3.35 \(\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^4} \, dx\)

Optimal. Leaf size=114 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.237801, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 41.9542, size = 128, normalized size = 1.12 \[ - \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{2 x^{3}} - \frac{b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 a x^{2}} + \frac{x \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8 a^{\frac{3}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a*x**2 + b*x**3 + c*x**4)/(2*x**3) - b*sqrt(a*x**2 + b*x**3 + c*x**4)/(4*a
*x**2) + x*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*s
qrt(a + b*x + c*x**2)))/(8*a**(3/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.171924, size = 125, normalized size = 1.1 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (x^2 \log (x) \left (-\left (b^2-4 a c\right )\right )+x^2 \left (b^2-4 a c\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)}\right )}{8 a^{3/2} x^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^4,x]

[Out]

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

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Maple [B]  time = 0.009, size = 206, normalized size = 1.8 \[{\frac{1}{8\,{a}^{2}{x}^{3}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( -4\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) c{x}^{2}-2\,\sqrt{c{x}^{2}+bx+a}c{x}^{3}b+4\,\sqrt{c{x}^{2}+bx+a}c{x}^{2}a+\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){x}^{2}{b}^{2}+2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}xb-2\,\sqrt{c{x}^{2}+bx+a}{x}^{2}{b}^{2}-4\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}a \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292596, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{a} x^{3} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} -{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )}}{16 \, a^{2} x^{3}}, -\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-a} x^{3} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )}}{8 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((b^2 - 4*a*c)*sqrt(a)*x^3*log((4*sqrt(c*x^4 + b*x^3 + a*x^2)*(a*b*x + 2*
a^2) - (8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x)*sqrt(a))/x^3) + 4*sqrt(c*x^4 +
b*x^3 + a*x^2)*(a*b*x + 2*a^2))/(a^2*x^3), -1/8*((b^2 - 4*a*c)*sqrt(-a)*x^3*arct
an(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x^4 + b*x^3 + a*x^2)*a)) + 2*sqrt(c*x^4
+ b*x^3 + a*x^2)*(a*b*x + 2*a^2))/(a^2*x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out