3.246 \(\int \frac{x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=153 \[ -\frac{d^2 e \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}+\frac{d^2 e \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{d \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a e+c d x^2}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \]

[Out]

-(a*e + c*d*x^2)/(4*c*(c*d^2 + a*e^2)*(a + c*x^4)) + (d*(c*d^2 - a*e^2)*ArcTan[(
Sqrt[c]*x^2)/Sqrt[a]])/(4*Sqrt[a]*Sqrt[c]*(c*d^2 + a*e^2)^2) + (d^2*e*Log[d + e*
x^2])/(2*(c*d^2 + a*e^2)^2) - (d^2*e*Log[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 0.482651, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{d^2 e \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}+\frac{d^2 e \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{d \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{a e+c d x^2}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

-(a*e + c*d*x^2)/(4*c*(c*d^2 + a*e^2)*(a + c*x^4)) + (d*(c*d^2 - a*e^2)*ArcTan[(
Sqrt[c]*x^2)/Sqrt[a]])/(4*Sqrt[a]*Sqrt[c]*(c*d^2 + a*e^2)^2) + (d^2*e*Log[d + e*
x^2])/(2*(c*d^2 + a*e^2)^2) - (d^2*e*Log[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)

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Rubi in Sympy [A]  time = 55.5867, size = 165, normalized size = 1.08 \[ - \frac{d^{2} e \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{d^{2} e \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} - \frac{a e + c d x^{2}}{4 c \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{c} d^{3} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{a} \sqrt{c} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

-d**2*e*log(a + c*x**4)/(4*(a*e**2 + c*d**2)**2) + d**2*e*log(d + e*x**2)/(2*(a*
e**2 + c*d**2)**2) - (a*e + c*d*x**2)/(4*c*(a + c*x**4)*(a*e**2 + c*d**2)) + sqr
t(c)*d**3*atan(sqrt(c)*x**2/sqrt(a))/(2*sqrt(a)*(a*e**2 + c*d**2)**2) - d*atan(s
qrt(c)*x**2/sqrt(a))/(4*sqrt(a)*sqrt(c)*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.274019, size = 120, normalized size = 0.78 \[ \frac{\frac{d \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}-\frac{\left (a e^2+c d^2\right ) \left (a e+c d x^2\right )}{c \left (a+c x^4\right )}-d^2 e \log \left (a+c x^4\right )+2 d^2 e \log \left (d+e x^2\right )}{4 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((d + e*x^2)*(a + c*x^4)^2),x]

[Out]

(-(((c*d^2 + a*e^2)*(a*e + c*d*x^2))/(c*(a + c*x^4))) + (d*(c*d^2 - a*e^2)*ArcTa
n[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + 2*d^2*e*Log[d + e*x^2] - d^2*e*Log
[a + c*x^4])/(4*(c*d^2 + a*e^2)^2)

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Maple [A]  time = 0.023, size = 252, normalized size = 1.7 \[ -{\frac{{e}^{2}{x}^{2}da}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{c{x}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{a}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}-{\frac{ae{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{d}^{2}e\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}-{\frac{{e}^{2}da}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{c{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{2}e\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*x^2*e^2*d*a-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*x^2*c*d
^3-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*a^2*e^3/c-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*a*e*d^2
-1/4*d^2*e*ln(c*x^4+a)/(a*e^2+c*d^2)^2-1/4/(a*e^2+c*d^2)^2*d/(a*c)^(1/2)*arctan(
c*x^2/(a*c)^(1/2))*a*e^2+1/4/(a*e^2+c*d^2)^2*d^3/(a*c)^(1/2)*arctan(c*x^2/(a*c)^
(1/2))*c+1/2*d^2*e*ln(e*x^2+d)/(a*e^2+c*d^2)^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^5/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.84151, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a c^{2} d^{3} - a^{2} c d e^{2} +{\left (c^{3} d^{3} - a c^{2} d e^{2}\right )} x^{4}\right )} \log \left (-\frac{2 \, a c x^{2} -{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) + 2 \,{\left (a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} +{\left (c^{2} d^{2} e x^{4} + a c d^{2} e\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (c^{2} d^{2} e x^{4} + a c d^{2} e\right )} \log \left (e x^{2} + d\right )\right )} \sqrt{-a c}}{8 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )} \sqrt{-a c}}, -\frac{{\left (a c^{2} d^{3} - a^{2} c d e^{2} +{\left (c^{3} d^{3} - a c^{2} d e^{2}\right )} x^{4}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) +{\left (a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} +{\left (c^{2} d^{2} e x^{4} + a c d^{2} e\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (c^{2} d^{2} e x^{4} + a c d^{2} e\right )} \log \left (e x^{2} + d\right )\right )} \sqrt{a c}}{4 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="fricas")

[Out]

[-1/8*((a*c^2*d^3 - a^2*c*d*e^2 + (c^3*d^3 - a*c^2*d*e^2)*x^4)*log(-(2*a*c*x^2 -
(c*x^4 - a)*sqrt(-a*c))/(c*x^4 + a)) + 2*(a*c*d^2*e + a^2*e^3 + (c^2*d^3 + a*c*
d*e^2)*x^2 + (c^2*d^2*e*x^4 + a*c*d^2*e)*log(c*x^4 + a) - 2*(c^2*d^2*e*x^4 + a*c
*d^2*e)*log(e*x^2 + d))*sqrt(-a*c))/((a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4
+ (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sqrt(-a*c)), -1/4*((a*c^2*d^3 -
a^2*c*d*e^2 + (c^3*d^3 - a*c^2*d*e^2)*x^4)*arctan(a/(sqrt(a*c)*x^2)) + (a*c*d^2
*e + a^2*e^3 + (c^2*d^3 + a*c*d*e^2)*x^2 + (c^2*d^2*e*x^4 + a*c*d^2*e)*log(c*x^4
+ a) - 2*(c^2*d^2*e*x^4 + a*c*d^2*e)*log(e*x^2 + d))*sqrt(a*c))/((a*c^3*d^4 + 2
*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4)*sq
rt(a*c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275551, size = 297, normalized size = 1.94 \[ -\frac{d^{2} e{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{d^{2} e^{2}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{{\left (c d^{3} - a d e^{2}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a c}} + \frac{c^{2} d^{2} x^{4} e - c^{2} d^{3} x^{2} - a c d x^{2} e^{2} - a^{2} e^{3}}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (c x^{4} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="giac")

[Out]

-1/4*d^2*e*ln(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/2*d^2*e^2*ln(ab
s(x^2*e + d))/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 1/4*(c*d^3 - a*d*e^2)*arct
an(c*x^2/sqrt(a*c))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*c)) + 1/4*(c^2*d
^2*x^4*e - c^2*d^3*x^2 - a*c*d*x^2*e^2 - a^2*e^3)/((c^3*d^4 + 2*a*c^2*d^2*e^2 +
a^2*c*e^4)*(c*x^4 + a))