3.2.31 \(\int \frac {x^3 (a+b x^2+c x^4)}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\)

Optimal. Leaf size=159 \[ \frac {(d-e x)^{3/2} (d+e x)^{3/2} \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8}-\frac {d^2 \sqrt {d-e x} \sqrt {d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^8}-\frac {(d-e x)^{5/2} (d+e x)^{5/2} \left (b e^2+3 c d^2\right )}{5 e^8}+\frac {c (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^8} \]

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Rubi [A]  time = 0.19, antiderivative size = 213, normalized size of antiderivative = 1.34, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {520, 1251, 771} \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^2 \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right )^3 \left (b e^2+3 c d^2\right )}{5 e^8 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^4}{7 e^8 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*(c*d^4 + b*d^2*e^2 + a*e^4)*(d^2 - e^2*x^2))/(e^8*Sqrt[d - e*x]*Sqrt[d + e*x])) + ((3*c*d^4 + 2*b*d^2*e
^2 + a*e^4)*(d^2 - e^2*x^2)^2)/(3*e^8*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((3*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^3)/(5*
e^8*Sqrt[d - e*x]*Sqrt[d + e*x]) + (c*(d^2 - e^2*x^2)^4)/(7*e^8*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 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [C]  time = 1.07, size = 232, normalized size = 1.46 \begin {gather*} -\frac {\frac {210 d^{5/2} \sqrt {d+e x} \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt {\frac {e x}{d}+1}}+\sqrt {d-e x} \sqrt {d+e x} \left (35 a e^4 \left (2 d^2+e^2 x^2\right )+7 b \left (8 d^4 e^2+4 d^2 e^4 x^2+3 e^6 x^4\right )+3 c \left (16 d^6+8 d^4 e^2 x^2+6 d^2 e^4 x^4+5 e^6 x^6\right )\right )-210 d^3 \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 )}{105 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*a*e^4*(2*d^2 + e^2*x^2) + 7*b*(8*d^4*e^2 + 4*d^2*e^4*x^2 + 3*e^6*x^4)
+ 3*c*(16*d^6 + 8*d^4*e^2*x^2 + 6*d^2*e^4*x^4 + 5*e^6*x^6)) + (210*d^(5/2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d
+ e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 + (e*x)/d] - 210*d^3*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan
[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^8

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IntegrateAlgebraic [B]  time = 0.29, size = 477, normalized size = 3.00 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (\frac {105 a d^3 e^4 (d+e x)^6}{(d-e x)^6}+\frac {490 a d^3 e^4 (d+e x)^5}{(d-e x)^5}+\frac {1015 a d^3 e^4 (d+e x)^4}{(d-e x)^4}+\frac {1260 a d^3 e^4 (d+e x)^3}{(d-e x)^3}+\frac {1015 a d^3 e^4 (d+e x)^2}{(d-e x)^2}+\frac {490 a d^3 e^4 (d+e x)}{d-e x}+105 a d^3 e^4+\frac {105 b d^5 e^2 (d+e x)^6}{(d-e x)^6}+\frac {350 b d^5 e^2 (d+e x)^5}{(d-e x)^5}+\frac {791 b d^5 e^2 (d+e x)^4}{(d-e x)^4}+\frac {1092 b d^5 e^2 (d+e x)^3}{(d-e x)^3}+\frac {791 b d^5 e^2 (d+e x)^2}{(d-e x)^2}+\frac {350 b d^5 e^2 (d+e x)}{d-e x}+105 b d^5 e^2+\frac {105 c d^7 (d+e x)^6}{(d-e x)^6}+\frac {210 c d^7 (d+e x)^5}{(d-e x)^5}+\frac {903 c d^7 (d+e x)^4}{(d-e x)^4}+\frac {636 c d^7 (d+e x)^3}{(d-e x)^3}+\frac {903 c d^7 (d+e x)^2}{(d-e x)^2}+\frac {210 c d^7 (d+e x)}{d-e x}+105 c d^7\right )}{105 e^8 \sqrt {d-e x} \left (\frac {d+e x}{d-e x}+1\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-2*Sqrt[d + e*x]*(105*c*d^7 + 105*b*d^5*e^2 + 105*a*d^3*e^4 + (210*c*d^7*(d + e*x))/(d - e*x) + (350*b*d^5*e^
2*(d + e*x))/(d - e*x) + (490*a*d^3*e^4*(d + e*x))/(d - e*x) + (903*c*d^7*(d + e*x)^2)/(d - e*x)^2 + (791*b*d^
5*e^2*(d + e*x)^2)/(d - e*x)^2 + (1015*a*d^3*e^4*(d + e*x)^2)/(d - e*x)^2 + (636*c*d^7*(d + e*x)^3)/(d - e*x)^
3 + (1092*b*d^5*e^2*(d + e*x)^3)/(d - e*x)^3 + (1260*a*d^3*e^4*(d + e*x)^3)/(d - e*x)^3 + (903*c*d^7*(d + e*x)
^4)/(d - e*x)^4 + (791*b*d^5*e^2*(d + e*x)^4)/(d - e*x)^4 + (1015*a*d^3*e^4*(d + e*x)^4)/(d - e*x)^4 + (210*c*
d^7*(d + e*x)^5)/(d - e*x)^5 + (350*b*d^5*e^2*(d + e*x)^5)/(d - e*x)^5 + (490*a*d^3*e^4*(d + e*x)^5)/(d - e*x)
^5 + (105*c*d^7*(d + e*x)^6)/(d - e*x)^6 + (105*b*d^5*e^2*(d + e*x)^6)/(d - e*x)^6 + (105*a*d^3*e^4*(d + e*x)^
6)/(d - e*x)^6))/(105*e^8*Sqrt[d - e*x]*(1 + (d + e*x)/(d - e*x))^7)

