3.293 \(\int \frac{1}{x^3 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=205 \[ \frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac{\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{\log (x) (a e+b d)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac{1}{2 a d x^2} \]

[Out]

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

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Rubi [A]  time = 0.787316, antiderivative size = 205, 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 \[ \frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac{\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{\log (x) (a e+b d)}{a^2 d^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac{1}{2 a d x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 157.915, size = 194, normalized size = 0.95 \[ \frac{e^{3} \log{\left (d + e x^{2} \right )}}{2 d^{2} \left (a e^{2} - b d e + c d^{2}\right )} - \frac{1}{2 a d x^{2}} - \frac{\left (- a c e + b^{2} e - b c d\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{2} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (- 3 a b c e + 2 a c^{2} d + b^{3} e - b^{2} c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} \sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )} - \frac{\left (a e + b d\right ) \log{\left (x^{2} \right )}}{2 a^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(e*x**2+d)/(c*x**4+b*x**2+a),x)

[Out]

e**3*log(d + e*x**2)/(2*d**2*(a*e**2 - b*d*e + c*d**2)) - 1/(2*a*d*x**2) - (-a*c
*e + b**2*e - b*c*d)*log(a + b*x**2 + c*x**4)/(4*a**2*(a*e**2 - b*d*e + c*d**2))
+ (-3*a*b*c*e + 2*a*c**2*d + b**3*e - b**2*c*d)*atanh((b + 2*c*x**2)/sqrt(-4*a*
c + b**2))/(2*a**2*sqrt(-4*a*c + b**2)*(a*e**2 - b*d*e + c*d**2)) - (a*e + b*d)*
log(x**2)/(2*a**2*d**2)

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Mathematica [A]  time = 0.574493, size = 331, normalized size = 1.61 \[ \frac{1}{4} \left (\frac{\left (b^2 \left (e \sqrt{b^2-4 a c}-c d\right )-b c \left (d \sqrt{b^2-4 a c}+3 a e\right )+a c \left (2 c d-e \sqrt{b^2-4 a c}\right )+b^3 e\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}+\frac{\left (b^2 \left (e \sqrt{b^2-4 a c}+c d\right )+b c \left (3 a e-d \sqrt{b^2-4 a c}\right )-a c \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 (-e)\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right )}-\frac{4 \log (x) (a e+b d)}{a^2 d^2}+\frac{2 e^3 \log \left (d+e x^2\right )}{d^2 e (a e-b d)+c d^4}-\frac{2}{a d x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [B]  time = 0.021, size = 430, normalized size = 2.1 \[{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) a}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{ \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ){a}^{2}}}+{\frac{3\,bce}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{c}^{2}d}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}cd}{ \left ( 2\,a{e}^{2}-2\,bde+2\,c{d}^{2} \right ){a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{2\,ad{x}^{2}}}-{\frac{\ln \left ( x \right ) e}{a{d}^{2}}}-{\frac{\ln \left ( x \right ) b}{{a}^{2}d}}+{\frac{{e}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\,{d}^{2} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302586, size = 320, normalized size = 1.56 \[ \frac{{\left (b c d - b^{2} e + a c e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )}} + \frac{{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (b d + a e\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2} d^{2}} + \frac{b d x^{2} + a x^{2} e - a d}{2 \, a^{2} d^{2} x^{2}} \]

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^3),x, algorithm="giac")

[Out]

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