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

Optimal. Leaf size=124 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*c^2) + ((b + 2*c*x
^2)*(a + b*x^2 + c*x^4)^(3/2))/(16*c) + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^2)
/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(256*c^(5/2))

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Rubi [A]  time = 0.181564, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2}}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{16 c} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*c^2) + ((b + 2*c*x
^2)*(a + b*x^2 + c*x^4)^(3/2))/(16*c) + (3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^2)
/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(256*c^(5/2))

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Rubi in Sympy [A]  time = 14.045, size = 116, normalized size = 0.94 \[ \frac{\left (b + 2 c x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{16 c} - \frac{3 \left (b + 2 c x^{2}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{128 c^{2}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{256 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x**2)*(a + b*x**2 + c*x**4)**(3/2)/(16*c) - 3*(b + 2*c*x**2)*(-4*a*c +
b**2)*sqrt(a + b*x**2 + c*x**4)/(128*c**2) + 3*(-4*a*c + b**2)**2*atanh((b + 2*c
*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(256*c**(5/2))

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Mathematica [A]  time = 0.102101, size = 111, normalized size = 0.9 \[ \frac{2 \sqrt{c} \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} \left (4 c \left (5 a+2 c x^4\right )-3 b^2+8 b c x^2\right )+3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{256 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4]*(-3*b^2 + 8*b*c*x^2 + 4*c*(5*a
+ 2*c*x^4)) + 3*(b^2 - 4*a*c)^2*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x
^4]])/(256*c^(5/2))

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Maple [B]  time = 0.019, size = 242, normalized size = 2. \[{\frac{3\,{a}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{c{x}^{6}}{8}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,b{x}^{4}}{16}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}{x}^{2}}{64\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{3}}{128\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{4}}{256}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{3\,a{b}^{2}}{32}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{5\,ab}{32\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,a{x}^{2}}{16}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+b*x^2+a)^(3/2),x)

[Out]

3/16*a^2*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+1/8*c*x^6*(c*x^
4+b*x^2+a)^(1/2)+3/16*b*x^4*(c*x^4+b*x^2+a)^(1/2)+1/64/c*b^2*x^2*(c*x^4+b*x^2+a)
^(1/2)-3/128/c^2*b^3*(c*x^4+b*x^2+a)^(1/2)+3/256/c^(5/2)*b^4*ln((1/2*b+c*x^2)/c^
(1/2)+(c*x^4+b*x^2+a)^(1/2))-3/32/c^(3/2)*b^2*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+
b*x^2+a)^(1/2))+5/32/c*b*a*(c*x^4+b*x^2+a)^(1/2)+5/16*a*x^2*(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 = 0.299633, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (16 \, c^{3} x^{6} + 24 \, b c^{2} x^{4} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{512 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (16 \, c^{3} x^{6} + 24 \, b c^{2} x^{4} - 3 \, b^{3} + 20 \, a b c + 2 \,{\left (b^{2} c + 20 \, a c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} + 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{256 \, \sqrt{-c} c^{2}}\right ] \]

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]

[1/512*(4*(16*c^3*x^6 + 24*b*c^2*x^4 - 3*b^3 + 20*a*b*c + 2*(b^2*c + 20*a*c^2)*x
^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(c) + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*log(-4*sq
rt(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^(5/2), 1/256*(2*(16*c^3*x^6 + 24*b*c^2*x^4 - 3*b^3 + 20*a*b*c + 2*(b
^2*c + 20*a*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) + 3*(b^4 - 8*a*b^2*c + 16
*a^2*c^2)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)))/(sqrt(
-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.290358, size = 182, normalized size = 1.47 \[ \frac{1}{128} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (2 \, c x^{2} + 3 \, b\right )} x^{2} + \frac{b^{2} c^{2} + 20 \, a c^{3}}{c^{3}}\right )} x^{2} - \frac{3 \, b^{3} c - 20 \, a b c^{2}}{c^{3}}\right )} - \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{5}{2}}} \]

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]

1/128*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(2*c*x^2 + 3*b)*x^2 + (b^2*c^2 + 20*a*c^3)/c
^3)*x^2 - (3*b^3*c - 20*a*b*c^2)/c^3) - 3/256*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*ln(
abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c^(5/2)