3.237 \(\int \frac{x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=359 \[ -\frac{a^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{a^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}-\frac{a^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{a^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} \left (a e^2+c d^2\right )}-\frac{d x}{c e^2}+\frac{x^3}{3 c e} \]

[Out]

-((d*x)/(c*e^2)) + x^3/(3*c*e) + (d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(5/2)*
(c*d^2 + a*e^2)) - (a^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) + (a^(5/4)*(Sqrt[c]*d - Sqrt[a]
*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2))
- (a^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) + (a^(5/4)*(Sqrt[c]*d + Sqrt[a]*
e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c
*d^2 + a*e^2))

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Rubi [A]  time = 0.653974, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{a^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{a^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}-\frac{a^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{a^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} c^{7/4} \left (a e^2+c d^2\right )}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} \left (a e^2+c d^2\right )}-\frac{d x}{c e^2}+\frac{x^3}{3 c e} \]

Antiderivative was successfully verified.

[In]  Int[x^8/((d + e*x^2)*(a + c*x^4)),x]

[Out]

-((d*x)/(c*e^2)) + x^3/(3*c*e) + (d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(5/2)*
(c*d^2 + a*e^2)) - (a^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) + (a^(5/4)*(Sqrt[c]*d - Sqrt[a]
*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2))
- (a^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) + (a^(5/4)*(Sqrt[c]*d + Sqrt[a]*
e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c
*d^2 + a*e^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{2} a^{\frac{5}{4}} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 c^{\frac{7}{4}} \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{2} a^{\frac{5}{4}} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 c^{\frac{7}{4}} \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{2} a^{\frac{5}{4}} \left (\sqrt{a} e + \sqrt{c} d\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 c^{\frac{7}{4}} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{2} a^{\frac{5}{4}} \left (\sqrt{a} e + \sqrt{c} d\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 c^{\frac{7}{4}} \left (a e^{2} + c d^{2}\right )} + \frac{d^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{e^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )} + \frac{x^{3}}{3 c e} - \frac{\int d\, dx}{c e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(e*x**2+d)/(c*x**4+a),x)

[Out]

sqrt(2)*a**(5/4)*(sqrt(a)*e - sqrt(c)*d)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(
4*c**(7/4)*(a*e**2 + c*d**2)) - sqrt(2)*a**(5/4)*(sqrt(a)*e - sqrt(c)*d)*atan(1
+ sqrt(2)*c**(1/4)*x/a**(1/4))/(4*c**(7/4)*(a*e**2 + c*d**2)) - sqrt(2)*a**(5/4)
*(sqrt(a)*e + sqrt(c)*d)*log(-sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*
x**2)/(8*c**(7/4)*(a*e**2 + c*d**2)) + sqrt(2)*a**(5/4)*(sqrt(a)*e + sqrt(c)*d)*
log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2)/(8*c**(7/4)*(a*e**2
+ c*d**2)) + d**(7/2)*atan(sqrt(e)*x/sqrt(d))/(e**(5/2)*(a*e**2 + c*d**2)) + x**
3/(3*c*e) - Integral(d, x)/(c*e**2)

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Mathematica [A]  time = 0.864583, size = 344, normalized size = 0.96 \[ \frac{-3 \sqrt{2} a e^{5/2} \left (a^{3/4} e+\sqrt [4]{a} \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} a e^{5/2} \left (a^{3/4} e+\sqrt [4]{a} \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+6 \sqrt{2} a^{5/4} e^{5/2} \left (\sqrt{a} e-\sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-6 \sqrt{2} a^{5/4} e^{5/2} \left (\sqrt{a} e-\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-24 c^{3/4} d \sqrt{e} x \left (a e^2+c d^2\right )+8 c^{3/4} e^{3/2} x^3 \left (a e^2+c d^2\right )+24 c^{7/4} d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{24 c^{7/4} e^{5/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((d + e*x^2)*(a + c*x^4)),x]

[Out]

