3.928 \(\int \frac{\sqrt{a+b x^2+c x^4}}{x^9} \, dx\)

Optimal. Leaf size=161 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]

[Out]

-((5*b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*a^3*x^4) - (a + b*
x^2 + c*x^4)^(3/2)/(8*a*x^8) + (5*b*(a + b*x^2 + c*x^4)^(3/2))/(48*a^2*x^6) + ((
b^2 - 4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c
*x^4])])/(256*a^(7/2))

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Rubi [A]  time = 0.386265, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2 + c*x^4]/x^9,x]

[Out]

-((5*b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*a^3*x^4) - (a + b*
x^2 + c*x^4)^(3/2)/(8*a*x^8) + (5*b*(a + b*x^2 + c*x^4)^(3/2))/(48*a^2*x^6) + ((
b^2 - 4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c
*x^4])])/(256*a^(7/2))

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Rubi in Sympy [A]  time = 30.3439, size = 148, normalized size = 0.92 \[ - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{8 a x^{8}} + \frac{5 b \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{48 a^{2} x^{6}} - \frac{\left (2 a + b x^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{128 a^{3} x^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{256 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(1/2)/x**9,x)

[Out]

-(a + b*x**2 + c*x**4)**(3/2)/(8*a*x**8) + 5*b*(a + b*x**2 + c*x**4)**(3/2)/(48*
a**2*x**6) - (2*a + b*x**2)*(-4*a*c + 5*b**2)*sqrt(a + b*x**2 + c*x**4)/(128*a**
3*x**4) + (-4*a*c + b**2)*(-4*a*c + 5*b**2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt
(a + b*x**2 + c*x**4)))/(256*a**(7/2))

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Mathematica [A]  time = 0.228904, size = 143, normalized size = 0.89 \[ \frac{-\frac{2 \sqrt{a} \sqrt{a+b x^2+c x^4} \left (48 a^3+8 a^2 x^2 \left (b+3 c x^2\right )-2 a b x^4 \left (5 b+26 c x^2\right )+15 b^3 x^6\right )}{x^8}-3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{768 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2 + c*x^4]/x^9,x]

[Out]

((-2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]*(48*a^3 + 15*b^3*x^6 + 8*a^2*x^2*(b + 3*c*x
^2) - 2*a*b*x^4*(5*b + 26*c*x^2)))/x^8 - 3*(b^2 - 4*a*c)*(5*b^2 - 4*a*c)*(Log[x^
2] - Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]]))/(768*a^(7/2))

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Maple [B]  time = 0.023, size = 387, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{8}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{4}}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{4}}{256}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,{x}^{2}{b}^{3}c}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{2}c}{64\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}c}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{16\,{a}^{2}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{bc}{32\,{a}^{3}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b{c}^{2}{x}^{2}}{32\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{c}^{2}}{16\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(1/2)/x^9,x)

[Out]

-1/8*(c*x^4+b*x^2+a)^(3/2)/a/x^8+5/48*b*(c*x^4+b*x^2+a)^(3/2)/a^2/x^6-5/64*b^2/a
^3/x^4*(c*x^4+b*x^2+a)^(3/2)+5/128*b^3/a^4/x^2*(c*x^4+b*x^2+a)^(3/2)-5/128*b^4/a
^4*(c*x^4+b*x^2+a)^(1/2)+5/256*b^4/a^(7/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+
a)^(1/2))/x^2)-5/128*b^3/a^4*c*(c*x^4+b*x^2+a)^(1/2)*x^2+7/64*b^2/a^3*c*(c*x^4+b
*x^2+a)^(1/2)-3/32*b^2/a^(5/2)*c*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/
x^2)+1/16*c/a^2/x^4*(c*x^4+b*x^2+a)^(3/2)-1/32*c/a^3*b/x^2*(c*x^4+b*x^2+a)^(3/2)
+1/32*c^2/a^3*b*(c*x^4+b*x^2+a)^(1/2)*x^2-1/16*c^2/a^2*(c*x^4+b*x^2+a)^(1/2)+1/1
6*c^2/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^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^2 + a)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310012, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{8} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \,{\left ({\left (15 \, b^{3} - 52 \, a b c\right )} x^{6} + 8 \, a^{2} b x^{2} - 2 \,{\left (5 \, a b^{2} - 12 \, a^{2} c\right )} x^{4} + 48 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{1536 \, a^{\frac{7}{2}} x^{8}}, \frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{8} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{3} - 52 \, a b c\right )} x^{6} + 8 \, a^{2} b x^{2} - 2 \,{\left (5 \, a b^{2} - 12 \, a^{2} c\right )} x^{4} + 48 \, a^{3}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{768 \, \sqrt{-a} a^{3} x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^9,x, algorithm="fricas")

[Out]

[1/1536*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x^8*log(-(4*sqrt(c*x^4 + b*x^2 + a)
*(a*b*x^2 + 2*a^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) - 4*(
(15*b^3 - 52*a*b*c)*x^6 + 8*a^2*b*x^2 - 2*(5*a*b^2 - 12*a^2*c)*x^4 + 48*a^3)*sqr
t(c*x^4 + b*x^2 + a)*sqrt(a))/(a^(7/2)*x^8), 1/768*(3*(5*b^4 - 24*a*b^2*c + 16*a
^2*c^2)*x^8*arctan(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*(
(15*b^3 - 52*a*b*c)*x^6 + 8*a^2*b*x^2 - 2*(5*a*b^2 - 12*a^2*c)*x^4 + 48*a^3)*sqr
t(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^8)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2} + c x^{4}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(1/2)/x**9,x)

[Out]

Integral(sqrt(a + b*x**2 + c*x**4)/x**9, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2} + a}}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^9,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)/x^9, x)