3.2.7 \(\int \frac {d+e x+f x^2+g x^3}{(a+b x^2+c x^4)^{5/2}} \, dx\) [107]

Optimal. Leaf size=680 \[ \frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}-\frac {b e-2 a g+(2 c e-b g) x^2}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {4 (2 c e-b g) \left (b+2 c x^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {x \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+a b^3 f+4 a^2 b c f+c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x \sqrt {a+b x^2+c x^4}}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (2 b^2 d-3 \sqrt {a} b \sqrt {c} d-10 a c d+a b f+6 a^{3/2} \sqrt {c} f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{7/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]

[Out]

1/3*x*(b^2*d-2*a*c*d-a*b*f+c*(-2*a*f+b*d)*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(3/2)+1/3*(-b*e+2*a*g-(-b*g+2*c*
e)*x^2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(3/2)+4/3*(-b*g+2*c*e)*(2*c*x^2+b)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)+1
/3*x*(2*b^4*d-17*a*b^2*c*d+20*a^2*c^2*d+a*b^3*f+4*a^2*b*c*f+c*(12*a^2*c*f+a*b^2*f-16*a*b*c*d+2*b^3*d)*x^2)/a^2
/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/3*(12*a^2*c*f+a*b^2*f-16*a*b*c*d+2*b^3*d)*x*c^(1/2)*(c*x^4+b*x^2+a)^(1
/2)/a^2/(-4*a*c+b^2)^2/(a^(1/2)+x^2*c^(1/2))+1/3*c^(1/4)*(12*a^2*c*f+a*b^2*f-16*a*b*c*d+2*b^3*d)*(cos(2*arctan
(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*
(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(7/4)/(-4
*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/6*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*
x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*
(2*b^2*d-10*a*c*d+a*b*f+6*a^(3/2)*f*c^(1/2)-3*b*d*a^(1/2)*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(
1/2)/a^(7/4)/(-4*a*c+b^2)/(b-2*a^(1/2)*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1687, 1192, 1211, 1117, 1209, 1261, 652, 627} \begin {gather*} -\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (6 a^{3/2} \sqrt {c} f-3 \sqrt {a} b \sqrt {c} d+a b f-10 a c d+2 b^2 d\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{7/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} x \sqrt {a+b x^2+c x^4} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {4 \left (b+2 c x^2\right ) (2 c e-b g)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {-2 a g+x^2 (2 c e-b g)+b e}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]

[Out]

(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(3*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^(3/2)) - (b*e - 2*a
*g + (2*c*e - b*g)*x^2)/(3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^(3/2)) + (4*(2*c*e - b*g)*(b + 2*c*x^2))/(3*(b^2
- 4*a*c)^2*Sqrt[a + b*x^2 + c*x^4]) + (x*(2*b^4*d - 17*a*b^2*c*d + 20*a^2*c^2*d + a*b^3*f + 4*a^2*b*c*f + c*(2
*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*x^2))/(3*a^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x^2 + c*x^4]) - (Sqrt[c]*(
2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*x*Sqrt[a + b*x^2 + c*x^4])/(3*a^2*(b^2 - 4*a*c)^2*(Sqrt[a] + Sqrt
[c]*x^2)) + (c^(1/4)*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c
*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(3*a^(
7/4)*(b^2 - 4*a*c)^2*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(2*b^2*d - 3*Sqrt[a]*b*Sqrt[c]*d - 10*a*c*d + a*b*f +
6*a^(3/2)*Sqrt[c]*f)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*
ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(6*a^(7/4)*(b - 2*Sqrt[a]*Sqrt[c])*(b^2 - 4*a*c)*Sq
rt[a + b*x^2 + c*x^4])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps

