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

Optimal. Leaf size=203 \[ \frac{x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{4 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c} \]

[Out]

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

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Rubi [A]  time = 0.850191, antiderivative size = 203, 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 \[ \frac{x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{4 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 0.262327, size = 193, normalized size = 0.95 \[ \frac{6 c x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )-3 \log \left (a+b x^2+c x^4\right ) \left (b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right )+\frac{6 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)+b^4 f-b^3 c e\right )}{\sqrt{4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 474, normalized size = 2.3 \[{\frac{f{x}^{6}}{6\,c}}-{\frac{b{x}^{4}f}{4\,{c}^{2}}}+{\frac{{x}^{4}e}{4\,c}}-{\frac{{x}^{2}af}{2\,{c}^{2}}}+{\frac{{b}^{2}f{x}^{2}}{2\,{c}^{3}}}-{\frac{{x}^{2}be}{2\,{c}^{2}}}+{\frac{d{x}^{2}}{2\,c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abf}{2\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ae}{4\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}f}{4\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{4\,{c}^{2}}}+{\frac{{a}^{2}f}{{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{a{b}^{2}f}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,abe}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{ad}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}f}{2\,{c}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((f*x^4 + e*x^2 + d)*x^5/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.457989, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (b^{3} c - 3 \, a b c^{2}\right )} e +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (2 \, c^{3} f x^{6} + 3 \,{\left (c^{3} e - b c^{2} f\right )} x^{4} + 6 \,{\left (c^{3} d - b c^{2} e +{\left (b^{2} c - a c^{2}\right )} f\right )} x^{2} - 3 \,{\left (b c^{2} d -{\left (b^{2} c - a c^{2}\right )} e +{\left (b^{3} - 2 \, a b c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{12 \, \sqrt{b^{2} - 4 \, a c} c^{4}}, \frac{6 \,{\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (b^{3} c - 3 \, a b c^{2}\right )} e +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c^{3} f x^{6} + 3 \,{\left (c^{3} e - b c^{2} f\right )} x^{4} + 6 \,{\left (c^{3} d - b c^{2} e +{\left (b^{2} c - a c^{2}\right )} f\right )} x^{2} - 3 \,{\left (b c^{2} d -{\left (b^{2} c - a c^{2}\right )} e +{\left (b^{3} - 2 \, a b c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{12 \, \sqrt{-b^{2} + 4 \, a c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 110.105, size = 1044, normalized size = 5.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d
+ b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2
*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4))*log(x**2 + (-3*a**2*b*c*f + 2*a**2*
c**2*e + a*b**3*f - a*b**2*c*e + a*b*c**2*d + 8*a*c**4*(-sqrt(-4*a*c + b**2)*(2*
a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**
2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e -
b*c**2*d)/(4*c**4)) - 2*b**2*c**3*(-sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**
2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*
a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4)))/
(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e +
b**2*c**2*d)) + (sqrt(-4*a*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*
e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a
*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d)/(4*c**4))*log(x**2 + (-3*a**2*
b*c*f + 2*a**2*c**2*e + a*b**3*f - a*b**2*c*e + a*b*c**2*d + 8*a*c**4*(sqrt(-4*a
*c + b**2)*(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f -
b**3*c*e + b**2*c**2*d)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f
+ b**2*c*e - b*c**2*d)/(4*c**4)) - 2*b**2*c**3*(sqrt(-4*a*c + b**2)*(2*a**2*c**
2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d
)/(4*c**4*(4*a*c - b**2)) + (2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e - b*c**2*d
)/(4*c**4)))/(2*a**2*c**2*f - 4*a*b**2*c*f + 3*a*b*c**2*e - 2*a*c**3*d + b**4*f
- b**3*c*e + b**2*c**2*d)) + f*x**6/(6*c) - x**4*(b*f - c*e)/(4*c**2) - x**2*(a*
c*f - b**2*f + b*c*e - c**2*d)/(2*c**3)

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GIAC/XCAS [A]  time = 0.30386, size = 289, normalized size = 1.42 \[ \frac{2 \, c^{2} f x^{6} - 3 \, b c f x^{4} + 3 \, c^{2} x^{4} e + 6 \, c^{2} d x^{2} + 6 \, b^{2} f x^{2} - 6 \, a c f x^{2} - 6 \, b c x^{2} e}{12 \, c^{3}} - \frac{{\left (b c^{2} d + b^{3} f - 2 \, a b c f - b^{2} c e + a c^{2} e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac{{\left (b^{2} c^{2} d - 2 \, a c^{3} d + b^{4} f - 4 \, a b^{2} c f + 2 \, a^{2} c^{2} f - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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