3.37 \(\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^6} \, dx\)

Optimal. Leaf size=205 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(4*x^5) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a*x^4
) + ((5*b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(96*a^2*x^3) - (b*(15*b^2 - 5
2*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*a^3*x^2) + ((b^2 - 4*a*c)*(5*b^2 - 4*a*
c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(7/2
))

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Rubi [A]  time = 0.619445, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{128 a^{7/2}}-\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 a^3 x^2}+\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 a^2 x^3}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{24 a x^4}-\frac{\sqrt{a x^2+b x^3+c x^4}}{4 x^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^6,x]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(4*x^5) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a*x^4
) + ((5*b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(96*a^2*x^3) - (b*(15*b^2 - 5
2*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*a^3*x^2) + ((b^2 - 4*a*c)*(5*b^2 - 4*a*
c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(128*a^(7/2
))

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Rubi in Sympy [A]  time = 84.7958, size = 214, normalized size = 1.04 \[ - \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 x^{5}} - \frac{b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{24 a x^{4}} + \frac{\left (- 12 a c + 5 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{96 a^{2} x^{3}} - \frac{b \left (- 52 a c + 15 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{192 a^{3} x^{2}} + \frac{x \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{7}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

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**6,x)

[Out]

-sqrt(a*x**2 + b*x**3 + c*x**4)/(4*x**5) - b*sqrt(a*x**2 + b*x**3 + c*x**4)/(24*
a*x**4) + (-12*a*c + 5*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(96*a**2*x**3) - b*(
-52*a*c + 15*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(192*a**3*x**2) + x*(-4*a*c +
b**2)*(-4*a*c + 5*b**2)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt
(a + b*x + c*x**2)))/(128*a**(7/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.392298, size = 186, normalized size = 0.91 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (-3 x^4 \log (x) \left (16 a^2 c^2-24 a b^2 c+5 b^4\right )+3 x^4 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (48 a^3+8 a^2 x (b+3 c x)-2 a b x^2 (5 b+26 c x)+15 b^3 x^3\right )\right )}{384 a^{7/2} x^5 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^6,x]

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(48*a^3 + 15*b^3*
x^3 + 8*a^2*x*(b + 3*c*x) - 2*a*b*x^2*(5*b + 26*c*x)) - 3*(5*b^4 - 24*a*b^2*c +
16*a^2*c^2)*x^4*Log[x] + 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^4*Log[2*a + b*x +
2*Sqrt[a]*Sqrt[a + x*(b + c*x)]]))/(384*a^(7/2)*x^5*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.011, size = 387, normalized size = 1.9 \[ -{\frac{1}{384\,{x}^{5}{a}^{4}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( -48\,{a}^{5/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){c}^{2}{x}^{4}-24\,\sqrt{c{x}^{2}+bx+a}{c}^{2}{x}^{5}ab+72\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) c{x}^{4}{b}^{2}+48\,\sqrt{c{x}^{2}+bx+a}{c}^{2}{x}^{4}{a}^{2}+30\,\sqrt{c{x}^{2}+bx+a}c{x}^{5}{b}^{3}+24\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}c{x}^{3}ab-84\,\sqrt{c{x}^{2}+bx+a}c{x}^{4}a{b}^{2}-15\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{4}{b}^{4}-48\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}c{x}^{2}{a}^{2}-30\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{3}{b}^{3}+30\,\sqrt{c{x}^{2}+bx+a}{x}^{4}{b}^{4}+60\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}a{b}^{2}-80\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}x{a}^{2}b+96\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^3+a*x^2)^(1/2)/x^6,x)

[Out]

-1/384*(c*x^4+b*x^3+a*x^2)^(1/2)*(-48*a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a
)^(1/2))/x)*c^2*x^4-24*(c*x^2+b*x+a)^(1/2)*c^2*x^5*a*b+72*a^(3/2)*ln((2*a+b*x+2*
a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*c*x^4*b^2+48*(c*x^2+b*x+a)^(1/2)*c^2*x^4*a^2+30*
(c*x^2+b*x+a)^(1/2)*c*x^5*b^3+24*(c*x^2+b*x+a)^(3/2)*c*x^3*a*b-84*(c*x^2+b*x+a)^
(1/2)*c*x^4*a*b^2-15*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^4*b
^4-48*(c*x^2+b*x+a)^(3/2)*c*x^2*a^2-30*(c*x^2+b*x+a)^(3/2)*x^3*b^3+30*(c*x^2+b*x
+a)^(1/2)*x^4*b^4+60*(c*x^2+b*x+a)^(3/2)*x^2*a*b^2-80*(c*x^2+b*x+a)^(3/2)*x*a^2*
b+96*(c*x^2+b*x+a)^(3/2)*a^3)/x^5/(c*x^2+b*x+a)^(1/2)/a^4

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30804, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{5} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} +{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) - 4 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{768 \, a^{4} x^{5}}, -\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{5} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) + 2 \,{\left (8 \, a^{3} b x + 48 \, a^{4} +{\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{384 \, a^{4} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x^6,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*sqrt(a)*x^5*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*(8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5*a^2*b^2 - 12*
a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^4*x^5), -1/384*(3*(5*b^4 - 24*a*b^2*
c + 16*a^2*c^2)*sqrt(-a)*x^5*arctan(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x^4 + b
*x^3 + a*x^2)*a)) + 2*(8*a^3*b*x + 48*a^4 + (15*a*b^3 - 52*a^2*b*c)*x^3 - 2*(5*a
^2*b^2 - 12*a^3*c)*x^2)*sqrt(c*x^4 + b*x^3 + a*x^2))/(a^4*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x**2*(a + b*x + c*x**2))/x**6, 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^6,x, algorithm="giac")

[Out]

Timed out