3.1080 \(\int \frac{x^{15/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=621 \[ -\frac{3 \left (\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 x^{5/2} \left (x^2 \left (12 a c+b^2\right )+8 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \sqrt{x} \left (12 a c+b^2\right )}{16 c \left (b^2-4 a c\right )^2} \]

[Out]

(-3*(b^2 + 12*a*c)*Sqrt[x])/(16*c*(b^2 - 4*a*c)^2) + (x^(9/2)*(2*a + b*x^2))/(4*
(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x^(5/2)*(8*a*b + (b^2 + 12*a*c)*x^2))/
(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(b^3 - 28*a*b*c + (b^4 - 30*a*b^2*
c - 24*a^2*c^2)/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)*c^(5/4)*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c]
)^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)/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)
*c^(5/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c +
(b^4 - 30*a*b^2*c - 24*a^2*c^2)/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)*c^(5/4)*(b^2 - 4*a*c)^2*(-b -
Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)
/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)*c^(5/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 3.75799, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{3 \left (\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{3 \left (-\frac{-24 a^2 c^2-30 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-28 a b c+b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{32 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right )^2 \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x^{9/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 x^{5/2} \left (x^2 \left (12 a c+b^2\right )+8 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \sqrt{x} \left (12 a c+b^2\right )}{16 c \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(15/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(-3*(b^2 + 12*a*c)*Sqrt[x])/(16*c*(b^2 - 4*a*c)^2) + (x^(9/2)*(2*a + b*x^2))/(4*
(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x^(5/2)*(8*a*b + (b^2 + 12*a*c)*x^2))/
(16*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (3*(b^3 - 28*a*b*c + (b^4 - 30*a*b^2*
c - 24*a^2*c^2)/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)*c^(5/4)*(b^2 - 4*a*c)^2*(-b - Sqrt[b^2 - 4*a*c]
)^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)/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)
*c^(5/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c +
(b^4 - 30*a*b^2*c - 24*a^2*c^2)/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)*c^(5/4)*(b^2 - 4*a*c)^2*(-b -
Sqrt[b^2 - 4*a*c])^(3/4)) - (3*(b^3 - 28*a*b*c - (b^4 - 30*a*b^2*c - 24*a^2*c^2)
/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)*c^(5/4)*(b^2 - 4*a*c)^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**(15/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.67206, size = 254, normalized size = 0.41 \[ \frac{3 c \left (a+b x^2+c x^4\right )^2 \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-28 \text{$\#$1}^4 a b c \log \left (\sqrt{x}-\text{$\#$1}\right )+\text{$\#$1}^4 b^3 \log \left (\sqrt{x}-\text{$\#$1}\right )+12 a^2 c \log \left (\sqrt{x}-\text{$\#$1}\right )+a b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]+16 \sqrt{x} \left (b^2-4 a c\right ) \left (-2 a^2 c+a b \left (b-3 c x^2\right )+b^3 x^2\right )+4 \sqrt{x} \left (-68 a^2 c^2+21 a b^2 c-28 a b c^2 x^2-4 b^4+b^3 c x^2\right ) \left (a+b x^2+c x^4\right )}{64 c^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(15/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(4*Sqrt[x]*(-4*b^4 + 21*a*b^2*c - 68*a^2*c^2 + b^3*c*x^2 - 28*a*b*c^2*x^2)*(a +
b*x^2 + c*x^4) + 16*(b^2 - 4*a*c)*Sqrt[x]*(-2*a^2*c + b^3*x^2 + a*b*(b - 3*c*x^2
)) + 3*c*(a + b*x^2 + c*x^4)^2*RootSum[a + b*#1^4 + c*#1^8 & , (a*b^2*Log[Sqrt[x
] - #1] + 12*a^2*c*Log[Sqrt[x] - #1] + b^3*Log[Sqrt[x] - #1]*#1^4 - 28*a*b*c*Log
[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(64*c^2*(b^2 - 4*a*c)^2*(a + b*x^2
+ c*x^4)^2)

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Maple [C]  time = 0.078, size = 275, normalized size = 0.4 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{3\,{a}^{2} \left ( 12\,ac+{b}^{2} \right ) \sqrt{x}}{ \left ( 512\,{a}^{2}{c}^{2}-256\,a{b}^{2}c+32\,{b}^{4} \right ) c}}-3/16\,{\frac{ab \left ( 8\,ac+{b}^{2} \right ){x}^{5/2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}-1/32\,{\frac{ \left ( 68\,{a}^{2}{c}^{2}+7\,a{b}^{2}c+3\,{b}^{4} \right ){x}^{9/2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}-1/32\,{\frac{b \left ( 28\,ac-{b}^{2} \right ){x}^{13/2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}} \right ) }+{\frac{3}{64\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{b \left ( -28\,ac+{b}^{2} \right ){{\it \_R}}^{4}+12\,{a}^{2}c+a{b}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \left ( 2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b \right ) }\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)/(c*x^4+b*x^2+a)^3,x)

[Out]

2*(-3/32*a^2*(12*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)/c*x^(1/2)-3/16*a/c*b*(8*a*c
+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)-1/32*(68*a^2*c^2+7*a*b^2*c+3*b^4)/(16*a
^2*c^2-8*a*b^2*c+b^4)/c*x^(9/2)-1/32*b*(28*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^(13/2))/(c*x^4+b*x^2+a)^2+3/64/c*sum((b*(-28*a*c+b^2)*_R^4+12*a^2*c+a*b^2)/(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. \[ \frac{3 \,{\left (b^{2} c + 12 \, a c^{2}\right )} x^{\frac{17}{2}} +{\left (7 \, b^{3} + 44 \, a b c\right )} x^{\frac{13}{2}} + 24 \, a^{2} b x^{\frac{5}{2}} +{\left (35 \, a b^{2} + 4 \, a^{2} c\right )} x^{\frac{9}{2}}}{16 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )}} - \int \frac{3 \,{\left ({\left (b^{2} + 12 \, a c\right )} x^{\frac{7}{2}} + 40 \, a b x^{\frac{3}{2}}\right )}}{32 \,{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/16*(3*(b^2*c + 12*a*c^2)*x^(17/2) + (7*b^3 + 44*a*b*c)*x^(13/2) + 24*a^2*b*x^(
5/2) + (35*a*b^2 + 4*a^2*c)*x^(9/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 +
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2
+ (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x
^2) - integrate(3/32*((b^2 + 12*a*c)*x^(7/2) + 40*a*b*x^(3/2))/(a*b^4 - 8*a^2*b^
2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 1
6*a^2*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^(15/2)/(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**(15/2)/(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. \[ \int \frac{x^{\frac{15}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

integrate(x^(15/2)/(c*x^4 + b*x^2 + a)^3, x)