(-24*c^(3/4)*d*Sqrt[e]*(c*d^2 + a*e^2)*x + 8*c^(3/4)*e^(3/2)*(c*d^2 + a*e^2)*x^3
+ 24*c^(7/4)*d^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 6*Sqrt[2]*a^(5/4)*e^(5/2)*(-
(Sqrt[c]*d) + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 6*Sqrt[2]*a^(
5/4)*e^(5/2)*(-(Sqrt[c]*d) + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]
- 3*Sqrt[2]*a*e^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*c^(1/4)*x + Sqrt[c]*x^2] + 3*Sqrt[2]*a*e^(5/2)*(a^(1/4)*Sqrt[c]*d + a^(3/4)*
e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(24*c^(7/4)*e^(5/2)*(
c*d^2 + a*e^2))

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Maple [A]  time = 0.014, size = 405, normalized size = 1.1 \[{\frac{{x}^{3}}{3\,ce}}-{\frac{dx}{{e}^{2}c}}+{\frac{ad\sqrt{2}}{ \left ( 8\,a{e}^{2}+8\,c{d}^{2} \right ) c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{ad\sqrt{2}}{ \left ( 4\,a{e}^{2}+4\,c{d}^{2} \right ) c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{ad\sqrt{2}}{ \left ( 4\,a{e}^{2}+4\,c{d}^{2} \right ) c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{{a}^{2}e\sqrt{2}}{ \left ( 8\,a{e}^{2}+8\,c{d}^{2} \right ){c}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{{a}^{2}e\sqrt{2}}{ \left ( 4\,a{e}^{2}+4\,c{d}^{2} \right ){c}^{2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{{a}^{2}e\sqrt{2}}{ \left ( 4\,a{e}^{2}+4\,c{d}^{2} \right ){c}^{2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{d}^{4}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(e*x^2+d)/(c*x^4+a),x)

[Out]

1/3*x^3/c/e-d*x/e^2/c+1/8*a/(a*e^2+c*d^2)/c*d*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c
*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+
1/4*a/(a*e^2+c*d^2)/c*d*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+
1/4*a/(a*e^2+c*d^2)/c*d*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)-
1/8*a^2/(a*e^2+c*d^2)/c^2*e/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2
)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))-1/4*a^2/(a*e^2+c*d
^2)/c^2*e/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)-1/4*a^2/(a*e^2
+c*d^2)/c^2*e/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)+1/e^2*d^4/
(a*e^2+c*d^2)/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))

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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^8/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.75617, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

