3.1071 \(\int \frac{x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=544 \[ -\frac{b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac{x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left (-\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\left (\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left (-\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}} \]

[Out]

-(b*x^(3/2))/(2*c*(b^2 - 4*a*c)) + (x^(7/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)) + ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*c)*Sqrt[b^2 - 4*a*c])*ArcT
an[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(4*2^(3/4)*c^(7/4)
*(b^2 - 4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ((3*b^3 - 20*a*b*c - (3*b
^2 - 14*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c]
)^(1/4)) - ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 -
4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + ((3*b^3 - 20*a*b*c - (3*b^2 - 14
*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4
))

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Rubi [A]  time = 4.76164, antiderivative size = 544, 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 \[ -\frac{b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac{x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\left (-\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{\left (\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\left (-\left (3 b^2-14 a c\right ) \sqrt{b^2-4 a c}-20 a b c+3 b^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt{b^2-4 a c}-b}} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

-(b*x^(3/2))/(2*c*(b^2 - 4*a*c)) + (x^(7/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)) + ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*c)*Sqrt[b^2 - 4*a*c])*ArcT
an[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(4*2^(3/4)*c^(7/4)
*(b^2 - 4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ((3*b^3 - 20*a*b*c - (3*b
^2 - 14*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c]
)^(1/4)) - ((3*b^3 - 20*a*b*c + (3*b^2 - 14*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 -
4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + ((3*b^3 - 20*a*b*c - (3*b^2 - 14
*a*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)])/(4*2^(3/4)*c^(7/4)*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4
))

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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**(13/2)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.325075, size = 144, normalized size = 0.26 \[ \frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{-14 \text{$\#$1}^4 a c \log \left (\sqrt{x}-\text{$\#$1}\right )+3 \text{$\#$1}^4 b^2 \log \left (\sqrt{x}-\text{$\#$1}\right )+3 a b \log \left (\sqrt{x}-\text{$\#$1}\right )}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right ]-\frac{4 x^{3/2} \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{a+b x^2+c x^4}}{8 c \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

((-4*x^(3/2)*(b^2*x^2 + a*(b - 2*c*x^2)))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1
^4 + c*#1^8 & , (3*a*b*Log[Sqrt[x] - #1] + 3*b^2*Log[Sqrt[x] - #1]*#1^4 - 14*a*c
*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(8*c*(b^2 - 4*a*c))

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Maple [C]  time = 0.078, size = 149, normalized size = 0.3 \[ 2\,{\frac{1}{c{x}^{4}+b{x}^{2}+a} \left ( -1/4\,{\frac{ \left ( 2\,ac-{b}^{2} \right ){x}^{7/2}}{ \left ( 4\,ac-{b}^{2} \right ) c}}+1/4\,{\frac{ab{x}^{3/2}}{ \left ( 4\,ac-{b}^{2} \right ) c}} \right ) }+{\frac{1}{8\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( 14\,ac-3\,{b}^{2} \right ){{\it \_R}}^{6}-3\,{{\it \_R}}^{2}ab}{ \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b \right ) }\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(c*x^4+b*x^2+a)^2,x)

[Out]

2*(-1/4*(2*a*c-b^2)/c/(4*a*c-b^2)*x^(7/2)+1/4*a*b/c/(4*a*c-b^2)*x^(3/2))/(c*x^4+
b*x^2+a)+1/8/c*sum(((14*a*c-3*b^2)*_R^6-3*_R^2*a*b)/(4*a*c-b^2)/(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. \[ -\frac{{\left (b^{2} - 2 \, a c\right )} x^{\frac{7}{2}} + a b x^{\frac{3}{2}}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}} + \int \frac{{\left (3 \, b^{2} - 14 \, a c\right )} x^{\frac{5}{2}} + 3 \, a b \sqrt{x}}{4 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

-1/2*((b^2 - 2*a*c)*x^(7/2) + a*b*x^(3/2))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c -
4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2) + integrate(1/4*((3*b^2 - 14*a*c)*x^(5/2) +
3*a*b*sqrt(x))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*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^(13/2)/(c*x^4 + b*x^2 + a)^2,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**(13/2)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{13}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

integrate(x^(13/2)/(c*x^4 + b*x^2 + a)^2, x)