3.1086 \(\int \frac{x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=594 \[ -\frac{\sqrt{x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\sqrt{x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 c^{3/4} \left (\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

[Out]

-(Sqrt[x]*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(b*(
b^2 + 20*a*c) + c*(b^2 + 44*a*c)*x^2))/(16*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)
) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*a*b*c)/Sqrt[b^2 - 4*a
*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/
4)*a*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*
c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*a*c))*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqr
t[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c
])^(3/4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*a*b*c)/Sqrt[b
^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/
(32*2^(1/4)*a*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^3
- 68*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])
/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b
^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 4.68936, antiderivative size = 594, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt{x} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\sqrt{x} \left (c x^2 \left (44 a c+b^2\right )+b \left (20 a c+b^2\right )\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 c^{3/4} \left (\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\frac{68 a b c}{\sqrt{b^2-4 a c}}-\frac{b^3}{\sqrt{b^2-4 a c}}+44 a c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 c^{3/4} \left (\sqrt{b^2-4 a c} \left (44 a c+b^2\right )-68 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} a \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (Sqrt[x]*(b*(
b^2 + 20*a*c) + c*(b^2 + 44*a*c)*x^2))/(16*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)
) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*a*b*c)/Sqrt[b^2 - 4*a
*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/
4)*a*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^3 - 68*a*b*
c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*a*c))*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqr
t[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c
])^(3/4)) - (3*c^(3/4)*(b^2 + 44*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*a*b*c)/Sqrt[b
^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/
(32*2^(1/4)*a*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - (3*c^(3/4)*(b^3
- 68*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 + 44*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])
/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(1/4)*a*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b
^2 - 4*a*c])^(3/4))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.612484, size = 224, normalized size = 0.38 \[ \frac{3 \left (a+b x^2+c x^4\right )^2 \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{44 \text{$\#$1}^4 a c^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+\text{$\#$1}^4 b^2 c \log \left (\sqrt{x}-\text{$\#$1}\right )-12 a b c \log \left (\sqrt{x}-\text{$\#$1}\right )+b^3 \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]-16 a \sqrt{x} \left (b^2-4 a c\right ) \left (b+2 c x^2\right )+4 \sqrt{x} \left (20 a b c+44 a c^2 x^2+b^3+b^2 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-16*a*(b^2 - 4*a*c)*Sqrt[x]*(b + 2*c*x^2) + 4*Sqrt[x]*(b^3 + 20*a*b*c + b^2*c*x
^2 + 44*a*c^2*x^2)*(a + b*x^2 + c*x^4) + 3*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#
1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 12*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log
[Sqrt[x] - #1]*#1^4 + 44*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/
(64*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2)

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Maple [C]  time = 0.048, size = 270, normalized size = 0.5 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,b \left ( 12\,ac-{b}^{2} \right ) \sqrt{x}}{512\,{a}^{2}{c}^{2}-256\,a{b}^{2}c+32\,{b}^{4}}}+1/32\,{\frac{ \left ( 76\,{a}^{2}{c}^{2}+13\,a{b}^{2}c+{b}^{4} \right ){x}^{5/2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) a}}+1/16\,{\frac{bc \left ( 32\,ac+{b}^{2} \right ){x}^{9/2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) a}}+1/32\,{\frac{{c}^{2} \left ( 44\,ac+{b}^{2} \right ){x}^{13/2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) a}} \right ) }+{\frac{3}{64\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{c \left ( 44\,ac+{b}^{2} \right ){{\it \_R}}^{4}-12\,abc+{b}^{3}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \left ( 2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b \right ) }\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3/32*b*(12*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)+1/32*(76*a^2*c^2+13*a*
b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^(5/2)+1/16/a*c*b*(32*a*c+b^2)/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^(9/2)+1/32*c^2*(44*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^
(13/2))/(c*x^4+b*x^2+a)^2+3/64/a*sum((c*(44*a*c+b^2)*_R^4-12*a*b*c+b^3)/(16*a^2*
c^2-8*a*b^2*c+b^4)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (b^{3} c^{2} - 12 \, a b c^{3}\right )} x^{\frac{17}{2}} +{\left (6 \, b^{4} c - 71 \, a b^{2} c^{2} + 44 \, a^{2} c^{3}\right )} x^{\frac{13}{2}} +{\left (3 \, b^{5} - 28 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{\frac{9}{2}} +{\left (7 \, a b^{4} - 59 \, a^{2} b^{2} c + 76 \, a^{3} c^{2}\right )} x^{\frac{5}{2}}}{16 \,{\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \,{\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} +{\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} + \int -\frac{3 \,{\left ({\left (b^{3} c - 12 \, a b c^{2}\right )} x^{\frac{7}{2}} +{\left (b^{4} - 13 \, a b^{2} c - 44 \, a^{2} c^{2}\right )} x^{\frac{3}{2}}\right )}}{32 \,{\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} +{\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} +{\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(3*(b^3*c^2 - 12*a*b*c^3)*x^(17/2) + (6*b^4*c - 71*a*b^2*c^2 + 44*a^2*c^3)*
x^(13/2) + (3*b^5 - 28*a*b^3*c - 8*a^2*b*c^2)*x^(9/2) + (7*a*b^4 - 59*a^2*b^2*c
+ 76*a^3*c^2)*x^(5/2))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8 + a^4*b^4
- 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^6 +
(a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*
c^2)*x^2) + integrate(-3/32*((b^3*c - 12*a*b*c^2)*x^(7/2) + (b^4 - 13*a*b^2*c -
44*a^2*c^2)*x^(3/2))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^
2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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