[1/12*(6*c*d^3*sqrt(-d/e)*log((e*x^2 + 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + 4*(c
*d^2*e + a*e^3)*x^3 - 3*(c^2*d^2*e^2 + a*c*e^4)*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a
*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^1
1*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8))
)/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x + (a^2
*c^3*d^3 - a^3*c^2*d*e^2 + (c^7*d^4*e + 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^
5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*
d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a*c^
4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d
^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(
c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))) + 3*(c^2*d^2*e^2 + a*c*e^4)*sqrt((2*a
^3*d*e + (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*
d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*
d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))*log(-(a^3*c*
d^2 - a^4*e^2)*x - (a^2*c^3*d^3 - a^3*c^2*d*e^2 + (c^7*d^4*e + 2*a*c^6*d^2*e^3 +
a^2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^
10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))*sqrt((2*a^3*
d*e + (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2
*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2
*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))) - 3*(c^2*d^2*e
^2 + a*c*e^4)*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-
(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c
^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2
*c^3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x + (a^2*c^3*d^3 - a^3*c^2*d*e^2 - (c^7*d^
4*e + 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*
e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*
c^7*e^8)))*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^
5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*
d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^
3*e^4))) + 3*(c^2*d^2*e^2 + a*c*e^4)*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^
2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a
*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4
+ 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x - (a^2*c^3*d^3 -
a^3*c^2*d*e^2 - (c^7*d^4*e + 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 -
2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4
*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 +
a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^
10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2
*a*c^4*d^2*e^2 + a^2*c^3*e^4))) - 12*(c*d^3 + a*d*e^2)*x)/(c^2*d^2*e^2 + a*c*e^4
), 1/12*(12*c*d^3*sqrt(d/e)*arctan(x/sqrt(d/e)) + 4*(c*d^2*e + a*e^3)*x^3 - 3*(c
^2*d^2*e^2 + a*c*e^4)*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4
)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 +
6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e
^2 + a^2*c^3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x + (a^2*c^3*d^3 - a^3*c^2*d*e^2 +
(c^7*d^4*e + 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^
2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^
6 + a^4*c^7*e^8)))*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*s
qrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*
a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2
+ a^2*c^3*e^4))) + 3*(c^2*d^2*e^2 + a*c*e^4)*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a*c^
4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d
^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(
c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x - (a^2*c^
3*d^3 - a^3*c^2*d*e^2 + (c^7*d^4*e + 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^5*c
^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4
*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))*sqrt((2*a^3*d*e + (c^5*d^4 + 2*a*c^4*d
^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8
+ 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5
*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))) - 3*(c^2*d^2*e^2 + a*c*e^4)*sqrt((2*a^3*
d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2
*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2
*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))*log(-(a^3*c*d^2
- a^4*e^2)*x + (a^2*c^3*d^3 - a^3*c^2*d*e^2 - (c^7*d^4*e + 2*a*c^6*d^2*e^3 + a^
2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*
d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))*sqrt((2*a^3*d*e
- (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^
2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^
6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4))) + 3*(c^2*d^2*e^2
+ a*c*e^4)*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^
5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*
d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^
3*e^4))*log(-(a^3*c*d^2 - a^4*e^2)*x - (a^2*c^3*d^3 - a^3*c^2*d*e^2 - (c^7*d^4*e
+ 2*a*c^6*d^2*e^3 + a^2*c^5*e^5)*sqrt(-(a^5*c^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4
)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7
*e^8)))*sqrt((2*a^3*d*e - (c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)*sqrt(-(a^5*c
^2*d^4 - 2*a^6*c*d^2*e^2 + a^7*e^4)/(c^11*d^8 + 4*a*c^10*d^6*e^2 + 6*a^2*c^9*d^4
*e^4 + 4*a^3*c^8*d^2*e^6 + a^4*c^7*e^8)))/(c^5*d^4 + 2*a*c^4*d^2*e^2 + a^2*c^3*e
^4))) - 12*(c*d^3 + a*d*e^2)*x)/(c^2*d^2*e^2 + a*c*e^4)]

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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**8/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282771, size = 490, normalized size = 1.36 \[ \frac{d^{\frac{7}{2}} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{c d^{2} e^{2} + a e^{4}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d - \left (a c^{3}\right )^{\frac{3}{4}} a e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} c^{5} d^{2} + \sqrt{2} a c^{4} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d - \left (a c^{3}\right )^{\frac{3}{4}} a e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} c^{5} d^{2} + \sqrt{2} a c^{4} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d + \left (a c^{3}\right )^{\frac{3}{4}} a e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} c^{5} d^{2} + \sqrt{2} a c^{4} e^{2}\right )}} - \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d + \left (a c^{3}\right )^{\frac{3}{4}} a e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} c^{5} d^{2} + \sqrt{2} a c^{4} e^{2}\right )}} + \frac{{\left (c^{2} x^{3} e^{2} - 3 \, c^{2} d x e\right )} e^{\left (-3\right )}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

d^(7/2)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/(c*d^2*e^2 + a*e^4) + 1/2*((a*c^3)^(1
/4)*a*c^2*d - (a*c^3)^(3/4)*a*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/
(a/c)^(1/4))/(sqrt(2)*c^5*d^2 + sqrt(2)*a*c^4*e^2) + 1/2*((a*c^3)^(1/4)*a*c^2*d
- (a*c^3)^(3/4)*a*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))
/(sqrt(2)*c^5*d^2 + sqrt(2)*a*c^4*e^2) + 1/4*((a*c^3)^(1/4)*a*c^2*d + (a*c^3)^(3
/4)*a*e)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*c^5*d^2 + sqrt(2)*
a*c^4*e^2) - 1/4*((a*c^3)^(1/4)*a*c^2*d + (a*c^3)^(3/4)*a*e)*ln(x^2 - sqrt(2)*x*
(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*c^5*d^2 + sqrt(2)*a*c^4*e^2) + 1/3*(c^2*x^3*e^
2 - 3*c^2*d*x*e)*e^(-3)/c^3