### 3.139 $$\int \frac{x^2 (a+b x^2+c x^4)}{\sqrt{d-e x} \sqrt{d+e x}} \, dx$$

Optimal. Leaf size=216 $\frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6}-\frac{x^3 \sqrt{d-e x} \sqrt{d+e x} \left (6 b e^2+5 c d^2\right )}{24 e^4}+\frac{c x^5 (e x-d) \sqrt{d+e x}}{6 e^2 \sqrt{d-e x}}$

[Out]

-((5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*x*Sqrt[d - e*x]*Sqrt[d + e*x])/(16*e^6) - ((5*c*d^2 + 6*b*e^2)*x^3*Sqrt[d
- e*x]*Sqrt[d + e*x])/(24*e^4) + (c*x^5*(-d + e*x)*Sqrt[d + e*x])/(6*e^2*Sqrt[d - e*x]) + (d^2*(5*c*d^4 + 6*b*
d^2*e^2 + 8*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^7*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.205081, antiderivative size = 245, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.171, Rules used = {520, 1267, 459, 321, 217, 203} $-\frac{x \left (d^2-e^2 x^2\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x^3 \left (d^2-e^2 x^2\right ) \left (6 b e^2+5 c d^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-((5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*x*(d^2 - e^2*x^2))/(16*e^6*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((5*c*d^2 + 6*b*
e^2)*x^3*(d^2 - e^2*x^2))/(24*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (c*x^5*(d^2 - e^2*x^2))/(6*e^2*Sqrt[d - e*x]*
Sqrt[d + e*x]) + (d^2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])
/(16*e^7*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b x^2+c x^4\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{x^2 \left (a+b x^2+c x^4\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{x^2 \left (-6 a e^2-\left (5 c d^2+6 b e^2\right ) x^2\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.808645, size = 202, normalized size = 0.94 $-\frac{e x \sqrt{d-e x} \sqrt{d+e x} \left (6 \left (4 a e^4+3 b d^2 e^2+2 b e^4 x^2\right )+c \left (10 d^2 e^2 x^2+15 d^4+8 e^4 x^4\right )\right )-\frac{6 d^{3/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (8 a e^4+10 b d^2 e^2+11 c d^4\right )}{\sqrt{\frac{e x}{d}+1}}+96 d^2 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{48 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(3*b*d^2*e^2 + 4*a*e^4 + 2*b*e^4*x^2) + c*(15*d^4 + 10*d^2*e^2*x^2 + 8*e^
4*x^4)) - (6*d^(3/2)*(11*c*d^4 + 10*b*d^2*e^2 + 8*a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])]
)/Sqrt[1 + (e*x)/d] + 96*d^2*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/(48*e^7)

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Maple [C]  time = 0.035, size = 273, normalized size = 1.3 \begin{align*} -{\frac{{\it csgn} \left ( e \right ) }{48\,{e}^{7}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 8\,{\it csgn} \left ( e \right ){x}^{5}c{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( e \right ){x}^{3}b{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+10\,{\it csgn} \left ( e \right ){x}^{3}c{d}^{2}{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+24\,{\it csgn} \left ( e \right ){e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xa+18\,{\it csgn} \left ( e \right ){e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xb{d}^{2}+15\,{\it csgn} \left ( e \right ) e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xc{d}^{4}-24\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) a{d}^{2}{e}^{4}-18\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) b{d}^{4}{e}^{2}-15\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) c{d}^{6} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/48*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(8*csgn(e)*x^5*c*e^5*(-e^2*x^2+d^2)^(1/2)+12*csgn(e)*x^3*b*e^5*(-e^2*x^2+d^
2)^(1/2)+10*csgn(e)*x^3*c*d^2*e^3*(-e^2*x^2+d^2)^(1/2)+24*csgn(e)*e^5*(-e^2*x^2+d^2)^(1/2)*x*a+18*csgn(e)*e^3*
(-e^2*x^2+d^2)^(1/2)*x*b*d^2+15*csgn(e)*e*(-e^2*x^2+d^2)^(1/2)*x*c*d^4-24*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1
/2))*a*d^2*e^4-18*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2))*b*d^4*e^2-15*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2
))*c*d^6)*csgn(e)/e^7/(-e^2*x^2+d^2)^(1/2)

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Maxima [A]  time = 1.56008, size = 309, normalized size = 1.43 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{5}}{6 \, e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{3}}{24 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{3}}{4 \, e^{2}} + \frac{5 \, c d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{6}} + \frac{3 \, b d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{a d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x}{16 \, e^{6}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x}{8 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x}{2 \, e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(-e^2*x^2 + d^2)*c*x^5/e^2 - 5/24*sqrt(-e^2*x^2 + d^2)*c*d^2*x^3/e^4 - 1/4*sqrt(-e^2*x^2 + d^2)*b*x^3
/e^2 + 5/16*c*d^6*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^6) + 3/8*b*d^4*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^
2)*e^4) + 1/2*a*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) - 5/16*sqrt(-e^2*x^2 + d^2)*c*d^4*x/e^6 - 3/8*
sqrt(-e^2*x^2 + d^2)*b*d^2*x/e^4 - 1/2*sqrt(-e^2*x^2 + d^2)*a*x/e^2

