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

Optimal. Leaf size=268 \[ \frac{\log (x) \left (a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 d^3}-\frac{\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (a e^2-b d e+c d^2\right )}+\frac{a e+b d}{2 a^2 d^2 x^2}+\frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2-b d e+c d^2\right )}-\frac{1}{4 a d x^4} \]

[Out]

-1/(4*a*d*x^4) + (b*d + a*e)/(2*a^2*d^2*x^2) + ((b^3*c*d - 3*a*b*c^2*d - b^4*e +
4*a*b^2*c*e - 2*a^2*c^2*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqr
t[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + ((b^2*d^2 + a*b*d*e - a*(c*d^2 - a*e^2
))*Log[x])/(a^3*d^3) - (e^4*Log[d + e*x^2])/(2*d^3*(c*d^2 - b*d*e + a*e^2)) - ((
b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*Log[a + b*x^2 + c*x^4])/(4*a^3*(c*d^2 - b
*d*e + a*e^2))

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Rubi [A]  time = 1.06887, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\log (x) \left (a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 d^3}-\frac{\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3 \left (a e^2-b d e+c d^2\right )}+\frac{a e+b d}{2 a^2 d^2 x^2}+\frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2-b d e+c d^2\right )}-\frac{1}{4 a d x^4} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*a*d*x^4) + (b*d + a*e)/(2*a^2*d^2*x^2) + ((b^3*c*d - 3*a*b*c^2*d - b^4*e +
4*a*b^2*c*e - 2*a^2*c^2*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqr
t[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + ((b^2*d^2 + a*b*d*e - a*(c*d^2 - a*e^2
))*Log[x])/(a^3*d^3) - (e^4*Log[d + e*x^2])/(2*d^3*(c*d^2 - b*d*e + a*e^2)) - ((
b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*Log[a + b*x^2 + c*x^4])/(4*a^3*(c*d^2 - b
*d*e + a*e^2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.795197, size = 426, normalized size = 1.59 \[ \frac{1}{4} \left (\frac{4 \log (x) \left (a b d e+a \left (a e^2-c d^2\right )+b^2 d^2\right )}{a^3 d^3}-\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}+2 a e\right )-b^2 c \left (d \sqrt{b^2-4 a c}+4 a e\right )+a b c \left (3 c d-2 e \sqrt{b^2-4 a c}\right )+b^3 \left (e \sqrt{b^2-4 a c}-c d\right )+b^4 e\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^3 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}-\frac{\left (a c^2 \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^2 c \left (4 a e-d \sqrt{b^2-4 a c}\right )-a b c \left (2 e \sqrt{b^2-4 a c}+3 c d\right )+b^3 \left (e \sqrt{b^2-4 a c}+c d\right )+b^4 (-e)\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^3 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}+\frac{2 (a e+b d)}{a^2 d^2 x^2}-\frac{2 e^4 \log \left (d+e x^2\right )}{d^3 e (a e-b d)+c d^5}-\frac{1}{a d x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.028, size = 584, normalized size = 2.2 \[ -{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) be}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{2}}}+{\frac{{c}^{2}\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}e}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ){a}^{3}}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ){a}^{3}}}+{\frac{{c}^{2}e}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{{b}^{2}ce}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ){a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,b{c}^{2}d}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}e}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}cd}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{4\,ad{x}^{4}}}+{\frac{e}{2\,a{d}^{2}{x}^{2}}}+{\frac{b}{2\,{a}^{2}d{x}^{2}}}+{\frac{\ln \left ( x \right ){e}^{2}}{a{d}^{3}}}+{\frac{\ln \left ( x \right ) be}{{a}^{2}{d}^{2}}}-{\frac{\ln \left ( x \right ) c}{{a}^{2}d}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}d}}-{\frac{{e}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302801, size = 448, normalized size = 1.67 \[ -\frac{{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )}} - \frac{e^{5}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{5} e - b d^{4} e^{2} + a d^{3} e^{3}\right )}} - \frac{{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} - a c d^{2} + a b d e + a^{2} e^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3} d^{3}} - \frac{3 \, b^{2} d^{2} x^{4} - 3 \, a c d^{2} x^{4} + 3 \, a b d x^{4} e + 3 \, a^{2} x^{4} e^{2} - 2 \, a b d^{2} x^{2} - 2 \, a^{2} d x^{2} e + a^{2} d^{2}}{4 \, a^{3} d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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