\begin {align*} \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx &=\int \frac {d+f x^2}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx+\int \frac {x \left (e+g x^2\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {1}{2} \text {Subst}\left (\int \frac {e+g x}{\left (a+b x+c x^2\right )^{5/2}} \, dx,x,x^2\right )-\frac {\int \frac {-2 b^2 d+10 a c d-a b f-3 c (b d-2 a f) x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{3 a \left (b^2-4 a c\right )}\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}-\frac {b e-2 a g+(2 c e-b g) x^2}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {x \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+a b^3 f+4 a^2 b c f+c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\int \frac {-a c \left (b^2 d-20 a c d+8 a b f\right )-c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a^2 \left (b^2-4 a c\right )^2}-\frac {(2 (2 c e-b g)) \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{3 \left (b^2-4 a c\right )}\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}-\frac {b e-2 a g+(2 c e-b g) x^2}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {4 (2 c e-b g) \left (b+2 c x^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {x \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+a b^3 f+4 a^2 b c f+c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a^{3/2} \left (b^2-4 a c\right )^2}-\frac {\left (\sqrt {c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f+\sqrt {a} \sqrt {c} \left (b^2 d-20 a c d+8 a b f\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a^{3/2} \left (b^2-4 a c\right )^2}\\ &=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}-\frac {b e-2 a g+(2 c e-b g) x^2}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {4 (2 c e-b g) \left (b+2 c x^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {x \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+a b^3 f+4 a^2 b c f+c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x \sqrt {a+b x^2+c x^4}}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f+\sqrt {a} \sqrt {c} \left (b^2 d-20 a c d+8 a b f\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.61, size = 598, normalized size = 0.88 \begin {gather*} \frac {-4 a \left (b^2-4 a c\right ) \left (-2 a^2 g-b d x \left (b+c x^2\right )+2 a c x (d+x (e+f x))+a b (e+x (f-g x))\right )+4 \left (a+b x^2+c x^4\right ) \left (2 b^3 d x \left (b+c x^2\right )+a b x \left (-17 b c d+b^2 f-16 c^2 d x^2+b c f x^2\right )+4 a^2 \left (-b^2 g+c^2 x (5 d+x (4 e+3 f x))+b c (2 e+x (f-2 g x))\right )\right )+\frac {i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (a+b x^2+c x^4\right ) \left (-\left (\left (-b+\sqrt {b^2-4 a c}\right ) \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )+\left (-2 b^4 d+b^3 \left (2 \sqrt {b^2-4 a c} d-a f\right )+4 a b c \left (-4 \sqrt {b^2-4 a c} d+a f\right )+a b^2 \left (18 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (-10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{12 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]

[Out]

(-4*a*(b^2 - 4*a*c)*(-2*a^2*g - b*d*x*(b + c*x^2) + 2*a*c*x*(d + x*(e + f*x)) + a*b*(e + x*(f - g*x))) + 4*(a
+ b*x^2 + c*x^4)*(2*b^3*d*x*(b + c*x^2) + a*b*x*(-17*b*c*d + b^2*f - 16*c^2*d*x^2 + b*c*f*x^2) + 4*a^2*(-(b^2*
g) + c^2*x*(5*d + x*(4*e + 3*f*x)) + b*c*(2*e + x*(f - 2*g*x)))) + (I*Sqrt[2]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*
c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4)*(-((-b + Sqrt[
b^2 - 4*a*c])*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 -
4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (-2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d - a*f
) + 4*a*b*c*(-4*Sqrt[b^2 - 4*a*c]*d + a*f) + a*b^2*(18*c*d + Sqrt[b^2 - 4*a*c]*f) + 4*a^2*c*(-10*c*d + 3*Sqrt[
b^2 - 4*a*c]*f))*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b -
Sqrt[b^2 - 4*a*c])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(12*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1394\) vs. \(2(654)=1308\).
time = 0.09, size = 1395, normalized size = 2.05