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Fricas [A]  time = 1.43201, size = 298, normalized size = 1.38 \begin{align*} -\frac{{\left (8 \, c e^{5} x^{5} + 2 \,{\left (5 \, c d^{2} e^{3} + 6 \, b e^{5}\right )} x^{3} + 3 \,{\left (5 \, c d^{4} e + 6 \, b d^{2} e^{3} + 8 \, a e^{5}\right )} x\right )} \sqrt{e x + d} \sqrt{-e x + d} + 6 \,{\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{48 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/48*((8*c*e^5*x^5 + 2*(5*c*d^2*e^3 + 6*b*e^5)*x^3 + 3*(5*c*d^4*e + 6*b*d^2*e^3 + 8*a*e^5)*x)*sqrt(e*x + d)*s
qrt(-e*x + d) + 6*(5*c*d^6 + 6*b*d^4*e^2 + 8*a*d^2*e^4)*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)))/e^7

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Sympy [C]  time = 125.927, size = 362, normalized size = 1.68 \begin{align*} - \frac{i a d^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} + \frac{a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} - \frac{i b d^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} + \frac{b d^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} - \frac{i c d^{6}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{11}{4}, - \frac{9}{4} & - \frac{5}{2}, - \frac{5}{2}, -2, 1 \\-3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{7}} + \frac{c d^{6}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, - \frac{13}{4}, -3, - \frac{11}{4}, - \frac{5}{2}, 1 & \\- \frac{13}{4}, - \frac{11}{4} & - \frac{7}{2}, -3, -3, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-I*a*d**2*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), d**2/(e**2*x**2))/(4
*pi**(3/2)*e**3) + a*d**2*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), d*
*2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**3) - I*b*d**4*meijerg(((-7/4, -5/4), (-3/2, -3/2, -1, 1)),
((-2, -7/4, -3/2, -5/4, -1, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**5) + b*d**4*meijerg(((-5/2, -9/4, -2, -
7/4, -3/2, 1), ()), ((-9/4, -7/4), (-5/2, -2, -2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**5)
- I*c*d**6*meijerg(((-11/4, -9/4), (-5/2, -5/2, -2, 1)), ((-3, -11/4, -5/2, -9/4, -2, 0), ()), d**2/(e**2*x**
2))/(4*pi**(3/2)*e**7) + c*d**6*meijerg(((-7/2, -13/4, -3, -11/4, -5/2, 1), ()), ((-13/4, -11/4), (-7/2, -3, -
3, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**7)

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Giac [A]  time = 1.1521, size = 257, normalized size = 1.19 \begin{align*} \frac{1}{34603008} \,{\left ({\left (33 \, c d^{5} e^{36} + 30 \, b d^{3} e^{38} + 24 \, a d e^{40} -{\left (85 \, c d^{4} e^{36} + 54 \, b d^{2} e^{38} - 2 \,{\left (55 \, c d^{3} e^{36} + 18 \, b d e^{38} -{\left (45 \, c d^{2} e^{36} + 4 \,{\left ({\left (x e + d\right )} c e^{36} - 5 \, c d e^{36}\right )}{\left (x e + d\right )} + 6 \, b e^{38}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 24 \, a e^{40}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} + 6 \,{\left (5 \, c d^{6} e^{36} + 6 \, b d^{4} e^{38} + 8 \, a d^{2} e^{40}\right )} \arcsin \left (\frac{\sqrt{2} \sqrt{x e + d}}{2 \, \sqrt{d}}\right )\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/34603008*((33*c*d^5*e^36 + 30*b*d^3*e^38 + 24*a*d*e^40 - (85*c*d^4*e^36 + 54*b*d^2*e^38 - 2*(55*c*d^3*e^36 +
18*b*d*e^38 - (45*c*d^2*e^36 + 4*((x*e + d)*c*e^36 - 5*c*d*e^36)*(x*e + d) + 6*b*e^38)*(x*e + d))*(x*e + d) +
24*a*e^40)*(x*e + d))*sqrt(x*e + d)*sqrt(-x*e + d) + 6*(5*c*d^6*e^36 + 6*b*d^4*e^38 + 8*a*d^2*e^40)*arcsin(1/
2*sqrt(2)*sqrt(x*e + d)/sqrt(d)))*e^(-1)