3.1.26 \(\int \frac {(A+B x^2) (d+e x^2)}{(a+b x^2+c x^4)^{3/2}} \, dx\) [26]

Optimal. Leaf size=481 \[ -\frac {x \left (a B (b d-2 a e)-A \left (b^2 d-2 a c d-a b e\right )-(A c (b d-2 a e)-a B (2 c d-b e)) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {(A c (b d-2 a e)-a B (2 c d-b e)) x \sqrt {a+b x^2+c x^4}}{a \sqrt {c} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(A c (b d-2 a e)-a B (2 c d-b e)) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{3/4} \sqrt {a+b x^2+c x^4}} \]

[Out]

-x*(a*B*(-2*a*e+b*d)-A*(-a*b*e-2*a*c*d+b^2*d)-(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*c*d))*x^2)/a/(-4*a*c+b^2)/(c*x^4+b
*x^2+a)^(1/2)-(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*c*d))*x*(c*x^4+b*x^2+a)^(1/2)/a/(-4*a*c+b^2)/c^(1/2)/(a^(1/2)+x^2*
c^(1/2))+(A*c*(-2*a*e+b*d)-a*B*(-b*e+2*c*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^(3/4)/c^(3/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*(cos(2*a
rctan(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))*(B*a^(1/2)-A*c^(1/2))*(-e*a^(1/2)+d*c^(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^(3/4)/c^(3/4)/(b-2*a^(1/2)*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 481, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1692, 1211, 1117, 1209} \begin {gather*} \frac {\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 \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 ) (A c (b d-2 a e)-a B (2 c d-b e))}{a^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {c} d-\sqrt {a} e\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 )}{2 a^{3/4} c^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4} (A c (b d-2 a e)-a B (2 c d-b e))}{a \sqrt {c} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {x \left (-A \left (-a b e-2 a c d+b^2 d\right )-\left (x^2 (A c (b d-2 a e)-a B (2 c d-b e))\right )+a B (b d-2 a e)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

-((x*(a*B*(b*d - 2*a*e) - A*(b^2*d - 2*a*c*d - a*b*e) - (A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*x^2))/(a*(b^2
- 4*a*c)*Sqrt[a + b*x^2 + c*x^4])) - ((A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*x*Sqrt[a + b*x^2 + c*x^4])/(a*Sq
rt[c]*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)) + ((A*c*(b*d - 2*a*e) - a*B*(2*c*d - b*e))*(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])/(a^(3/4)*c^(3/4)*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]*d -
Sqrt[a]*e)*(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])/(2*a^(3/4)*(b - 2*Sqrt[a]*Sqrt[c])*c^(3/4)*Sqrt[a + b*x^2 + c
*x^4])

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 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 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin {align*} \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {x \left (a B (b d-2 a e)-A \left (b^2 d-2 a c d-a b e\right )-(A c (b d-2 a e)-a B (2 c d-b e)) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {-a (b B d-2 A c d+A b e-2 a B e)+(A c (b d-2 a e)-a B (2 c d-b e)) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac {x \left (a B (b d-2 a e)-A \left (b^2 d-2 a c d-a b e\right )-(A c (b d-2 a e)-a B (2 c d-b e)) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {c}}+\frac {(A c (b d-2 a e)-a B (2 c d-b e)) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \sqrt {c} \left (b^2-4 a c\right )}\\ &=-\frac {x \left (a B (b d-2 a e)-A \left (b^2 d-2 a c d-a b e\right )-(A c (b d-2 a e)-a B (2 c d-b e)) x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {(A c (b d-2 a e)-a B (2 c d-b e)) x \sqrt {a+b x^2+c x^4}}{a \sqrt {c} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {(A c (b d-2 a e)-a B (2 c d-b e)) \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 )}{a^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\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 )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{3/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.69, size = 597, normalized size = 1.24 \begin {gather*} \frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (a B \left (-2 a e+2 c d x^2+b \left (d-e x^2\right )\right )+A \left (-b^2 d+b \left (a e-c d x^2\right )+2 a c \left (d+e x^2\right )\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) (A c (b d-2 a e)+a B (-2 c d+b e)) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} 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 )-i \left (A c \left (-b^2 d+4 a c d+b \sqrt {b^2-4 a c} d-2 a \sqrt {b^2-4 a c} e\right )+a B \left (b \left (-b+\sqrt {b^2-4 a c}\right ) e+c \left (-2 \sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} 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 )}{4 a c \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(a*B*(-2*a*e + 2*c*d*x^2 + b*(d - e*x^2)) + A*(-(b^2*d) + b*(a*e - c*d*
x^2) + 2*a*c*(d + e*x^2))) + I*(-b + Sqrt[b^2 - 4*a*c])*(A*c*(b*d - 2*a*e) + a*B*(-2*c*d + b*e))*Sqrt[(b + Sqr
t[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 -
4*a*c])]*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])] - I*(A*c*(-(b^2*d) + 4*a*c*d + b*Sqrt[b^2 - 4*a*c]*d - 2*a*Sqrt[b^2 - 4*a*c]*e) + a*B*(b*(-b + Sqr
t[b^2 - 4*a*c])*e + c*(-2*Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2
- 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqr
t[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(4*a*c*(-b^2 + 4*a*c)*Sqrt[
c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1389\) vs. \(2(471)=942\).
time = 0.04, size = 1390, normalized size = 2.89

method	result	size
elliptic	\(-\frac {2 c \left (-\frac {\left (2 a c e A -A b c d -B a b e +2 a c d B \right ) x^{3}}{2 c a \left (4 a c -b^{2}\right )}-\frac {\left (A a b e +2 a c d A -A \,b^{2} d -2 e \,a^{2} B +B a b d \right ) x}{2 c a \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {B e}{c}+\frac {A c d -a B e}{a c}-\frac {A a b e +2 a c d A -A \,b^{2} d -2 e \,a^{2} B +B a b d}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (2 a c e A -A b c d -B a b e +2 a c d B \right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(593\)
default	\(\text {Expression too large to display}\)	\(1390\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*e*(-2*c*(1/2*b/c/(4*a*c-b^2)*x^3+a/c/(4*a*c-b^2)*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/2*a/(4*a*c-b^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))-1/2*b/(4*a*c-b^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))*(Ellipti
cF(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))))+(A*e+B*d)*(-2*c
*(-1/(4*a*c-b^2)*x^3-1/2*b/(4*a*c-b^2)/c*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)-1/4*b/(4*a*c-b^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))+c/(4*a*c-b^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))))+A*d*(-2*c*(1/2/a*b/(4*a*c-b^2
)*x^3-1/2*(2*a*c-b^2)/a/(4*a*c-b^2)/c*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/4*(1/a-(2*a*c-b^2)/a/(4*a*c-b^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))-1/2*b*c/(4*a*c-b^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))*(El
lipticF(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))-Ellipti
cE(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((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(x^2*e + d)/(c*x^4 + b*x^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral((A + B*x**2)*(d + e*x**2)/(a + b*x**2 + c*x**4)**(3/2), x)

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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((B*x^2+A)*(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(x^2*e + d)/(c*x^4 + b*x^2 + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x)

[Out]

int(((A + B*x^2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2), x)

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