method	result	size
elliptic	\(\frac {\left (\frac {\left (2 f a -b d \right ) x^{3}}{3 c a \left (4 a c -b^{2}\right )}-\frac {\left (b g -2 c e \right ) x^{2}}{3 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\left (a b f +2 a c d -b^{2} d \right ) x}{3 a \left (4 a c -b^{2}\right ) c^{2}}-\frac {2 a g -e b}{3 \left (4 a c -b^{2}\right ) c^{2}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right )^{2}}-\frac {2 c \left (-\frac {\left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) x^{3}}{6 a^{2} \left (4 a c -b^{2}\right )^{2}}+\frac {4 \left (b g -2 c e \right ) x^{2}}{3 \left (4 a c -b^{2}\right )^{2}}-\frac {\left (4 a^{2} b c f +20 a^{2} c^{2} d +a \,b^{3} f -17 a \,b^{2} c d +2 b^{4} d \right ) x}{6 a^{2} \left (4 a c -b^{2}\right )^{2} c}+\frac {2 b \left (b g -2 c e \right )}{3 \left (4 a c -b^{2}\right )^{2} c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {a b f -10 a c d +2 b^{2} d}{3 \left (4 a c -b^{2}\right ) a^{2}}-\frac {4 a^{2} b c f +20 a^{2} c^{2} d +a \,b^{3} f -17 a \,b^{2} c d +2 b^{4} d}{3 a^{2} \left (4 a c -b^{2}\right )^{2}}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {c \left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \left (4 a c -b^{2}\right )^{2} a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(829\)
default	\(\text {Expression too large to display}\)	\(1395\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*g*(8*b*c^2*x^6+12*b^2*c*x^4+12*a*b*c*x^2+3*b^3*x^2+8*a^2*c+2*a*b^2)/(c*x^4+b*x^2+a)^(3/2)/(16*a^2*c^2-8*a
*b^2*c+b^4)+f*((2/3/c/(4*a*c-b^2)*x^3+1/3*b/(4*a*c-b^2)/c^2*x)*(c*x^4+b*x^2+a)^(1/2)/(x^4+b/c*x^2+a/c)^2-2*c*(
-1/6*(12*a*c+b^2)/(4*a*c-b^2)^2/a*x^3-1/6*b*(4*a*c+b^2)/a/(4*a*c-b^2)^2/c*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/4*(
-1/3*b/a/(4*a*c-b^2)-1/3*b*(4*a*c+b^2)/a/(4*a*c-b^2)^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4
*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2
^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/6*c*(12*a*c+b^2)/(4*
a*c-b^2)^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c
+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b
^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(
1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))))+1/3*e*(16*c^3*x^6+24*b*c^2*x^4+24*a*c^2*x^2+6*
b^2*c*x^2+12*a*b*c-b^3)/(c*x^4+b*x^2+a)^(3/2)/(16*a^2*c^2-8*a*b^2*c+b^4)+d*((-1/3*b/a/(4*a*c-b^2)/c*x^3+1/3*(2
*a*c-b^2)/a/(4*a*c-b^2)/c^2*x)*(c*x^4+b*x^2+a)^(1/2)/(x^4+b/c*x^2+a/c)^2-2*c*(1/3*b*(8*a*c-b^2)/(4*a*c-b^2)^2/
a^2*x^3-1/6*(20*a^2*c^2-17*a*b^2*c+2*b^4)/a^2/(4*a*c-b^2)^2/c*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/4*(2/3*(5*a*c-b
^2)/(4*a*c-b^2)/a^2-1/3*(20*a^2*c^2-17*a*b^2*c+2*b^4)/a^2/(4*a*c-b^2)^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(
1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*
EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/3*b
*c*(8*a*c-b^2)/(4*a*c-b^2)^2/a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(
1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^
(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)
*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((g*x^3 + f*x^2 + x*e + d)/(c*x^4 + b*x^2 + a)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (652) = 1304\).
time = 0.12, size = 1948, normalized size = 2.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(sqrt(1/2)*((2*(b^4*c^2 - 8*a*b^2*c^3)*d + (a*b^3*c^2 + 12*a^2*b*c^3)*f)*x^8 + 2*(2*(b^5*c - 8*a*b^3*c^2)
