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

Optimal. Leaf size=150 \[ -\frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}-\frac{b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

[Out]

(3*b*(b^2 - 4*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) - (b*(b + 2*
c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*c^2) + (a + b*x^2 + c*x^4)^(5/2)/(10*c) -
(3*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])
/(512*c^(7/2))

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Rubi [A]  time = 0.259659, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{3 b \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}-\frac{b \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*b*(b^2 - 4*a*c)*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*c^3) - (b*(b + 2*
c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*c^2) + (a + b*x^2 + c*x^4)^(5/2)/(10*c) -
(3*b*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])
/(512*c^(7/2))

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Rubi in Sympy [A]  time = 21.5849, size = 141, normalized size = 0.94 \[ - \frac{b \left (b + 2 c x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 c^{2}} + \frac{3 b \left (b + 2 c x^{2}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{256 c^{3}} - \frac{3 b \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 c^{\frac{7}{2}}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(b + 2*c*x**2)*(a + b*x**2 + c*x**4)**(3/2)/(32*c**2) + 3*b*(b + 2*c*x**2)*(-
4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)/(256*c**3) - 3*b*(-4*a*c + b**2)**2*atan
h((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(512*c**(7/2)) + (a + b*
x**2 + c*x**4)**(5/2)/(10*c)

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Mathematica [A]  time = 0.14079, size = 142, normalized size = 0.95 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left (4 b^2 c \left (2 c x^4-25 a\right )+8 b c^2 x^2 \left (7 a+22 c x^4\right )+128 c^2 \left (a+c x^4\right )^2+15 b^4-10 b^3 c x^2\right )-15 b \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{2560 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(15*b^4 - 10*b^3*c*x^2 + 128*c^2*(a + c*x^4)^
2 + 4*b^2*c*(-25*a + 2*c*x^4) + 8*b*c^2*x^2*(7*a + 22*c*x^4)) - 15*b*(b^2 - 4*a*
c)^2*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(2560*c^(7/2))

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Maple [B]  time = 0.025, size = 316, normalized size = 2.1 \[{\frac{{a}^{2}}{10\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{a}^{2}b}{32}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{c{x}^{8}}{10}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{11\,b{x}^{6}}{80}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}{x}^{4}}{160\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{3}{x}^{2}}{128\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{4}}{256\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{5}}{512}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,a{b}^{3}}{64}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,a{b}^{2}}{64\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,ab{x}^{2}}{160\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{a{x}^{4}}{5}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*a^2/c*(c*x^4+b*x^2+a)^(1/2)-3/32*a^2*b/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*
x^4+b*x^2+a)^(1/2))+1/10*c*x^8*(c*x^4+b*x^2+a)^(1/2)+11/80*b*x^6*(c*x^4+b*x^2+a)
^(1/2)+1/160/c*b^2*x^4*(c*x^4+b*x^2+a)^(1/2)-1/128/c^2*b^3*x^2*(c*x^4+b*x^2+a)^(
1/2)+3/256/c^3*b^4*(c*x^4+b*x^2+a)^(1/2)-3/512/c^(7/2)*b^5*ln((1/2*b+c*x^2)/c^(1
/2)+(c*x^4+b*x^2+a)^(1/2))+3/64/c^(5/2)*b^3*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*
x^2+a)^(1/2))-5/64/c^2*b^2*a*(c*x^4+b*x^2+a)^(1/2)+7/160/c*b*a*x^2*(c*x^4+b*x^2+
a)^(1/2)+1/5*a*x^4*(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304455, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (128 \, c^{4} x^{8} + 176 \, b c^{3} x^{6} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{4} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \log \left (4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{5120 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (128 \, c^{4} x^{8} + 176 \, b c^{3} x^{6} + 8 \,{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} x^{4} + 15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2} - 2 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} - 15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{2560 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/5120*(4*(128*c^4*x^8 + 176*b*c^3*x^6 + 8*(b^2*c^2 + 32*a*c^3)*x^4 + 15*b^4 -
100*a*b^2*c + 128*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a
)*sqrt(c) + 15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*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^(7/2), 1/256
0*(2*(128*c^4*x^8 + 176*b*c^3*x^6 + 8*(b^2*c^2 + 32*a*c^3)*x^4 + 15*b^4 - 100*a*
b^2*c + 128*a^2*c^2 - 2*(5*b^3*c - 28*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt
(-c) - 15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sq
rt(c*x^4 + b*x^2 + a)*c)))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288637, size = 232, normalized size = 1.55 \[ \frac{1}{1280} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x^{2} + 11 \, b\right )} x^{2} + \frac{b^{2} c^{3} + 32 \, a c^{4}}{c^{4}}\right )} x^{2} - \frac{5 \, b^{3} c^{2} - 28 \, a b c^{3}}{c^{4}}\right )} x^{2} + \frac{15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}}{c^{4}}\right )} + \frac{3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{512 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*sqrt(c*x^4 + b*x^2 + a)*(2*(4*(2*(8*c*x^2 + 11*b)*x^2 + (b^2*c^3 + 32*a*c
^4)/c^4)*x^2 - (5*b^3*c^2 - 28*a*b*c^3)/c^4)*x^2 + (15*b^4*c - 100*a*b^2*c^2 + 1
28*a^2*c^3)/c^4) + 3/512*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*ln(abs(-2*(sqrt(c)*x^2
- sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c^(7/2)