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

Optimal. Leaf size=273 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-b^3 c (c d-5 a f)-4 a b^2 c^2 e+a b c^2 (3 c d-5 a f)+b^5 (-f)+b^4 c e\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c} \]

[Out]

((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*x^2)/(2*c^4) + ((c^2*d + b^2*f
- c*(b*e + a*f))*x^4)/(4*c^3) + ((c*e - b*f)*x^6)/(6*c^2) + (f*x^8)/(8*c) - ((b^
4*c*e - 4*a*b^2*c^2*e + 2*a^2*c^3*e - b^5*f - b^3*c*(c*d - 5*a*f) + a*b*c^2*(3*c
*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^5*Sqrt[b^2 - 4*a*c])
- ((b^3*c*e - 2*a*b*c^2*e - b^4*f - b^2*c*(c*d - 3*a*f) + a*c^2*(c*d - a*f))*Lo
g[a + b*x^2 + c*x^4])/(4*c^5)

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Rubi [A]  time = 1.63638, antiderivative size = 273, 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 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-b^3 c (c d-5 a f)-4 a b^2 c^2 e+a b c^2 (3 c d-5 a f)+b^5 (-f)+b^4 c e\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^4 (-f)+b^3 c e\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f))*x^2)/(2*c^4) + ((c^2*d + b^2*f
- c*(b*e + a*f))*x^4)/(4*c^3) + ((c*e - b*f)*x^6)/(6*c^2) + (f*x^8)/(8*c) - ((b^
4*c*e - 4*a*b^2*c^2*e + 2*a^2*c^3*e - b^5*f - b^3*c*(c*d - 5*a*f) + a*b*c^2*(3*c
*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^5*Sqrt[b^2 - 4*a*c])
- ((b^3*c*e - 2*a*b*c^2*e - b^4*f - b^2*c*(c*d - 3*a*f) + a*c^2*(c*d - a*f))*Lo
g[a + b*x^2 + c*x^4])/(4*c^5)

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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**7*(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 = 0.393585, size = 260, normalized size = 0.95 \[ \frac{-\frac{12 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-2 a^2 c^3 e+b^3 c (c d-5 a f)+4 a b^2 c^2 e+a b c^2 (5 a f-3 c d)+b^5 f-b^4 c e\right )}{\sqrt{4 a c-b^2}}+6 c^2 x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )-12 c x^2 \left (b c (c d-2 a f)+a c^2 e+b^3 f-b^2 c e\right )+6 \log \left (a+b x^2+c x^4\right ) \left (b^2 c (c d-3 a f)+2 a b c^2 e+a c^2 (a f-c d)+b^4 f-b^3 c e\right )+4 c^3 x^6 (c e-b f)+3 c^4 f x^8}{24 c^5} \]

Antiderivative was successfully verified.

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

[Out]

(-12*c*(-(b^2*c*e) + a*c^2*e + b^3*f + b*c*(c*d - 2*a*f))*x^2 + 6*c^2*(c^2*d + b
^2*f - c*(b*e + a*f))*x^4 + 4*c^3*(c*e - b*f)*x^6 + 3*c^4*f*x^8 - (12*(-(b^4*c*e
) + 4*a*b^2*c^2*e - 2*a^2*c^3*e + b^5*f + b^3*c*(c*d - 5*a*f) + a*b*c^2*(-3*c*d
+ 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^
3*c*e) + 2*a*b*c^2*e + b^4*f + b^2*c*(c*d - 3*a*f) + a*c^2*(-(c*d) + a*f))*Log[a
+ b*x^2 + c*x^4])/(24*c^5)

