3.54 \(\int \frac{d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=244 \[ -\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{b d-a e}{4 a^2 x^4}+\frac{\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac{\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{d}{6 a x^6} \]

[Out]

-d/(6*a*x^6) + (b*d - a*e)/(4*a^2*x^4) - (b^2*d - a*b*e - a*(c*d - a*f))/(2*a^3*
x^2) - ((b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^2*(4*c*d - a*
f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*Sqrt[b^2 - 4*a*c]) - ((b^3*
d - a*b^2*e + a^2*c*e - a*b*(2*c*d - a*f))*Log[x])/a^4 + ((b^3*d - a*b^2*e + a^2
*c*e - a*b*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

_______________________________________________________________________________________

Rubi [A]  time = 1.1775, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{b d-a e}{4 a^2 x^4}+\frac{\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac{\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{d}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]

[Out]

-d/(6*a*x^6) + (b*d - a*e)/(4*a^2*x^4) - (b^2*d - a*b*e - a*(c*d - a*f))/(2*a^3*
x^2) - ((b^4*d - a*b^3*e + 3*a^2*b*c*e + 2*a^2*c*(c*d - a*f) - a*b^2*(4*c*d - a*
f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*Sqrt[b^2 - 4*a*c]) - ((b^3*
d - a*b^2*e + a^2*c*e - a*b*(2*c*d - a*f))*Log[x])/a^4 + ((b^3*d - a*b^2*e + a^2
*c*e - a*b*(2*c*d - a*f))*Log[a + b*x^2 + c*x^4])/(4*a^4)

_______________________________________________________________________________________

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((f*x**4+e*x**2+d)/x**7/(c*x**4+b*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.789086, size = 416, normalized size = 1.7 \[ \frac{-\frac{2 a^3 d}{x^6}-12 \log (x) \left (a^2 c e-a b^2 e+a b (a f-2 c d)+b^3 d\right )+\frac{3 \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}-2 a f+2 c d\right )+a b^2 \left (-e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}-a e\right )+b^4 d\right )}{\sqrt{b^2-4 a c}}+\frac{3 \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}+2 a f-2 c d\right )-a b^2 \left (e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}+a e\right )+b^4 (-d)\right )}{\sqrt{b^2-4 a c}}+\frac{3 a^2 (b d-a e)}{x^4}+\frac{6 a \left (a b e+a (c d-a f)+b^2 (-d)\right )}{x^2}}{12 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2 + f*x^4)/(x^7*(a + b*x^2 + c*x^4)),x]

[Out]

((-2*a^3*d)/x^6 + (3*a^2*(b*d - a*e))/x^4 + (6*a*(-(b^2*d) + a*b*e + a*(c*d - a*
f)))/x^2 - 12*(b^3*d - a*b^2*e + a^2*c*e + a*b*(-2*c*d + a*f))*Log[x] + (3*(b^4*
d + b^3*(Sqrt[b^2 - 4*a*c]*d - a*e) + a^2*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f
) + a*b^2*(-4*c*d - Sqrt[b^2 - 4*a*c]*e + a*f) + a*b*(-2*c*Sqrt[b^2 - 4*a*c]*d +
3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^
2 - 4*a*c] + (3*(-(b^4*d) + b^3*(Sqrt[b^2 - 4*a*c]*d + a*e) - a*b^2*(-4*c*d + Sq
rt[b^2 - 4*a*c]*e + a*f) + a^2*c*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + a*b*(-
2*c*Sqrt[b^2 - 4*a*c]*d - 3*a*c*e + a*Sqrt[b^2 - 4*a*c]*f))*Log[b + Sqrt[b^2 - 4
*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(12*a^4)

_______________________________________________________________________________________

Maple [B]  time = 0.019, size = 523, normalized size = 2.1 \[{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{a}^{3}}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{2\,{a}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}d}{4\,{a}^{4}}}-{\frac{cf}{a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}f}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,bce}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{c}^{2}d}{{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{{b}^{2}cd}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}d}{2\,{a}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{d}{6\,a{x}^{6}}}-{\frac{e}{4\,a{x}^{4}}}+{\frac{bd}{4\,{a}^{2}{x}^{4}}}-{\frac{f}{2\,a{x}^{2}}}+{\frac{be}{2\,{a}^{2}{x}^{2}}}+{\frac{cd}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{2}d}{2\,{a}^{3}{x}^{2}}}-{\frac{\ln \left ( x \right ) bf}{{a}^{2}}}-{\frac{\ln \left ( x \right ) ce}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}e}{{a}^{3}}}+2\,{\frac{\ln \left ( x \right ) bcd}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}d}{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^4+e*x^2+d)/x^7/(c*x^4+b*x^2+a),x)

[Out]

1/4/a^2*ln(c*x^4+b*x^2+a)*b*f+1/4/a^2*c*ln(c*x^4+b*x^2+a)*e-1/4/a^3*ln(c*x^4+b*x
^2+a)*b^2*e-1/2/a^3*c*ln(c*x^4+b*x^2+a)*b*d+1/4/a^4*ln(c*x^4+b*x^2+a)*b^3*d-1/a/
(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c*f+1/2/a^2/(4*a*c-b^2)^
(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*f+3/2/a^2/(4*a*c-b^2)^(1/2)*arct
an((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*e+1/a^2/(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^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-
b^2)^(1/2))*b^3*e-2/a^3/(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^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^4*d-1/
6*d/a/x^6-1/4/a/x^4*e+1/4/a^2/x^4*b*d-1/2/a/x^2*f+1/2/a^2/x^2*b*e+1/2/a^2/x^2*c*
d-1/2/a^3/x^2*b^2*d-1/a^2*ln(x)*b*f-1/a^2*ln(x)*c*e+1/a^3*ln(x)*b^2*e+2/a^3*ln(x
)*b*c*d-1/a^4*ln(x)*b^3*d

