3.118 \(\int \frac{\sqrt{x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.128364, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{x} \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a*x + b*x^3 + c*x^5)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.5493, size = 122, normalized size = 1.18 \[ \frac{\sqrt{x} \left (- 2 a c + b^{2} + b c x^{2}\right )}{a \left (- 4 a c + b^{2}\right ) \sqrt{a x + b x^{3} + c x^{5}}} - \frac{\sqrt{x} \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 a^{\frac{3}{2}} \sqrt{a x + b x^{3} + c x^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(c*x**5+b*x**3+a*x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.369491, size = 154, normalized size = 1.5 \[ \frac{\sqrt{x} \left (-2 \sqrt{a} \left (-2 a c+b^2+b c x^2\right )-\log \left (x^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}+\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{2 a^{3/2} \left (4 a c-b^2\right ) \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a*x + b*x^3 + c*x^5)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.018, size = 179, normalized size = 1.7 \[ -{\frac{1}{ \left ( 2\,c{x}^{4}+2\,b{x}^{2}+2\,a \right ) \left ( 4\,ac-{b}^{2} \right ) }\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( 2\,{x}^{2}bc\sqrt{a}+4\,\ln \left ({\frac{2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a}}{{x}^{2}}} \right ) ac\sqrt{c{x}^{4}+b{x}^{2}+a}-\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){b}^{2}\sqrt{c{x}^{4}+b{x}^{2}+a}-4\,{a}^{3/2}c+2\,{b}^{2}\sqrt{a} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^5+b*x^3+a*x)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^5 + b*x^3 + a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x)/(c*x^5 + b*x^3 + a*x)^(3/2), x)

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Fricas [A]  time = 0.310398, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{a} \sqrt{x} +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{5} +{\left (b^{3} - 4 \, a b c\right )} x^{3} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \log \left (\frac{4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt{x} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{5}}\right )}{4 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{5} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x\right )} \sqrt{a}}, \frac{2 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{-a} \sqrt{x} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{5} +{\left (b^{3} - 4 \, a b c\right )} x^{3} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \arctan \left (\frac{{\left (b x^{3} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{5} + b x^{3} + a x} a \sqrt{x}}\right )}{2 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{5} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^5 + b*x^3 + a*x)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(c*x**5+b*x**3+a*x)**(3/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^5 + b*x^3 + a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(x)/(c*x^5 + b*x^3 + a*x)^(3/2), x)