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

Optimal. Leaf size=223 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 x^3}{32 a^2 \left (a+c x^4\right )}+\frac{x^3}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^3/(8*a*(a + c*x^4)^2) + (5*x^3)/(32*a^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*
c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(9/4)*c^(3/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/
4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(9/4)*c^(3/4)) + (5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(9/4)*c^(3/4)) - (5*Log[Sqrt[a] + Sqrt[
2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(9/4)*c^(3/4))

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Rubi [A]  time = 0.286353, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{9/4} c^{3/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{9/4} c^{3/4}}+\frac{5 x^3}{32 a^2 \left (a+c x^4\right )}+\frac{x^3}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + c*x^4)^3,x]

[Out]

x^3/(8*a*(a + c*x^4)^2) + (5*x^3)/(32*a^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*
c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(9/4)*c^(3/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/
4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(9/4)*c^(3/4)) + (5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(9/4)*c^(3/4)) - (5*Log[Sqrt[a] + Sqrt[
2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(9/4)*c^(3/4))

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Rubi in Sympy [A]  time = 60.191, size = 211, normalized size = 0.95 \[ \frac{x^{3}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{5 x^{3}}{32 a^{2} \left (a + c x^{4}\right )} + \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{9}{4}} c^{\frac{3}{4}}} - \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{9}{4}} c^{\frac{3}{4}}} - \frac{5 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{9}{4}} c^{\frac{3}{4}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{9}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/(8*a*(a + c*x**4)**2) + 5*x**3/(32*a**2*(a + c*x**4)) + 5*sqrt(2)*log(-sqrt
(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(9/4)*c**(3/4)) - 5*sq
rt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(9/4)*c*
*(3/4)) - 5*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(9/4)*c**(3/4)
) + 5*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(9/4)*c**(3/4))

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Mathematica [A]  time = 0.183884, size = 204, normalized size = 0.91 \[ \frac{\frac{32 a^{5/4} x^3}{\left (a+c x^4\right )^2}+\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{40 \sqrt [4]{a} x^3}{a+c x^4}}{256 a^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + c*x^4)^3,x]

[Out]

((32*a^(5/4)*x^3)/(a + c*x^4)^2 + (40*a^(1/4)*x^3)/(a + c*x^4) - (10*Sqrt[2]*Arc
Tan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (10*Sqrt[2]*ArcTan[1 + (Sqrt[2]*
c^(1/4)*x)/a^(1/4)])/c^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*
x + Sqrt[c]*x^2])/c^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/c^(3/4))/(256*a^(9/4))

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Maple [A]  time = 0.007, size = 171, normalized size = 0.8 \[{\frac{{x}^{3}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{5\,{x}^{3}}{32\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{5\,\sqrt{2}}{256\,{a}^{2}c}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,\sqrt{2}}{128\,{a}^{2}c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^4+a)^3,x)

[Out]

1/8*x^3/a/(c*x^4+a)^2+5/32*x^3/a^2/(c*x^4+a)+5/256/a^2/c/(a/c)^(1/4)*2^(1/2)*ln(
(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))
+5/128/a^2/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+5/128/a^2/c/(a/
c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)

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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(x^2/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250395, size = 311, normalized size = 1.39 \[ \frac{20 \, c x^{7} + 36 \, a x^{3} + 20 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{7} c^{2} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{3}{4}}}{x + \sqrt{-a^{5} c \sqrt{-\frac{1}{a^{9} c^{3}}} + x^{2}}}\right ) + 5 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \log \left (a^{7} c^{2} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{3}{4}} + x\right ) - 5 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{1}{4}} \log \left (-a^{7} c^{2} \left (-\frac{1}{a^{9} c^{3}}\right )^{\frac{3}{4}} + x\right )}{128 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(20*c*x^7 + 36*a*x^3 + 20*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^9*c^3))
^(1/4)*arctan(a^7*c^2*(-1/(a^9*c^3))^(3/4)/(x + sqrt(-a^5*c*sqrt(-1/(a^9*c^3)) +
x^2))) + 5*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^9*c^3))^(1/4)*log(a^7*c^2*(
-1/(a^9*c^3))^(3/4) + x) - 5*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*(-1/(a^9*c^3))^(1
/4)*log(-a^7*c^2*(-1/(a^9*c^3))^(3/4) + x))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)

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Sympy [A]  time = 4.92303, size = 71, normalized size = 0.32 \[ \frac{9 a x^{3} + 5 c x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{9} c^{3} + 625, \left ( t \mapsto t \log{\left (\frac{2097152 t^{3} a^{7} c^{2}}{125} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**4+a)**3,x)

[Out]

(9*a*x**3 + 5*c*x**7)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8) + RootSum(2
68435456*_t**4*a**9*c**3 + 625, Lambda(_t, _t*log(2097152*_t**3*a**7*c**2/125 +
x)))

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GIAC/XCAS [A]  time = 0.226927, size = 278, normalized size = 1.25 \[ \frac{5 \, c x^{7} + 9 \, a x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} a^{2}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} c^{3}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(5*c*x^7 + 9*a*x^3)/((c*x^4 + a)^2*a^2) + 5/128*sqrt(2)*(a*c^3)^(3/4)*arcta
n(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 5/128*sqrt(2)
*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*
c^3) - 5/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(
a^3*c^3) + 5/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c
))/(a^3*c^3)