3.251 \(\int \frac{1}{x^5 \left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=265 \[ \frac{c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}-\frac{\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}-\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e}{2 a^2 d^2 x^2}-\frac{1}{4 a^2 d x^4}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.664066, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{c^{3/2} e \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{5/2} \left (a e^2+c d^2\right )}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 a^3 \left (a e^2+c d^2\right )^2}-\frac{\log (x) \left (2 c d^2-a e^2\right )}{a^3 d^3}-\frac{c^2 \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e}{2 a^2 d^2 x^2}-\frac{1}{4 a^2 d x^4}-\frac{e^6 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 96.4187, size = 246, normalized size = 0.93 \[ - \frac{e^{6} \log{\left (d + e x^{2} \right )}}{2 d^{3} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{c^{2} \left (d - e x^{2}\right )}{4 a^{2} \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} - \frac{1}{4 a^{2} d x^{4}} + \frac{e}{2 a^{2} d^{2} x^{2}} + \frac{c^{2} d \left (3 a e^{2} + 2 c d^{2}\right ) \log{\left (a + c x^{4} \right )}}{4 a^{3} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{\left (a e^{2} - 2 c d^{2}\right ) \log{\left (x^{2} \right )}}{2 a^{3} d^{3}} + \frac{c^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )} + \frac{c^{\frac{3}{2}} e \left (2 a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

-e**6*log(d + e*x**2)/(2*d**3*(a*e**2 + c*d**2)**2) - c**2*(d - e*x**2)/(4*a**2*
(a + c*x**4)*(a*e**2 + c*d**2)) - 1/(4*a**2*d*x**4) + e/(2*a**2*d**2*x**2) + c**
2*d*(3*a*e**2 + 2*c*d**2)*log(a + c*x**4)/(4*a**3*(a*e**2 + c*d**2)**2) + (a*e**
2 - 2*c*d**2)*log(x**2)/(2*a**3*d**3) + c**(3/2)*e*atan(sqrt(c)*x**2/sqrt(a))/(4
*a**(5/2)*(a*e**2 + c*d**2)) + c**(3/2)*e*(2*a*e**2 + c*d**2)*atan(sqrt(c)*x**2/
sqrt(a))/(2*a**(5/2)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 0.80328, size = 278, normalized size = 1.05 \[ \frac{1}{4} \left (-\frac{c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}-\frac{c^{3/2} e \left (5 a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{5/2} \left (a e^2+c d^2\right )^2}+\frac{c^2 \left (3 a d e^2+2 c d^3\right ) \log \left (a+c x^4\right )}{a^3 \left (a e^2+c d^2\right )^2}+\frac{4 \log (x) \left (a e^2-2 c d^2\right )}{a^3 d^3}+\frac{c^2 \left (e x^2-d\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{2 e}{a^2 d^2 x^2}-\frac{1}{a^2 d x^4}-\frac{2 e^6 \log \left (d+e x^2\right )}{d^3 \left (a e^2+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(d + e*x^2)*(a + c*x^4)^2),x]

[Out]

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

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Maple [A]  time = 0.035, size = 363, normalized size = 1.4 \[ -{\frac{1}{4\,{a}^{2}d{x}^{4}}}+{\frac{\ln \left ( x \right ){e}^{2}}{{d}^{3}{a}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{a}^{3}d}}+{\frac{e}{2\,{a}^{2}{d}^{2}{x}^{2}}}+{\frac{{c}^{2}{x}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{3}{x}^{2}{d}^{2}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{2}d{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a \left ( c{x}^{4}+a \right ) }}-{\frac{{c}^{3}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3\,{c}^{2}\ln \left ( c{x}^{4}+a \right ) d{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}}+{\frac{{c}^{3}\ln \left ( c{x}^{4}+a \right ){d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{3}}}+{\frac{5\,{c}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{c}^{3}{d}^{2}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{6}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(e*x^2+d)/(c*x^4+a)^2,x)

[Out]

-1/4/a^2/d/x^4+1/a^2/d^3*ln(x)*e^2-2/a^3/d*ln(x)*c+1/2*e/a^2/d^2/x^2+1/4/(a*e^2+
c*d^2)^2*c^2/a/(c*x^4+a)*x^2*e^3+1/4/(a*e^2+c*d^2)^2*c^3/a^2/(c*x^4+a)*x^2*d^2*e
-1/4/(a*e^2+c*d^2)^2*c^2/a/(c*x^4+a)*d*e^2-1/4/(a*e^2+c*d^2)^2*c^3/a^2/(c*x^4+a)
*d^3+3/4/(a*e^2+c*d^2)^2*c^2/a^2*ln(c*x^4+a)*d*e^2+1/2/(a*e^2+c*d^2)^2*c^3/a^3*l
n(c*x^4+a)*d^3+5/4/(a*e^2+c*d^2)^2*c^2/a/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))*e
^3+3/4/(a*e^2+c*d^2)^2*c^3/a^2/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))*d^2*e-1/2*e
^6*ln(e*x^2+d)/d^3/(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(1/((c*x^4 + a)^2*(e*x^2 + d)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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(1/((c*x^4 + a)^2*(e*x^2 + d)*x^5),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(1/x**5/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278368, size = 473, normalized size = 1.78 \[ \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}} - \frac{e^{7}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{7} e + 2 \, a c d^{5} e^{3} + a^{2} d^{3} e^{5}\right )}} + \frac{{\left (3 \, c^{3} d^{2} e + 5 \, a c^{2} e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} \sqrt{a c}} - \frac{2 \, c^{4} d^{3} x^{4} + 3 \, a c^{3} d x^{4} e^{2} - a c^{3} d^{2} x^{2} e + 3 \, a c^{3} d^{3} - a^{2} c^{2} x^{2} e^{3} + 4 \, a^{2} c^{2} d e^{2}}{4 \,{\left (a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}\right )}{\left (c x^{4} + a\right )}} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3} d^{3}} + \frac{6 \, c d^{2} x^{4} - 3 \, a x^{4} e^{2} + 2 \, a d x^{2} e - a d^{2}}{4 \, a^{3} d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x^2 + d)*x^5),x, algorithm="giac")

[Out]

1/4*(2*c^3*d^3 + 3*a*c^2*d*e^2)*ln(c*x^4 + a)/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a
^5*e^4) - 1/2*e^7*ln(abs(x^2*e + d))/(c^2*d^7*e + 2*a*c*d^5*e^3 + a^2*d^3*e^5) +
1/4*(3*c^3*d^2*e + 5*a*c^2*e^3)*arctan(c*x^2/sqrt(a*c))/((a^2*c^2*d^4 + 2*a^3*c
*d^2*e^2 + a^4*e^4)*sqrt(a*c)) - 1/4*(2*c^4*d^3*x^4 + 3*a*c^3*d*x^4*e^2 - a*c^3*
d^2*x^2*e + 3*a*c^3*d^3 - a^2*c^2*x^2*e^3 + 4*a^2*c^2*d*e^2)/((a^3*c^2*d^4 + 2*a
^4*c*d^2*e^2 + a^5*e^4)*(c*x^4 + a)) - 1/2*(2*c*d^2 - a*e^2)*ln(x^2)/(a^3*d^3) +
1/4*(6*c*d^2*x^4 - 3*a*x^4*e^2 + 2*a*d*x^2*e - a*d^2)/(a^3*d^3*x^4)