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

Optimal. Leaf size=155 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2} \]

[Out]

((b^2 + 8*a*c + 2*b*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*c) + (a + b*x^2 + c*x^4)
^(3/2)/6 - (a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/
2 - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]
)/(32*c^(3/2))

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Rubi [A]  time = 0.462733, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 + 8*a*c + 2*b*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(16*c) + (a + b*x^2 + c*x^4)
^(3/2)/6 - (a^(3/2)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/
2 - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]
)/(32*c^(3/2))

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Rubi in Sympy [A]  time = 36.3782, size = 139, normalized size = 0.9 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2} - \frac{b \left (- 12 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{3}{2}}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6} + \frac{\sqrt{a + b x^{2} + c x^{4}} \left (4 a c + \frac{b^{2}}{2} + b c x^{2}\right )}{8 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4)))/2 - b*(-12
*a*c + b**2)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(32*c**
(3/2)) + (a + b*x**2 + c*x**4)**(3/2)/6 + sqrt(a + b*x**2 + c*x**4)*(4*a*c + b**
2/2 + b*c*x**2)/(8*c)

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Mathematica [A]  time = 0.622195, size = 150, normalized size = 0.97 \[ \frac{1}{96} \left (-48 a^{3/2} \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )+48 a^{3/2} \log \left (x^2\right )-\frac{3 \left (b^3-12 a b c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{c^{3/2}}+\frac{2 \sqrt{a+b x^2+c x^4} \left (8 c \left (4 a+c x^4\right )+3 b^2+14 b c x^2\right )}{c}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + b*x^2 + c*x^4]*(3*b^2 + 14*b*c*x^2 + 8*c*(4*a + c*x^4)))/c + 48*a^(
3/2)*Log[x^2] - 48*a^(3/2)*Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]]
- (3*(b^3 - 12*a*b*c)*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/c^(3
/2))/96

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Maple [A]  time = 0.023, size = 192, normalized size = 1.2 \[{\frac{{b}^{2}}{16\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{3}}{32}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{1}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ) }+{\frac{c{x}^{4}}{6}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,b{x}^{2}}{24}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,ab}{8}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{2\,a}{3}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x,x)

[Out]

1/16*b^2/c*(c*x^4+b*x^2+a)^(1/2)-1/32*b^3/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^
4+b*x^2+a)^(1/2))-1/2*a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2
)+1/6*c*x^4*(c*x^4+b*x^2+a)^(1/2)+7/24*b*x^2*(c*x^4+b*x^2+a)^(1/2)+3/8/c^(1/2)*b
*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+2/3*a*(c*x^4+b*x^2+a)^(1/2)

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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 + a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.417914, size = 1, normalized size = 0.01 \[ \left [\frac{48 \, a^{\frac{3}{2}} c^{\frac{3}{2}} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \,{\left (8 \, c^{2} x^{4} + 14 \, b c x^{2} + 3 \, b^{2} + 32 \, a c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} - 3 \,{\left (b^{3} - 12 \, a b c\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{192 \, c^{\frac{3}{2}}}, \frac{24 \, a^{\frac{3}{2}} \sqrt{-c} c \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) + 2 \,{\left (8 \, c^{2} x^{4} + 14 \, b c x^{2} + 3 \, b^{2} + 32 \, a c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} - 3 \,{\left (b^{3} - 12 \, a b c\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{96 \, \sqrt{-c} c}, -\frac{96 \, \sqrt{-a} a c^{\frac{3}{2}} \arctan \left (\frac{b x^{2} + 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}\right ) - 4 \,{\left (8 \, c^{2} x^{4} + 14 \, b c x^{2} + 3 \, b^{2} + 32 \, a c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 3 \,{\left (b^{3} - 12 \, a b c\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{192 \, c^{\frac{3}{2}}}, -\frac{48 \, \sqrt{-a} a \sqrt{-c} c \arctan \left (\frac{b x^{2} + 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}\right ) - 2 \,{\left (8 \, c^{2} x^{4} + 14 \, b c x^{2} + 3 \, b^{2} + 32 \, a c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} + 3 \,{\left (b^{3} - 12 \, a b c\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{96 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(48*a^(3/2)*c^(3/2)*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 - 4*sqrt(c*x^4 +
b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) + 4*(8*c^2*x^4 + 14*b*c*x^2 + 3*b
^2 + 32*a*c)*sqrt(c*x^4 + b*x^2 + a)*sqrt(c) - 3*(b^3 - 12*a*b*c)*log(-4*sqrt(c*
x^4 + b*x^2 + a)*(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*a*c)*sqrt(
c)))/c^(3/2), 1/96*(24*a^(3/2)*sqrt(-c)*c*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 -
4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) + 2*(8*c^2*x^4 + 1
4*b*c*x^2 + 3*b^2 + 32*a*c)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) - 3*(b^3 - 12*a*b*c
)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)))/(sqrt(-c)*c),
-1/192*(96*sqrt(-a)*a*c^(3/2)*arctan(1/2*(b*x^2 + 2*a)/(sqrt(c*x^4 + b*x^2 + a)*
sqrt(-a))) - 4*(8*c^2*x^4 + 14*b*c*x^2 + 3*b^2 + 32*a*c)*sqrt(c*x^4 + b*x^2 + a)
*sqrt(c) + 3*(b^3 - 12*a*b*c)*log(-4*sqrt(c*x^4 + b*x^2 + a)*(2*c^2*x^2 + b*c) -
(8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*a*c)*sqrt(c)))/c^(3/2), -1/96*(48*sqrt(-a)*a*s
qrt(-c)*c*arctan(1/2*(b*x^2 + 2*a)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))) - 2*(8*c^
2*x^4 + 14*b*c*x^2 + 3*b^2 + 32*a*c)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) + 3*(b^3 -
12*a*b*c)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)))/(sqrt
(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x, x)