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

Optimal. Leaf size=109 \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{7/2}}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{1}{b x^3 \sqrt{b x^2+c x^4}} \]

[Out]

1/(b*x^3*Sqrt[b*x^2 + c*x^4]) - (5*Sqrt[b*x^2 + c*x^4])/(4*b^2*x^5) + (15*c*Sqrt
[b*x^2 + c*x^4])/(8*b^3*x^3) - (15*c^2*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])
/(8*b^(7/2))

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Rubi [A]  time = 0.260075, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{7/2}}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{1}{b x^3 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

1/(b*x^3*Sqrt[b*x^2 + c*x^4]) - (5*Sqrt[b*x^2 + c*x^4])/(4*b^2*x^5) + (15*c*Sqrt
[b*x^2 + c*x^4])/(8*b^3*x^3) - (15*c^2*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])
/(8*b^(7/2))

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Rubi in Sympy [A]  time = 29.9249, size = 102, normalized size = 0.94 \[ \frac{1}{b x^{3} \sqrt{b x^{2} + c x^{4}}} - \frac{5 \sqrt{b x^{2} + c x^{4}}}{4 b^{2} x^{5}} + \frac{15 c \sqrt{b x^{2} + c x^{4}}}{8 b^{3} x^{3}} - \frac{15 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.100422, size = 115, normalized size = 1.06 \[ \frac{\sqrt{b} \left (-2 b^2+5 b c x^2+15 c^2 x^4\right )+15 c^2 x^4 \log (x) \sqrt{b+c x^2}-15 c^2 x^4 \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{8 b^{7/2} x^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 94, normalized size = 0.9 \[ -{\frac{c{x}^{2}+b}{8\,x} \left ( 15\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) \sqrt{c{x}^{2}+b}{x}^{4}b{c}^{2}-15\,{b}^{3/2}{x}^{4}{c}^{2}-5\,{b}^{5/2}{x}^{2}c+2\,{b}^{7/2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2)^(3/2)*x^2), x)

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Fricas [A]  time = 0.280649, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (c^{3} x^{7} + b c^{2} x^{5}\right )} \sqrt{b} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \,{\left (15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}, \frac{15 \,{\left (c^{3} x^{7} + b c^{2} x^{5}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^2),x, algorithm="fricas")

[Out]

[1/16*(15*(c^3*x^7 + b*c^2*x^5)*sqrt(b)*log(-((c*x^3 + 2*b*x)*sqrt(b) - 2*sqrt(c
*x^4 + b*x^2)*b)/x^3) + 2*(15*b*c^2*x^4 + 5*b^2*c*x^2 - 2*b^3)*sqrt(c*x^4 + b*x^
2))/(b^4*c*x^7 + b^5*x^5), 1/8*(15*(c^3*x^7 + b*c^2*x^5)*sqrt(-b)*arctan(sqrt(-b
)*x/sqrt(c*x^4 + b*x^2)) + (15*b*c^2*x^4 + 5*b^2*c*x^2 - 2*b^3)*sqrt(c*x^4 + b*x
^2))/(b^4*c*x^7 + b^5*x^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(x**2*(b + c*x**2))**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + b*x^2)^(3/2)*x^2), x)