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

Optimal. Leaf size=123 \[ -\frac{d^2 (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{5/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{16 c^2}+\frac{x^4 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{8 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{3/2}}{6 c} \]

[Out]

(d*(2*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(16*c^2) + ((2*b*c - a*d)*Sqrt[c + d/x^2]*
x^4)/(8*c) + (a*(c + d/x^2)^(3/2)*x^6)/(6*c) - (d^2*(2*b*c - a*d)*ArcTanh[Sqrt[c
+ d/x^2]/Sqrt[c]])/(16*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.274661, antiderivative size = 123, 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 \[ -\frac{d^2 (2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{5/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{16 c^2}+\frac{x^4 \sqrt{c+\frac{d}{x^2}} (2 b c-a d)}{8 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(d*(2*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(16*c^2) + ((2*b*c - a*d)*Sqrt[c + d/x^2]*
x^4)/(8*c) + (a*(c + d/x^2)^(3/2)*x^6)/(6*c) - (d^2*(2*b*c - a*d)*ArcTanh[Sqrt[c
+ d/x^2]/Sqrt[c]])/(16*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.6044, size = 107, normalized size = 0.87 \[ \frac{a x^{6} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{6 c} - \frac{x^{4} \sqrt{c + \frac{d}{x^{2}}} \left (\frac{a d}{2} - b c\right )}{4 c} - \frac{d x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (a d - 2 b c\right )}{16 c^{2}} + \frac{d^{2} \left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x**5*(c+d/x**2)**(1/2),x)

[Out]

a*x**6*(c + d/x**2)**(3/2)/(6*c) - x**4*sqrt(c + d/x**2)*(a*d/2 - b*c)/(4*c) - d
*x**2*sqrt(c + d/x**2)*(a*d - 2*b*c)/(16*c**2) + d**2*(a*d/2 - b*c)*atanh(sqrt(c
+ d/x**2)/sqrt(c))/(8*c**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.155007, size = 123, normalized size = 1. \[ \frac{d^2 x \sqrt{c+\frac{d}{x^2}} (a d-2 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{16 c^{5/2} \sqrt{c x^2+d}}+x \sqrt{c+\frac{d}{x^2}} \left (-\frac{d x (a d-2 b c)}{16 c^2}+\frac{x^3 (a d+6 b c)}{24 c}+\frac{a x^5}{6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 167, normalized size = 1.4 \[{\frac{x}{48}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 8\,a{x}^{3} \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{7/2}-6\,adx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{5/2}+12\,bx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{7/2}+3\,a{d}^{2}x\sqrt{c{x}^{2}+d}{c}^{5/2}-6\,bdx\sqrt{c{x}^{2}+d}{c}^{7/2}-6\,b{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{3}+3\,a{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^5*(c+d/x^2)^(1/2),x)

[Out]

1/48*((c*x^2+d)/x^2)^(1/2)*x*(8*a*x^3*(c*x^2+d)^(3/2)*c^(7/2)-6*a*d*x*(c*x^2+d)^
(3/2)*c^(5/2)+12*b*x*(c*x^2+d)^(3/2)*c^(7/2)+3*a*d^2*x*(c*x^2+d)^(1/2)*c^(5/2)-6
*b*d*x*(c*x^2+d)^(1/2)*c^(7/2)-6*b*d^2*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^3+3*a*d^3
*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^2)/(c*x^2+d)^(1/2)/c^(9/2)

_______________________________________________________________________________________

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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.251476, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, c^{3}}, \frac{3 \,{\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \,{\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 33.5933, size = 226, normalized size = 1.84 \[ \frac{a c x^{7}}{6 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{5 a \sqrt{d} x^{5}}{24 \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{\frac{3}{2}} x^{3}}{48 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{\frac{5}{2}} x}{16 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{16 c^{\frac{5}{2}}} + \frac{b c x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{b d^{\frac{3}{2}} x}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**5*(c+d/x**2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222024, size = 196, normalized size = 1.59 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, a x^{2}{\rm sign}\left (x\right ) + \frac{6 \, b c^{4}{\rm sign}\left (x\right ) + a c^{3} d{\rm sign}\left (x\right )}{c^{4}}\right )} x^{2} + \frac{3 \,{\left (2 \, b c^{3} d{\rm sign}\left (x\right ) - a c^{2} d^{2}{\rm sign}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + d} x + \frac{{\left (2 \, b c d^{2}{\rm sign}\left (x\right ) - a d^{3}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{16 \, c^{\frac{5}{2}}} - \frac{{\left (2 \, b c d^{2}{\rm ln}\left (\sqrt{d}\right ) - a d^{3}{\rm ln}\left (\sqrt{d}\right )\right )}{\rm sign}\left (x\right )}{16 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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