*d + (a*b^4*c + 12*a^2*b^2*c^2)*f)*x^6 + (2*(b^6 - 6*a*b^4*c - 16*a^2*b^2*c^2)*d + (a*b^5 + 14*a^2*b^3*c + 24*
a^3*b*c^2)*f)*x^4 + 2*(2*(a*b^5 - 8*a^2*b^3*c)*d + (a^2*b^4 + 12*a^3*b^2*c)*f)*x^2 + 2*(a^2*b^4 - 8*a^3*b^2*c)
*d + (a^3*b^3 + 12*a^4*b*c)*f - ((2*(a*b^3*c^2 - 8*a^2*b*c^3)*d + (a^2*b^2*c^2 + 12*a^3*c^3)*f)*x^8 + 2*(2*(a*
b^4*c - 8*a^2*b^2*c^2)*d + (a^2*b^3*c + 12*a^3*b*c^2)*f)*x^6 + (2*(a*b^5 - 6*a^2*b^3*c - 16*a^3*b*c^2)*d + (a^
2*b^4 + 14*a^3*b^2*c + 24*a^4*c^2)*f)*x^4 + 2*(2*(a^2*b^4 - 8*a^3*b^2*c)*d + (a^3*b^3 + 12*a^4*b*c)*f)*x^2 + 2
*(a^3*b^3 - 8*a^4*b*c)*d + (a^4*b^2 + 12*a^5*c)*f)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)
/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a
*c)/a^2) + b^2 - 2*a*c)/(a*c)) + sqrt(1/2)*(((4*(5*a^2*b + 4*a*b^2)*c^3 - (a*b^3 + 2*b^4)*c^2)*d - (12*a^2*b*c
^3 + (8*a^2*b^2 + a*b^3)*c^2)*f)*x^8 + 2*((4*(5*a^2*b^2 + 4*a*b^3)*c^2 - (a*b^4 + 2*b^5)*c)*d - (12*a^2*b^2*c^
2 + (8*a^2*b^3 + a*b^4)*c)*f)*x^6 - ((a*b^5 + 2*b^6 - 8*(5*a^3*b + 4*a^2*b^2)*c^2 - 6*(3*a^2*b^3 + 2*a*b^4)*c)
*d + (8*a^2*b^4 + a*b^5 + 24*a^3*b*c^2 + 2*(8*a^3*b^2 + 7*a^2*b^3)*c)*f)*x^4 - 2*((a^2*b^4 + 2*a*b^5 - 4*(5*a^
3*b^2 + 4*a^2*b^3)*c)*d + (8*a^3*b^3 + a^2*b^4 + 12*a^3*b^2*c)*f)*x^2 - (a^3*b^3 + 2*a^2*b^4 - 4*(5*a^4*b + 4*
a^3*b^2)*c)*d - (8*a^4*b^2 + a^3*b^3 + 12*a^4*b*c)*f + (((4*(5*a^3 - 4*a^2*b)*c^3 - (a^2*b^2 - 2*a*b^3)*c^2)*d
+ (12*a^3*c^3 - (8*a^3*b - a^2*b^2)*c^2)*f)*x^8 + 2*((4*(5*a^3*b - 4*a^2*b^2)*c^2 - (a^2*b^3 - 2*a*b^4)*c)*d
+ (12*a^3*b*c^2 - (8*a^3*b^2 - a^2*b^3)*c)*f)*x^6 - ((a^2*b^4 - 2*a*b^5 - 8*(5*a^4 - 4*a^3*b)*c^2 - 6*(3*a^3*b
^2 - 2*a^2*b^3)*c)*d + (8*a^3*b^3 - a^2*b^4 - 24*a^4*c^2 + 2*(8*a^4*b - 7*a^3*b^2)*c)*f)*x^4 - 2*((a^3*b^3 - 2
*a^2*b^4 - 4*(5*a^4*b - 4*a^3*b^2)*c)*d + (8*a^4*b^2 - a^3*b^3 - 12*a^4*b*c)*f)*x^2 - (a^4*b^2 - 2*a^3*b^3 - 4
*(5*a^5 - 4*a^4*b)*c)*d - (8*a^5*b - a^4*b^2 - 12*a^5*c)*f)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2
- 4*a*c)/a^2) - b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((
b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) - 2*((2*(a*b^3*c^2 - 8*a^2*b*c^3)*d + (a^2*b^2*c^2 + 12*a^3*c^3)*f)*x^
7 + 8*(2*a^3*c^3*e - a^3*b*c^2*g)*x^6 + ((4*a*b^4*c - 33*a^2*b^2*c^2 + 20*a^3*c^3)*d + 2*(a^2*b^3*c + 8*a^3*b*
c^2)*f)*x^5 + 12*(2*a^3*b*c^2*e - a^3*b^2*c*g)*x^4 + (2*(a*b^5 - 7*a^2*b^3*c)*d + (a^2*b^4 + 3*a^3*b^2*c + 20*
a^4*c^2)*f)*x^3 + 3*(2*(a^3*b^2*c + 4*a^4*c^2)*e - (a^3*b^3 + 4*a^4*b*c)*g)*x^2 - (a^3*b^3 - 12*a^4*b*c)*e - 2
*(a^4*b^2 + 4*a^5*c)*g + (8*a^4*b*c*f + (3*a^2*b^4 - 23*a^3*b^2*c + 28*a^4*c^2)*d)*x)*sqrt(c*x^4 + b*x^2 + a))
/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^8 + 2*(a^3*b^5*c - 8*a^4*b
^3*c^2 + 16*a^5*b*c^3)*x^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^4 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^
2)*x^2)

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Sympy [F(-1)] Timed out
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((g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**(5/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x^3 + f*x^2 + x*e + d)/(c*x^4 + b*x^2 + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {g\,x^3+f\,x^2+e\,x+d}{{\left (c\,x^4+b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x)

[Out]

int((d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2), x)

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