_______________________________________________________________________________________

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((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.04713, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 3 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (3 \,{\left (a^{2} b f +{\left (b^{3} - 2 \, a b c\right )} d -{\left (a b^{2} - a^{2} c\right )} e\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) - 12 \,{\left (a^{2} b f +{\left (b^{3} - 2 \, a b c\right )} d -{\left (a b^{2} - a^{2} c\right )} e\right )} x^{6} \log \left (x\right ) + 6 \,{\left (a^{2} b e - a^{3} f -{\left (a b^{2} - a^{2} c\right )} d\right )} x^{4} - 2 \, a^{3} d + 3 \,{\left (a^{2} b d - a^{3} e\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{12 \, \sqrt{b^{2} - 4 \, a c} a^{4} x^{6}}, \frac{6 \,{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 3 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (3 \,{\left (a^{2} b f +{\left (b^{3} - 2 \, a b c\right )} d -{\left (a b^{2} - a^{2} c\right )} e\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) - 12 \,{\left (a^{2} b f +{\left (b^{3} - 2 \, a b c\right )} d -{\left (a b^{2} - a^{2} c\right )} e\right )} x^{6} \log \left (x\right ) + 6 \,{\left (a^{2} b e - a^{3} f -{\left (a b^{2} - a^{2} c\right )} d\right )} x^{4} - 2 \, a^{3} d + 3 \,{\left (a^{2} b d - a^{3} e\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{12 \, \sqrt{-b^{2} + 4 \, a c} a^{4} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^7),x, algorithm="fricas")

[Out]

[-1/12*(3*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3*a^2*b*c)*e + (a^2*b^2 -
2*a^3*c)*f)*x^6*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 + 2*b*
c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - (3*(a^2*b*f + (b^
3 - 2*a*b*c)*d - (a*b^2 - a^2*c)*e)*x^6*log(c*x^4 + b*x^2 + a) - 12*(a^2*b*f + (
b^3 - 2*a*b*c)*d - (a*b^2 - a^2*c)*e)*x^6*log(x) + 6*(a^2*b*e - a^3*f - (a*b^2 -
a^2*c)*d)*x^4 - 2*a^3*d + 3*(a^2*b*d - a^3*e)*x^2)*sqrt(b^2 - 4*a*c))/(sqrt(b^2
- 4*a*c)*a^4*x^6), 1/12*(6*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d - (a*b^3 - 3*a^2*b*
c)*e + (a^2*b^2 - 2*a^3*c)*f)*x^6*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2
- 4*a*c)) + (3*(a^2*b*f + (b^3 - 2*a*b*c)*d - (a*b^2 - a^2*c)*e)*x^6*log(c*x^4 +
b*x^2 + a) - 12*(a^2*b*f + (b^3 - 2*a*b*c)*d - (a*b^2 - a^2*c)*e)*x^6*log(x) +
6*(a^2*b*e - a^3*f - (a*b^2 - a^2*c)*d)*x^4 - 2*a^3*d + 3*(a^2*b*d - a^3*e)*x^2)
*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4*x^6)]

_______________________________________________________________________________________

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((f*x**4+e*x**2+d)/x**7/(c*x**4+b*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295154, size = 423, normalized size = 1.73 \[ \frac{{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac{{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} + \frac{{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d + a^{2} b^{2} f - 2 \, a^{3} c f - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{4}} + \frac{11 \, b^{3} d x^{6} - 22 \, a b c d x^{6} + 11 \, a^{2} b f x^{6} - 11 \, a b^{2} x^{6} e + 11 \, a^{2} c x^{6} e - 6 \, a b^{2} d x^{4} + 6 \, a^{2} c d x^{4} - 6 \, a^{3} f x^{4} + 6 \, a^{2} b x^{4} e + 3 \, a^{2} b d x^{2} - 3 \, a^{3} x^{2} e - 2 \, a^{3} d}{12 \, a^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^7),x, algorithm="giac")

[Out]

1/4*(b^3*d - 2*a*b*c*d + a^2*b*f - a*b^2*e + a^2*c*e)*ln(c*x^4 + b*x^2 + a)/a^4
- 1/2*(b^3*d - 2*a*b*c*d + a^2*b*f - a*b^2*e + a^2*c*e)*ln(x^2)/a^4 + 1/2*(b^4*d
- 4*a*b^2*c*d + 2*a^2*c^2*d + a^2*b^2*f - 2*a^3*c*f - a*b^3*e + 3*a^2*b*c*e)*ar
ctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) + 1/12*(11*b^3*d
*x^6 - 22*a*b*c*d*x^6 + 11*a^2*b*f*x^6 - 11*a*b^2*x^6*e + 11*a^2*c*x^6*e - 6*a*b
^2*d*x^4 + 6*a^2*c*d*x^4 - 6*a^3*f*x^4 + 6*a^2*b*x^4*e + 3*a^2*b*d*x^2 - 3*a^3*x
^2*e - 2*a^3*d)/(a^4*x^6)