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

Optimal. Leaf size=120 \[ -\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{7/2}}+\frac{3 x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-5 a d)}{8 c^3}-\frac{x^2 (4 b c-5 a d)}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}} \]

[Out]

-((4*b*c - 5*a*d)*x^2)/(4*c^2*Sqrt[c + d/x^2]) + (3*(4*b*c - 5*a*d)*Sqrt[c + d/x
^2]*x^2)/(8*c^3) + (a*x^4)/(4*c*Sqrt[c + d/x^2]) - (3*d*(4*b*c - 5*a*d)*ArcTanh[
Sqrt[c + d/x^2]/Sqrt[c]])/(8*c^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.269688, antiderivative size = 120, 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 \[ -\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{7/2}}+\frac{3 x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-5 a d)}{8 c^3}-\frac{x^2 (4 b c-5 a d)}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-((4*b*c - 5*a*d)*x^2)/(4*c^2*Sqrt[c + d/x^2]) + (3*(4*b*c - 5*a*d)*Sqrt[c + d/x
^2]*x^2)/(8*c^3) + (a*x^4)/(4*c*Sqrt[c + d/x^2]) - (3*d*(4*b*c - 5*a*d)*ArcTanh[
Sqrt[c + d/x^2]/Sqrt[c]])/(8*c^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.0362, size = 114, normalized size = 0.95 \[ \frac{a x^{4}}{4 c \sqrt{c + \frac{d}{x^{2}}}} + \frac{x^{2} \left (5 a d - 4 b c\right )}{4 c^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{3 x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (5 a d - 4 b c\right )}{8 c^{3}} + \frac{3 d \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{8 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/(4*c*sqrt(c + d/x**2)) + x**2*(5*a*d - 4*b*c)/(4*c**2*sqrt(c + d/x**2)) -
3*x**2*sqrt(c + d/x**2)*(5*a*d - 4*b*c)/(8*c**3) + 3*d*(5*a*d - 4*b*c)*atanh(sq
rt(c + d/x**2)/sqrt(c))/(8*c**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.147545, size = 113, normalized size = 0.94 \[ \frac{\sqrt{c} x \left (a \left (2 c^2 x^4-5 c d x^2-15 d^2\right )+4 b c \left (c x^2+3 d\right )\right )+3 d \sqrt{c x^2+d} (5 a d-4 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{8 c^{7/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*x*(4*b*c*(3*d + c*x^2) + a*(-15*d^2 - 5*c*d*x^2 + 2*c^2*x^4)) + 3*d*(-4
*b*c + 5*a*d)*Sqrt[d + c*x^2]*Log[c*x + Sqrt[c]*Sqrt[d + c*x^2]])/(8*c^(7/2)*Sqr
t[c + d/x^2]*x)

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 142, normalized size = 1.2 \[{\frac{c{x}^{2}+d}{8\,{x}^{3}} \left ( 2\,a{x}^{5}{c}^{11/2}-5\,ad{x}^{3}{c}^{9/2}+4\,{x}^{3}b{c}^{11/2}-15\,a{d}^{2}x{c}^{7/2}+12\,xbd{c}^{9/2}-12\,bd\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{4}\sqrt{c{x}^{2}+d}+15\,a{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{3}\sqrt{c{x}^{2}+d} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(c*x^2+d)*(2*a*x^5*c^(11/2)-5*a*d*x^3*c^(9/2)+4*x^3*b*c^(11/2)-15*a*d^2*x*c^
(7/2)+12*x*b*d*c^(9/2)-12*b*d*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^4*(c*x^2+d)^(1/2)+
15*a*d^2*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^3*(c*x^2+d)^(1/2))/((c*x^2+d)/x^2)^(3/2
)/x^3/c^(13/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.244504, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (4 \, b c d^{2} - 5 \, a d^{3} +{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (2 \, a c^{3} x^{6} +{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \,{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \,{\left (c^{5} x^{2} + c^{4} d\right )}}, \frac{3 \,{\left (4 \, b c d^{2} - 5 \, a d^{3} +{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (2 \, a c^{3} x^{6} +{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \,{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \,{\left (c^{5} x^{2} + c^{4} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(3*(4*b*c*d^2 - 5*a*d^3 + (4*b*c^2*d - 5*a*c*d^2)*x^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^3*x^6 + (4*b*c^3 -
5*a*c^2*d)*x^4 + 3*(4*b*c^2*d - 5*a*c*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/(c^5*x^2
+ c^4*d), 1/8*(3*(4*b*c*d^2 - 5*a*d^3 + (4*b*c^2*d - 5*a*c*d^2)*x^2)*sqrt(-c)*ar
ctan(sqrt(-c)/sqrt((c*x^2 + d)/x^2)) + (2*a*c^3*x^6 + (4*b*c^3 - 5*a*c^2*d)*x^4
+ 3*(4*b*c^2*d - 5*a*c*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/(c^5*x^2 + c^4*d)]

_______________________________________________________________________________________

Sympy [A]  time = 24.2744, size = 177, normalized size = 1.48 \[ a \left (\frac{x^{5}}{4 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{5 \sqrt{d} x^{3}}{8 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{15 d^{\frac{3}{2}} x}{8 c^{3} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{15 d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{7}{2}}}\right ) + b \left (\frac{x^{3}}{2 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 \sqrt{d} x}{2 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{5}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(x**5/(4*c*sqrt(d)*sqrt(c*x**2/d + 1)) - 5*sqrt(d)*x**3/(8*c**2*sqrt(c*x**2/d
+ 1)) - 15*d**(3/2)*x/(8*c**3*sqrt(c*x**2/d + 1)) + 15*d**2*asinh(sqrt(c)*x/sqrt
(d))/(8*c**(7/2))) + b*(x**3/(2*c*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*sqrt(d)*x/(2*c
**2*sqrt(c*x**2/d + 1)) - 3*d*asinh(sqrt(c)*x/sqrt(d))/(2*c**(5/2)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.254722, size = 261, normalized size = 2.17 \[ \frac{1}{8} \, d^{2}{\left (\frac{3 \,{\left (4 \, b c - 5 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3} d} + \frac{8 \,{\left (b c - a d\right )}}{c^{3} d \sqrt{\frac{c x^{2} + d}{x^{2}}}} - \frac{4 \, b c^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - 9 \, a c d \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{4 \,{\left (c x^{2} + d\right )} b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}} + \frac{7 \,{\left (c x^{2} + d\right )} a d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )}^{2} c^{3} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*d^2*(3*(4*b*c - 5*a*d)*arctan(sqrt((c*x^2 + d)/x^2)/sqrt(-c))/(sqrt(-c)*c^3*
d) + 8*(b*c - a*d)/(c^3*d*sqrt((c*x^2 + d)/x^2)) - (4*b*c^2*sqrt((c*x^2 + d)/x^2
) - 9*a*c*d*sqrt((c*x^2 + d)/x^2) - 4*(c*x^2 + d)*b*c*sqrt((c*x^2 + d)/x^2)/x^2
+ 7*(c*x^2 + d)*a*d*sqrt((c*x^2 + d)/x^2)/x^2)/((c - (c*x^2 + d)/x^2)^2*c^3*d))