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Maple [B]  time = 0.009, size = 622, normalized size = 2.3 \[{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abe}{2\,{c}^{3}}}-{\frac{{x}^{6}bf}{6\,{c}^{2}}}-{\frac{{x}^{4}af}{4\,{c}^{2}}}-{\frac{b{x}^{2}d}{2\,{c}^{2}}}-{\frac{be{x}^{4}}{4\,{c}^{2}}}+{\frac{{b}^{2}{x}^{4}f}{4\,{c}^{3}}}+{\frac{{b}^{2}e{x}^{2}}{2\,{c}^{3}}}-{\frac{{b}^{3}f{x}^{2}}{2\,{c}^{4}}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{c}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){a}^{2}f}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ad}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{4}f}{4\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}e}{4\,{c}^{4}}}-{\frac{3\,\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a{b}^{2}f}{4\,{c}^{4}}}+{\frac{f{x}^{8}}{8\,c}}+{\frac{3\,abd}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{5\,a{b}^{3}f}{2\,{c}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{a{b}^{2}e}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{abf{x}^{2}}{{c}^{3}}}-{\frac{{b}^{3}d}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{a}^{2}e}{{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{5}f}{2\,{c}^{5}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}e}{2\,{c}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{5\,{a}^{2}bf}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{x}^{6}e}{6\,c}}+{\frac{d{x}^{4}}{4\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/c^3*ln(c*x^4+b*x^2+a)*a*b*e-1/6/c^2*x^6*b*f-1/4/c^2*x^4*a*f-1/2/c^2*x^2*b*d-
1/4/c^2*x^4*b*e+1/4/c^3*x^4*b^2*f+1/2/c^3*b^2*e*x^2-1/2/c^4*b^3*f*x^2-1/2/c^2*x^
2*a*e+1/4/c^3*ln(c*x^4+b*x^2+a)*b^2*d+1/4/c^3*ln(c*x^4+b*x^2+a)*a^2*f-1/4/c^2*ln
(c*x^4+b*x^2+a)*a*d+1/4/c^5*ln(c*x^4+b*x^2+a)*b^4*f-1/4/c^4*ln(c*x^4+b*x^2+a)*b^
3*e-3/4/c^4*ln(c*x^4+b*x^2+a)*a*b^2*f+1/8*f*x^8/c+3/2/c^2/(4*a*c-b^2)^(1/2)*arct
an((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a*b*d+5/2/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x^
2+b)/(4*a*c-b^2)^(1/2))*a*b^3*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*e+1/c^3*a*b*f*x^2-1/2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+
b)/(4*a*c-b^2)^(1/2))*b^3*d+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^
2)^(1/2))*e*a^2-1/2/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*
b^5*f+1/2/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^4*e-5/2/
c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a^2*b*f+1/6/c*x^6*e+
1/4/c*x^4*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^7/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.61195, size = 1, normalized size = 0. \[ \left [\frac{6 \,{\left ({\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d -{\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e +{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} f\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (3 \, c^{4} f x^{8} + 4 \,{\left (c^{4} e - b c^{3} f\right )} x^{6} + 6 \,{\left (c^{4} d - b c^{3} e +{\left (b^{2} c^{2} - a c^{3}\right )} f\right )} x^{4} - 12 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e +{\left (b^{3} c - 2 \, a b c^{2}\right )} f\right )} x^{2} + 6 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e +{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{24 \, \sqrt{b^{2} - 4 \, a c} c^{5}}, -\frac{12 \,{\left ({\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d -{\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} e +{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} f\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (3 \, c^{4} f x^{8} + 4 \,{\left (c^{4} e - b c^{3} f\right )} x^{6} + 6 \,{\left (c^{4} d - b c^{3} e +{\left (b^{2} c^{2} - a c^{3}\right )} f\right )} x^{4} - 12 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e +{\left (b^{3} c - 2 \, a b c^{2}\right )} f\right )} x^{2} + 6 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e +{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{24 \, \sqrt{-b^{2} + 4 \, a c} c^{5}}\right ] \]

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, algorithm="fricas")

[Out]

[1/24*(6*((b^3*c^2 - 3*a*b*c^3)*d - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e + (b^5 -
5*a*b^3*c + 5*a^2*b*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)) + (3*c
^4*f*x^8 + 4*(c^4*e - b*c^3*f)*x^6 + 6*(c^4*d - b*c^3*e + (b^2*c^2 - a*c^3)*f)*x
^4 - 12*(b*c^3*d - (b^2*c^2 - a*c^3)*e + (b^3*c - 2*a*b*c^2)*f)*x^2 + 6*((b^2*c^
2 - a*c^3)*d - (b^3*c - 2*a*b*c^2)*e + (b^4 - 3*a*b^2*c + a^2*c^2)*f)*log(c*x^4
+ b*x^2 + a))*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*c^5), -1/24*(12*((b^3*c^2 -
3*a*b*c^3)*d - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e + (b^5 - 5*a*b^3*c + 5*a^2*b*
c^2)*f)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (3*c^4*f*x^8 +
4*(c^4*e - b*c^3*f)*x^6 + 6*(c^4*d - b*c^3*e + (b^2*c^2 - a*c^3)*f)*x^4 - 12*(b
*c^3*d - (b^2*c^2 - a*c^3)*e + (b^3*c - 2*a*b*c^2)*f)*x^2 + 6*((b^2*c^2 - a*c^3)
*d - (b^3*c - 2*a*b*c^2)*e + (b^4 - 3*a*b^2*c + a^2*c^2)*f)*log(c*x^4 + b*x^2 +
a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)]

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Sympy [A]  time = 149.096, size = 1392, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2
*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2)
) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e +
b**2*c**2*d)/(4*c**5))*log(x**2 + (2*a**3*c**2*f - 4*a**2*b**2*c*f + 3*a**2*b*c*
*2*e - 2*a**2*c**3*d + a*b**4*f - a*b**3*c*e + a*b**2*c**2*d - 8*a*c**5*(-sqrt(-
4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e
- 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**2)) + (a**
2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e + b**2*c**
2*d)/(4*c**5)) + 2*b**2*c**4*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**
3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c
**2*d)/(4*c**5*(4*a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*
c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5)))/(5*a**2*b*c**2*f - 2*a**2*c
**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3
*c**2*d)) + (sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f
+ 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*
a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f -
b**3*c*e + b**2*c**2*d)/(4*c**5))*log(x**2 + (2*a**3*c**2*f - 4*a**2*b**2*c*f +
3*a**2*b*c**2*e - 2*a**2*c**3*d + a*b**4*f - a*b**3*c*e + a*b**2*c**2*d - 8*a*c*
*5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b*
*2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e + b**3*c**2*d)/(4*c**5*(4*a*c - b**
2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c**2*e - a*c**3*d + b**4*f - b**3*c*e
+ b**2*c**2*d)/(4*c**5)) + 2*b**2*c**4*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*f - 2
*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c*e
+ b**3*c**2*d)/(4*c**5*(4*a*c - b**2)) + (a**2*c**2*f - 3*a*b**2*c*f + 2*a*b*c*
*2*e - a*c**3*d + b**4*f - b**3*c*e + b**2*c**2*d)/(4*c**5)))/(5*a**2*b*c**2*f -
2*a**2*c**3*e - 5*a*b**3*c*f + 4*a*b**2*c**2*e - 3*a*b*c**3*d + b**5*f - b**4*c
*e + b**3*c**2*d)) + f*x**8/(8*c) - x**6*(b*f - c*e)/(6*c**2) - x**4*(a*c*f - b*
*2*f + b*c*e - c**2*d)/(4*c**3) + x**2*(2*a*b*c*f - a*c**2*e - b**3*f + b**2*c*e
- b*c**2*d)/(2*c**4)

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GIAC/XCAS [A]  time = 0.3175, size = 413, normalized size = 1.51 \[ \frac{3 \, c^{3} f x^{8} - 4 \, b c^{2} f x^{6} + 4 \, c^{3} x^{6} e + 6 \, c^{3} d x^{4} + 6 \, b^{2} c f x^{4} - 6 \, a c^{2} f x^{4} - 6 \, b c^{2} x^{4} e - 12 \, b c^{2} d x^{2} - 12 \, b^{3} f x^{2} + 24 \, a b c f x^{2} + 12 \, b^{2} c x^{2} e - 12 \, a c^{2} x^{2} e}{24 \, c^{4}} + \frac{{\left (b^{2} c^{2} d - a c^{3} d + b^{4} f - 3 \, a b^{2} c f + a^{2} c^{2} f - b^{3} c e + 2 \, a b c^{2} e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{5}} - \frac{{\left (b^{3} c^{2} d - 3 \, a b c^{3} d + b^{5} f - 5 \, a b^{3} c f + 5 \, a^{2} b c^{2} f - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{5}} \]

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, algorithm="giac")

[Out]

1/24*(3*c^3*f*x^8 - 4*b*c^2*f*x^6 + 4*c^3*x^6*e + 6*c^3*d*x^4 + 6*b^2*c*f*x^4 -
6*a*c^2*f*x^4 - 6*b*c^2*x^4*e - 12*b*c^2*d*x^2 - 12*b^3*f*x^2 + 24*a*b*c*f*x^2 +
12*b^2*c*x^2*e - 12*a*c^2*x^2*e)/c^4 + 1/4*(b^2*c^2*d - a*c^3*d + b^4*f - 3*a*b
^2*c*f + a^2*c^2*f - b^3*c*e + 2*a*b*c^2*e)*ln(c*x^4 + b*x^2 + a)/c^5 - 1/2*(b^3
*c^2*d - 3*a*b*c^3*d + b^5*f - 5*a*b^3*c*f + 5*a^2*b*c^2*f - b^4*c*e + 4*a*b^2*c
^2*e - 2*a^2*c^3*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)
*c^5)