.218334, size = 189, normalized size = 2.2 \[ \frac{b d^{2} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right ){\rm sign}\left (x\right )}{\sqrt{-d}} - \frac{{\left (15 \, b c d^{2} \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + 20 \, b c \sqrt{-d} d^{\frac{3}{2}} + 3 \, a \sqrt{-d} d^{\frac{5}{2}}\right )}{\rm sign}\left (x\right )}{15 \, c \sqrt{-d}} + \frac{3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} a c^{4}{\rm sign}\left (x\right ) + 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{5}{\rm sign}\left (x\right ) + 15 \, \sqrt{c x^{2} + d} b c^{5} d{\rm sign}\left (x\right )}{15 \, c^{5}} \]

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]

b*d^2*arctan(sqrt(c*x^2 + d)/sqrt(-d))*sign(x)/sqrt(-d) - 1/15*(15*b*c*d^2*arcta
n(sqrt(d)/sqrt(-d)) + 20*b*c*sqrt(-d)*d^(3/2) + 3*a*sqrt(-d)*d^(5/2))*sign(x)/(c
*sqrt(-d)) + 1/15*(3*(c*x^2 + d)^(5/2)*a*c^4*sign(x) + 5*(c*x^2 + d)^(3/2)*b*c^5
*sign(x) + 15*sqrt(c*x^2 + d)*b*c^5*d*sign(x))/c^5