3.284 \(\int \frac{1}{\left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{1}{b x \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.119543, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{1}{b x \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.8115, size = 73, normalized size = 0.9 \[ \frac{1}{b x \sqrt{b x^{2} + c x^{4}}} - \frac{3 \sqrt{b x^{2} + c x^{4}}}{2 b^{2} x^{3}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**4+b*x**2)**(3/2),x)

[Out]

1/(b*x*sqrt(b*x**2 + c*x**4)) - 3*sqrt(b*x**2 + c*x**4)/(2*b**2*x**3) + 3*c*atan
h(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(2*b**(5/2))

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Mathematica [A]  time = 0.0693013, size = 99, normalized size = 1.22 \[ \frac{-\sqrt{b} \left (b+3 c x^2\right )-3 c x^2 \log (x) \sqrt{b+c x^2}+3 c x^2 \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{2 b^{5/2} x \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(-3/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 79, normalized size = 1. \[{\frac{x \left ( c{x}^{2}+b \right ) }{2} \left ( 3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) \sqrt{c{x}^{2}+b}{x}^{2}bc-3\,{b}^{3/2}{x}^{2}c-{b}^{{\frac{5}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+b*x^2)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(-3/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(-3/2), x)

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Fricas [A]  time = 0.278149, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{b} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left (3 \, b c x^{2} + b^{2}\right )}}{4 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, -\frac{3 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}}{\left (3 \, b c x^{2} + b^{2}\right )}}{2 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(-3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral((b*x**2 + c*x**4)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.640204, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(-3/2),x, algorithm="giac")

[Out]

sage0*x