3.144 \(\int \frac{A+B x^2}{x^4 \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=103 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]

[Out]

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

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Rubi [A]  time = 0.288223, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^4*Sqrt[b*x^2 + c*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.3809, size = 94, normalized size = 0.91 \[ - \frac{A \sqrt{b x^{2} + c x^{4}}}{4 b x^{5}} + \frac{\left (3 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{8 b^{2} x^{3}} - \frac{c \left (3 A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**4/(c*x**4+b*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.172625, size = 133, normalized size = 1.29 \[ \frac{-\sqrt{b} \left (b+c x^2\right ) \left (2 A b-3 A c x^2+4 b B x^2\right )+c x^4 \log (x) \sqrt{b+c x^2} (3 A c-4 b B)+c x^4 \sqrt{b+c x^2} (4 b B-3 A c) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{8 b^{5/2} x^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^4*Sqrt[b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.021, size = 148, normalized size = 1.4 \[ -{\frac{1}{8\,{x}^{3}}\sqrt{c{x}^{2}+b} \left ( 4\,B\sqrt{c{x}^{2}+b}{x}^{2}{b}^{7/2}-3\,Ac\sqrt{c{x}^{2}+b}{x}^{2}{b}^{5/2}+3\,A{c}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{2}-4\,Bc\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{3}+2\,A\sqrt{c{x}^{2}+b}{b}^{7/2} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^4/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/8*(c*x^2+b)^(1/2)*(4*B*(c*x^2+b)^(1/2)*x^2*b^(7/2)-3*A*c*(c*x^2+b)^(1/2)*x^2*
b^(5/2)+3*A*c^2*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^4*b^2-4*B*c*ln(2*(b^(1/2)*
(c*x^2+b)^(1/2)+b)/x)*x^4*b^3+2*A*(c*x^2+b)^(1/2)*b^(7/2))/x^3/(c*x^4+b*x^2)^(1/
2)/b^(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((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239479, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{b} x^{5} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{16 \, b^{3} x^{5}}, -\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{8 \, b^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="fricas")

[Out]

[-1/16*((4*B*b*c - 3*A*c^2)*sqrt(b)*x^5*log(-((c*x^3 + 2*b*x)*sqrt(b) - 2*sqrt(c
*x^4 + b*x^2)*b)/x^3) + 2*sqrt(c*x^4 + b*x^2)*(2*A*b^2 + (4*B*b^2 - 3*A*b*c)*x^2
))/(b^3*x^5), -1/8*((4*B*b*c - 3*A*c^2)*sqrt(-b)*x^5*arctan(sqrt(-b)*x/sqrt(c*x^
4 + b*x^2)) + sqrt(c*x^4 + b*x^2)*(2*A*b^2 + (4*B*b^2 - 3*A*b*c)*x^2))/(b^3*x^5)
]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{4} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**4/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral((A + B*x**2)/(x**4*sqrt(x**2*(b + c*x**2))), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError