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

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

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Rubi [A]  time = 0.85, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1663, 1628, 634, 618, 206, 628} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c^3 e-4 a b^2 c^2 e-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)+b^4 c e+b^5 (-f)\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^2 c e+b^3 (-f)\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^3 c e+b^4 (-f)\right )}{4 c^5}+\frac {x^6 (c e-b f)}{6 c^2}+\frac {f x^8}{8 c} \end {gather*}

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))*Log[a + b*x^2 + c*x^4])/(4*c^5)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {x^7 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 \left (d+e x+f x^2\right )}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)}{c^4}+\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x}{c^3}+\frac {(c e-b f) x^2}{c^2}+\frac {f x^3}{c}+\frac {-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (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)\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac {(c e-b f) x^6}{6 c^2}+\frac {f x^8}{8 c}+\frac {\operatorname {Subst}\left (\int \frac {-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (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)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac {(c e-b f) x^6}{6 c^2}+\frac {f x^8}{8 c}+\frac {\left (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)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^5}-\frac {\left (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)\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^5}\\ &=\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac {(c e-b f) x^6}{6 c^2}+\frac {f x^8}{8 c}-\frac {\left (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)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}-\frac {\left (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)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^5}\\ &=\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac {(c e-b f) x^6}{6 c^2}+\frac {f x^8}{8 c}-\frac {\left (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)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^5 \sqrt {b^2-4 a c}}-\frac {\left (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)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 260, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.56, size = 900, normalized size = 3.30 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} f x^{8} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} e - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} f\right )} x^{6} + 6 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e + {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} f\right )} x^{4} - 12 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e + {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} f\right )} x^{2} + 6 \, \sqrt {b^{2} - 4 \, a c} {\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 {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + 6 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e + {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{24 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} f x^{8} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} e - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} f\right )} x^{6} + 6 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e + {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} f\right )} x^{4} - 12 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e + {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} f\right )} x^{2} + 12 \, \sqrt {-b^{2} + 4 \, a c} {\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 ) + 6 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e + {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{24 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}\right ] \end {gather*}

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, algorithm="fricas")

[Out]

[1/24*(3*(b^2*c^4 - 4*a*c^5)*f*x^8 + 4*((b^2*c^4 - 4*a*c^5)*e - (b^3*c^3 - 4*a*b*c^4)*f)*x^6 + 6*((b^2*c^4 - 4
*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*f)*x^4 - 12*((b^3*c^3 - 4*a*b*c^4)*d
- (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e + (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*f)*x^2 + 6*sqrt(b^2 - 4*a*c)*((
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((2*c^2*x
^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + 6*((b^4*c^2 - 5*a*b^2*c
^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*f)*
log(c*x^4 + b*x^2 + a))/(b^2*c^5 - 4*a*c^6), 1/24*(3*(b^2*c^4 - 4*a*c^5)*f*x^8 + 4*((b^2*c^4 - 4*a*c^5)*e - (b
^3*c^3 - 4*a*b*c^4)*f)*x^6 + 6*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e + (b^4*c^2 - 5*a*b^2*c^3 + 4*a
^2*c^4)*f)*x^4 - 12*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e + (b^5*c - 6*a*b^3*c^2 +
8*a^2*b*c^3)*f)*x^2 + 12*sqrt(-b^2 + 4*a*c)*((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)) + 6*((b^4*c^2 - 5*a*
b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3
)*f)*log(c*x^4 + b*x^2 + a))/(b^2*c^5 - 4*a*c^6)]

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giac [A]  time = 1.87, size = 306, normalized size = 1.12 \begin {gather*} \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 )} \log \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}} \end {gather*}

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, 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)*log(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)

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maple [B]  time = 0.01, size = 622, normalized size = 2.28 \begin {gather*} \frac {f \,x^{8}}{8 c}-\frac {b f \,x^{6}}{6 c^{2}}+\frac {e \,x^{6}}{6 c}-\frac {a f \,x^{4}}{4 c^{2}}+\frac {b^{2} f \,x^{4}}{4 c^{3}}-\frac {b e \,x^{4}}{4 c^{2}}+\frac {d \,x^{4}}{4 c}-\frac {5 a^{2} b f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{3}}+\frac {a^{2} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {5 a \,b^{3} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{4}}-\frac {2 a \,b^{2} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {3 a b d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {a b f \,x^{2}}{c^{3}}-\frac {a e \,x^{2}}{2 c^{2}}-\frac {b^{5} f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{5}}+\frac {b^{4} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{4}}-\frac {b^{3} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{3}}-\frac {b^{3} f \,x^{2}}{2 c^{4}}+\frac {b^{2} e \,x^{2}}{2 c^{3}}-\frac {b d \,x^{2}}{2 c^{2}}+\frac {a^{2} f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}}-\frac {3 a \,b^{2} f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{4}}+\frac {a b e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c^{3}}-\frac {a d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}}+\frac {b^{4} f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{5}}-\frac {b^{3} e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{4}}+\frac {b^{2} d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}} \end {gather*}

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 1.60, size = 2972, normalized size = 10.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^6*(e/(6*c) - (b*f)/(6*c^2)) - x^4*((b*(e/c - (b*f)/c^2))/(4*c) - d/(4*c) + (a*f)/(4*c^2)) - x^2*((a*(e/c - (
b*f)/c^2))/(2*c) - (b*((b*(e/c - (b*f)/c^2))/c - d/c + (a*f)/c^2))/(2*c)) + (f*x^8)/(8*c) - (log(a + b*x^2 + c
*x^4)*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f - 10*a*
b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(2*(16*a*c^6 - 4*b^2*c^5)) + (atan((2*c^8*(4*a*c - b^2)*(x^2*(((
((4*a^2*c^8*e - 6*b^3*c^7*d + 6*b^4*c^6*e - 6*b^5*c^5*f + 10*a*b*c^8*d - 16*a*b^2*c^7*e + 22*a*b^3*c^6*f - 14*
a^2*b*c^7*f)/c^8 - (4*b*c^2*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f
- 14*a*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(16*a*c^6 - 4*b^2*c^5))*(b^5*f - 2*a^2*c^3
*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f))/(8*c^5*(4*a*c - b^2)^(1
/2)) - (b*(b^5*f - 2*a^2*c^3*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2
*f)*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f - 10*a*b^
2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(2*c^3*(4*a*c - b^2)^(1/2)*(16*a*c^6 - 4*b^2*c^5)))/a - (b*((((4*a
^2*c^8*e - 6*b^3*c^7*d + 6*b^4*c^6*e - 6*b^5*c^5*f + 10*a*b*c^8*d - 16*a*b^2*c^7*e + 22*a*b^3*c^6*f - 14*a^2*b
*c^7*f)/c^8 - (4*b*c^2*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*
a*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(16*a*c^6 - 4*b^2*c^5))*(2*b^6*f + 8*a^2*c^4*d
+ 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b^3*c^2*e -
16*a^2*b*c^3*e))/(2*(16*a*c^6 - 4*b^2*c^5)) - (b^9*f^2 + b^5*c^4*d^2 + b^7*c^2*e^2 - 3*a*b^3*c^5*d^2 + 2*a^2*b
*c^6*d^2 - 5*a*b^5*c^3*e^2 - 2*a^3*b*c^5*e^2 + 3*a^4*b*c^4*f^2 - 2*b^8*c*e*f + 7*a^2*b^3*c^4*e^2 + 16*a^2*b^5*
c^2*f^2 - 13*a^3*b^3*c^3*f^2 - 7*a*b^7*c*f^2 + a^3*c^6*d*e - 2*b^6*c^3*d*e - a^4*c^5*e*f + 2*b^7*c^2*d*f + 8*a
*b^4*c^4*d*e - 10*a*b^5*c^3*d*f - 5*a^3*b*c^5*d*f + 12*a*b^6*c^2*e*f - 8*a^2*b^2*c^5*d*e + 14*a^2*b^3*c^4*d*f
- 22*a^2*b^4*c^3*e*f + 12*a^3*b^2*c^4*e*f)/c^8 + (b*(b^5*f - 2*a^2*c^3*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d -
5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f)^2)/(2*c^8*(4*a*c - b^2))))/(2*a*(4*a*c - b^2)^(1/2))) - ((((8*a^
3*c^7*f - 8*a^2*c^8*d - 24*a^2*b^2*c^6*f + 8*a*b^2*c^7*d - 8*a*b^3*c^6*e + 16*a^2*b*c^7*e + 8*a*b^4*c^5*f)/c^8
+ (8*a*c^2*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f -
10*a*b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(16*a*c^6 - 4*b^2*c^5))*(b^5*f - 2*a^2*c^3*e + b^3*c^2*d -
b^4*c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f))/(8*c^5*(4*a*c - b^2)^(1/2)) + (a*(b^5*f
- 2*a^2*c^3*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f)*(2*b^6*f + 8
*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b
^3*c^2*e - 16*a^2*b*c^3*e))/(c^3*(4*a*c - b^2)^(1/2)*(16*a*c^6 - 4*b^2*c^5)))/a + (b*((a*b^8*f^2 + a^3*c^6*d^2
+ a^5*c^4*f^2 + a*b^4*c^4*d^2 + a*b^6*c^2*e^2 - 6*a^2*b^6*c*f^2 - 2*a^2*b^2*c^5*d^2 - 4*a^2*b^4*c^3*e^2 + 4*a
^3*b^2*c^4*e^2 + 11*a^3*b^4*c^2*f^2 - 6*a^4*b^2*c^3*f^2 - 2*a^4*c^5*d*f - 2*a*b^5*c^3*d*e - 4*a^3*b*c^5*d*e +
2*a*b^6*c^2*d*f + 4*a^4*b*c^4*e*f + 6*a^2*b^3*c^4*d*e - 8*a^2*b^4*c^3*d*f + 8*a^3*b^2*c^4*d*f + 10*a^2*b^5*c^2
*e*f - 14*a^3*b^3*c^3*e*f - 2*a*b^7*c*e*f)/c^8 + (((8*a^3*c^7*f - 8*a^2*c^8*d - 24*a^2*b^2*c^6*f + 8*a*b^2*c^7
*d - 8*a*b^3*c^6*e + 16*a^2*b*c^7*e + 8*a*b^4*c^5*f)/c^8 + (8*a*c^2*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a
^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(1
6*a*c^6 - 4*b^2*c^5))*(2*b^6*f + 8*a^2*c^4*d + 2*b^4*c^2*d - 8*a^3*c^3*f - 2*b^5*c*e + 26*a^2*b^2*c^2*f - 14*a
*b^4*c*f - 10*a*b^2*c^3*d + 12*a*b^3*c^2*e - 16*a^2*b*c^3*e))/(2*(16*a*c^6 - 4*b^2*c^5)) - (a*(b^5*f - 2*a^2*c
^3*e + b^3*c^2*d - b^4*c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f)^2)/(c^8*(4*a*c - b^2))
))/(2*a*(4*a*c - b^2)^(1/2))))/(b^10*f^2 + 4*a^4*c^6*e^2 + b^6*c^4*d^2 + b^8*c^2*e^2 - 6*a*b^4*c^5*d^2 - 8*a*b
^6*c^3*e^2 - 2*b^9*c*e*f + 9*a^2*b^2*c^6*d^2 + 20*a^2*b^4*c^4*e^2 - 16*a^3*b^2*c^5*e^2 + 35*a^2*b^6*c^2*f^2 -
50*a^3*b^4*c^3*f^2 + 25*a^4*b^2*c^4*f^2 - 10*a*b^8*c*f^2 - 2*b^7*c^3*d*e + 2*b^8*c^2*d*f + 14*a*b^5*c^4*d*e +
12*a^3*b*c^6*d*e - 16*a*b^6*c^3*d*f + 18*a*b^7*c^2*e*f - 20*a^4*b*c^5*e*f - 28*a^2*b^3*c^5*d*e + 40*a^2*b^4*c^
4*d*f - 30*a^3*b^2*c^5*d*f - 54*a^2*b^5*c^3*e*f + 60*a^3*b^3*c^4*e*f))*(b^5*f - 2*a^2*c^3*e + b^3*c^2*d - b^4*
c*e - 3*a*b*c^3*d - 5*a*b^3*c*f + 4*a*b^2*c^2*e + 5*a^2*b*c^2*f))/(2*c^5*(4*a*c - b^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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