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

Optimal. Leaf size=90 \[ -\frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{5/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{8 c^2}+\frac{a x^4 \sqrt{c+\frac{d}{x^2}}}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.218531, 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 \[ -\frac{d (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{5/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-3 a d)}{8 c^2}+\frac{a x^4 \sqrt{c+\frac{d}{x^2}}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5434, size = 82, normalized size = 0.91 \[ \frac{a x^{4} \sqrt{c + \frac{d}{x^{2}}}}{4 c} - \frac{x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (3 a d - 4 b c\right )}{8 c^{2}} + \frac{d \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.11793, size = 98, normalized size = 1.09 \[ \frac{\sqrt{c} x \left (c x^2+d\right ) \left (2 a c x^2-3 a d+4 b c\right )+d \sqrt{c x^2+d} (3 a d-4 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{8 c^{5/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 131, normalized size = 1.5 \[{\frac{1}{8\,x}\sqrt{c{x}^{2}+d} \left ( 2\,a{x}^{3}\sqrt{c{x}^{2}+d}{c}^{7/2}-3\,adx\sqrt{c{x}^{2}+d}{c}^{5/2}+4\,bx\sqrt{c{x}^{2}+d}{c}^{7/2}-4\,bd\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{3}+3\,a{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{c}^{-{\frac{9}{2}}}} \]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23529, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, b c d - 3 \, a d^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c^{3}}, \frac{{\left (4 \, b c d - 3 \, a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 24.6507, size = 150, normalized size = 1.67 \[ \frac{a x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a \sqrt{d} x^{3}}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 a d^{\frac{3}{2}} x}{8 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{5}{2}}} + \frac{b \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2 c} - \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{3}{2}}} \]

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

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GIAC/XCAS [A]  time = 0.248383, size = 220, normalized size = 2.44 \[ \frac{1}{8} \, d^{2}{\left (\frac{{\left (4 \, b c - 3 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2} d} - \frac{4 \, b c^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - 5 \, a c d \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{4 \,{\left (c x^{2} + d\right )} b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}} + \frac{3 \,{\left (c x^{2} + d\right )} a d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )}^{2} c^{2} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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