3.678 \(\int \frac{x^{10}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=223 \[ \frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac{x^7}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^7/(8*c*(a + c*x^4)^2) - (7*x^3)/(32*c^2*(a + c*x^4)) - (21*ArcTan[1 - (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(1/4)*c^(11/4)) + (21*ArcTan[1 + (Sqrt[2]*c
^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(1/4)*c^(11/4)) + (21*Log[Sqrt[a] - Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(1/4)*c^(11/4)) - (21*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(1/4)*c^(11/4))

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Rubi [A]  time = 0.303145, 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 \[ \frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac{x^7}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a + c*x^4)^3,x]

[Out]

-x^7/(8*c*(a + c*x^4)^2) - (7*x^3)/(32*c^2*(a + c*x^4)) - (21*ArcTan[1 - (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(1/4)*c^(11/4)) + (21*ArcTan[1 + (Sqrt[2]*c
^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(1/4)*c^(11/4)) + (21*Log[Sqrt[a] - Sqrt[2]*a^
(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(1/4)*c^(11/4)) - (21*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(1/4)*c^(11/4))

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Rubi in Sympy [A]  time = 60.6886, size = 211, normalized size = 0.95 \[ - \frac{x^{7}}{8 c \left (a + c x^{4}\right )^{2}} - \frac{7 x^{3}}{32 c^{2} \left (a + c x^{4}\right )} + \frac{21 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 \sqrt [4]{a} c^{\frac{11}{4}}} - \frac{21 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 \sqrt [4]{a} c^{\frac{11}{4}}} - \frac{21 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 \sqrt [4]{a} c^{\frac{11}{4}}} + \frac{21 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 \sqrt [4]{a} c^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(c*x**4+a)**3,x)

[Out]

-x**7/(8*c*(a + c*x**4)**2) - 7*x**3/(32*c**2*(a + c*x**4)) + 21*sqrt(2)*log(-sq
rt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(1/4)*c**(11/4)) - 2
1*sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(1/4
)*c**(11/4)) - 21*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(1/4)*c*
*(11/4)) + 21*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(1/4)*c**(11
/4))

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Mathematica [A]  time = 0.201928, size = 205, normalized size = 0.92 \[ \frac{-\frac{88 c^{3/4} x^3}{a+c x^4}+\frac{32 a c^{3/4} x^3}{\left (a+c x^4\right )^2}+\frac{21 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{21 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{42 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{42 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}}{256 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a + c*x^4)^3,x]

[Out]

((32*a*c^(3/4)*x^3)/(a + c*x^4)^2 - (88*c^(3/4)*x^3)/(a + c*x^4) - (42*Sqrt[2]*A
rcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(1/4) + (42*Sqrt[2]*ArcTan[1 + (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/a^(1/4) + (21*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/
4)*x + Sqrt[c]*x^2])/a^(1/4) - (21*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/a^(1/4))/(256*c^(11/4))

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Maple [A]  time = 0.017, size = 156, normalized size = 0.7 \[{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{11\,{x}^{7}}{32\,c}}-{\frac{7\,a{x}^{3}}{32\,{c}^{2}}} \right ) }+{\frac{21\,\sqrt{2}}{256\,{c}^{3}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,\sqrt{2}}{128\,{c}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,\sqrt{2}}{128\,{c}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(c*x^4+a)^3,x)

[Out]

(-11/32/c*x^7-7/32*a/c^2*x^3)/(c*x^4+a)^2+21/256/c^3/(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)))+21/
128/c^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+21/128/c^3/(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^10/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240513, size = 308, normalized size = 1.38 \[ -\frac{44 \, c x^{7} + 28 \, a x^{3} - 84 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c^{8} \left (-\frac{1}{a c^{11}}\right )^{\frac{3}{4}}}{x + \sqrt{-a c^{5} \sqrt{-\frac{1}{a c^{11}}} + x^{2}}}\right ) - 21 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \log \left (a c^{8} \left (-\frac{1}{a c^{11}}\right )^{\frac{3}{4}} + x\right ) + 21 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \log \left (-a c^{8} \left (-\frac{1}{a c^{11}}\right )^{\frac{3}{4}} + x\right )}{128 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

-1/128*(44*c*x^7 + 28*a*x^3 - 84*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a*c^11))
^(1/4)*arctan(a*c^8*(-1/(a*c^11))^(3/4)/(x + sqrt(-a*c^5*sqrt(-1/(a*c^11)) + x^2
))) - 21*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a*c^11))^(1/4)*log(a*c^8*(-1/(a*
c^11))^(3/4) + x) + 21*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a*c^11))^(1/4)*log
(-a*c^8*(-1/(a*c^11))^(3/4) + x))/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)

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Sympy [A]  time = 5.14377, size = 68, normalized size = 0.3 \[ - \frac{7 a x^{3} + 11 c x^{7}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a c^{11} + 194481, \left ( t \mapsto t \log{\left (\frac{2097152 t^{3} a c^{8}}{9261} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(c*x**4+a)**3,x)

[Out]

-(7*a*x**3 + 11*c*x**7)/(32*a**2*c**2 + 64*a*c**3*x**4 + 32*c**4*x**8) + RootSum
(268435456*_t**4*a*c**11 + 194481, Lambda(_t, _t*log(2097152*_t**3*a*c**8/9261 +
x)))

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GIAC/XCAS [A]  time = 0.226937, size = 278, normalized size = 1.25 \[ -\frac{11 \, c x^{7} + 7 \, a x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} c^{2}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{5}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{5}} - \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{5}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

-1/32*(11*c*x^7 + 7*a*x^3)/((c*x^4 + a)^2*c^2) + 21/128*sqrt(2)*(a*c^3)^(3/4)*ar
ctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^5) + 21/128*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*
c^5) - 21/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/
(a*c^5) + 21/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c
))/(a*c^5)