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

Optimal. Leaf size=73 \[ b^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\right )+\frac{b \sqrt{b x^2+c x^4}}{x}+\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.166862, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ b^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\right )+\frac{b \sqrt{b x^2+c x^4}}{x}+\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.8416, size = 61, normalized size = 0.84 \[ - b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )} + \frac{b \sqrt{b x^{2} + c x^{4}}}{x} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(3/2)*atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4)) + b*sqrt(b*x**2 + c*x**4)/x + (
b*x**2 + c*x**4)**(3/2)/(3*x**3)

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Mathematica [A]  time = 0.108358, size = 89, normalized size = 1.22 \[ \frac{x \sqrt{b+c x^2} \left (-3 b^{3/2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+3 b^{3/2} \log (x)+\sqrt{b+c x^2} \left (4 b+c x^2\right )\right )}{3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 78, normalized size = 1.1 \[ -{\frac{1}{3\,{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3/2}- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c{x}^{2}+b}b \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^4,x)

[Out]

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

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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((c*x^4 + b*x^2)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277891, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{\frac{3}{2}} x \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 4 \, b\right )}}{6 \, x}, -\frac{3 \, \sqrt{-b} b x \arctan \left (\frac{b x}{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}\right ) - \sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + 4 \, b\right )}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*b^(3/2)*x*log(-(c*x^3 + 2*b*x - 2*sqrt(c*x^4 + b*x^2)*sqrt(b))/x^3) + 2*
sqrt(c*x^4 + b*x^2)*(c*x^2 + 4*b))/x, -1/3*(3*sqrt(-b)*b*x*arctan(b*x/(sqrt(c*x^
4 + b*x^2)*sqrt(-b))) - sqrt(c*x^4 + b*x^2)*(c*x^2 + 4*b))/x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273278, size = 119, normalized size = 1.63 \[ \frac{1}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} +{\left (c x^{2} + b\right )}^{\frac{3}{2}} + 3 \, \sqrt{c x^{2} + b} b\right )}{\rm sign}\left (x\right ) - \frac{{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*b^2*arctan(sqrt(c*x^2 + b)/sqrt(-b))/sqrt(-b) + (c*x^2 + b)^(3/2) + 3*sqr
t(c*x^2 + b)*b)*sign(x) - 1/3*(3*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/
2))*sign(x)/sqrt(-b)