3.776 \(\int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x^5 \, dx\)

Optimal. Leaf size=123 \[ \frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{3/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (6 b c-a d)}{16 c}+\frac{x^4 \left (c+\frac{d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{5/2}}{6 c} \]

[Out]

(d*(6*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(16*c) + ((6*b*c - a*d)*(c + d/x^2)^(3/2)*
x^4)/(24*c) + (a*(c + d/x^2)^(5/2)*x^6)/(6*c) + (d^2*(6*b*c - a*d)*ArcTanh[Sqrt[
c + d/x^2]/Sqrt[c]])/(16*c^(3/2))

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Rubi [A]  time = 0.264568, 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 \[ \frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{3/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (6 b c-a d)}{16 c}+\frac{x^4 \left (c+\frac{d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{5/2}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x]

[Out]

(d*(6*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(16*c) + ((6*b*c - a*d)*(c + d/x^2)^(3/2)*
x^4)/(24*c) + (a*(c + d/x^2)^(5/2)*x^6)/(6*c) + (d^2*(6*b*c - a*d)*ArcTanh[Sqrt[
c + d/x^2]/Sqrt[c]])/(16*c^(3/2))

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Rubi in Sympy [A]  time = 19.1018, size = 105, normalized size = 0.85 \[ \frac{a x^{6} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{6 c} - \frac{d x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (a d - 6 b c\right )}{16 c} - \frac{x^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 6 b c\right )}{24 c} - \frac{d^{2} \left (a d - 6 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{16 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**5,x)

[Out]

a*x**6*(c + d/x**2)**(5/2)/(6*c) - d*x**2*sqrt(c + d/x**2)*(a*d - 6*b*c)/(16*c)
- x**4*(c + d/x**2)**(3/2)*(a*d - 6*b*c)/(24*c) - d**2*(a*d - 6*b*c)*atanh(sqrt(
c + d/x**2)/sqrt(c))/(16*c**(3/2))

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Mathematica [A]  time = 0.158235, size = 124, normalized size = 1.01 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{c x^2+d} \left (a \left (8 c^2 x^4+14 c d x^2+3 d^2\right )+6 b c \left (2 c x^2+5 d\right )\right )-3 d^2 (a d-6 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )\right )}{48 c^{3/2} \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x]

[Out]

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

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Maple [A]  time = 0.014, size = 165, normalized size = 1.3 \[{\frac{{x}^{3}}{48} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 8\,ax \left ( c{x}^{2}+d \right ) ^{5/2}{c}^{3/2}-2\,adx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{3/2}+12\,bx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{5/2}-3\,a{d}^{2}x\sqrt{c{x}^{2}+d}{c}^{3/2}+18\,bdx\sqrt{c{x}^{2}+d}{c}^{5/2}+18\,b{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-3\,a{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) c \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(3/2)*x^5,x)

[Out]

1/48*((c*x^2+d)/x^2)^(3/2)*x^3*(8*a*x*(c*x^2+d)^(5/2)*c^(3/2)-2*a*d*x*(c*x^2+d)^
(3/2)*c^(3/2)+12*b*x*(c*x^2+d)^(3/2)*c^(5/2)-3*a*d^2*x*(c*x^2+d)^(1/2)*c^(3/2)+1
8*b*d*x*(c*x^2+d)^(1/2)*c^(5/2)+18*b*d^2*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^2-3*a*d
^3*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c)/(c*x^2+d)^(3/2)/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)*(c + d/x^2)^(3/2)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253723, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \,{\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, c^{2}}, -\frac{3 \,{\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \,{\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 48.9777, size = 253, normalized size = 2.06 \[ \frac{a c^{2} x^{7}}{6 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{11 a c \sqrt{d} x^{5}}{24 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{17 a d^{\frac{3}{2}} x^{3}}{48 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{5}{2}} x}{16 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{16 c^{\frac{3}{2}}} + \frac{b c^{2} x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b c \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{b d^{\frac{3}{2}} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{b d^{\frac{3}{2}} x}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.235508, size = 197, normalized size = 1.6 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, a c x^{2}{\rm sign}\left (x\right ) + \frac{6 \, b c^{5}{\rm sign}\left (x\right ) + 7 \, a c^{4} d{\rm sign}\left (x\right )}{c^{4}}\right )} x^{2} + \frac{3 \,{\left (10 \, b c^{4} d{\rm sign}\left (x\right ) + a c^{3} d^{2}{\rm sign}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + d} x - \frac{{\left (6 \, b c d^{2}{\rm sign}\left (x\right ) - a d^{3}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{16 \, c^{\frac{3}{2}}} + \frac{{\left (6 \, b c d^{2}{\rm ln}\left (\sqrt{d}\right ) - a d^{3}{\rm ln}\left (\sqrt{d}\right )\right )}{\rm sign}\left (x\right )}{16 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x, algorithm="giac")

[Out]

1/48*(2*(4*a*c*x^2*sign(x) + (6*b*c^5*sign(x) + 7*a*c^4*d*sign(x))/c^4)*x^2 + 3*
(10*b*c^4*d*sign(x) + a*c^3*d^2*sign(x))/c^4)*sqrt(c*x^2 + d)*x - 1/16*(6*b*c*d^
2*sign(x) - a*d^3*sign(x))*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^(3/2) + 1/16*
(6*b*c*d^2*ln(sqrt(d)) - a*d^3*ln(sqrt(d)))*sign(x)/c^(3/2)