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fricas [A]  time = 1.25, size = 104, normalized size = 0.65 \begin {gather*} -\frac {{\left (15 \, c e^{6} x^{6} + 48 \, c d^{6} + 56 \, b d^{4} e^{2} + 70 \, a d^{2} e^{4} + 3 \, {\left (6 \, c d^{2} e^{4} + 7 \, b e^{6}\right )} x^{4} + {\left (24 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 35 \, a e^{6}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{105 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(15*c*e^6*x^6 + 48*c*d^6 + 56*b*d^4*e^2 + 70*a*d^2*e^4 + 3*(6*c*d^2*e^4 + 7*b*e^6)*x^4 + (24*c*d^4*e^2
+ 28*b*d^2*e^4 + 35*a*e^6)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d)/e^8

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giac [A]  time = 0.75, size = 194, normalized size = 1.22 \begin {gather*} -\frac {1}{105} \, {\left ({\left ({\left (3 \, {\left ({\left (5 \, {\left ({\left (x e + d\right )} c e^{\left (-7\right )} - 6 \, c d e^{\left (-7\right )}\right )} {\left (x e + d\right )} + {\left (81 \, c d^{2} e^{49} + 7 \, b e^{51}\right )} e^{\left (-56\right )}\right )} {\left (x e + d\right )} - 4 \, {\left (31 \, c d^{3} e^{49} + 7 \, b d e^{51}\right )} e^{\left (-56\right )}\right )} {\left (x e + d\right )} + 7 \, {\left (51 \, c d^{4} e^{49} + 22 \, b d^{2} e^{51} + 5 \, a e^{53}\right )} e^{\left (-56\right )}\right )} {\left (x e + d\right )} - 70 \, {\left (3 \, c d^{5} e^{49} + 2 \, b d^{3} e^{51} + a d e^{53}\right )} e^{\left (-56\right )}\right )} {\left (x e + d\right )} + 105 \, {\left (c d^{6} e^{49} + b d^{4} e^{51} + a d^{2} e^{53}\right )} e^{\left (-56\right )}\right )} \sqrt {x e + d} \sqrt {-x e + d} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/105*(((3*((5*((x*e + d)*c*e^(-7) - 6*c*d*e^(-7))*(x*e + d) + (81*c*d^2*e^49 + 7*b*e^51)*e^(-56))*(x*e + d)
- 4*(31*c*d^3*e^49 + 7*b*d*e^51)*e^(-56))*(x*e + d) + 7*(51*c*d^4*e^49 + 22*b*d^2*e^51 + 5*a*e^53)*e^(-56))*(x
*e + d) - 70*(3*c*d^5*e^49 + 2*b*d^3*e^51 + a*d*e^53)*e^(-56))*(x*e + d) + 105*(c*d^6*e^49 + b*d^4*e^51 + a*d^
2*e^53)*e^(-56))*sqrt(x*e + d)*sqrt(-x*e + d)*e^(-1)

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maple [A]  time = 0.01, size = 109, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (15 c \,x^{6} e^{6}+21 b \,e^{6} x^{4}+18 c \,d^{2} e^{4} x^{4}+35 a \,e^{6} x^{2}+28 b \,d^{2} e^{4} x^{2}+24 c \,d^{4} e^{2} x^{2}+70 a \,d^{2} e^{4}+56 b \,d^{4} e^{2}+48 c \,d^{6}\right )}{105 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(15*c*e^6*x^6+21*b*e^6*x^4+18*c*d^2*e^4*x^4+35*a*e^6*x^2+28*b*d^2*e^4*x^2+
24*c*d^4*e^2*x^2+70*a*d^2*e^4+56*b*d^4*e^2+48*c*d^6)/e^8

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maxima [A]  time = 1.06, size = 217, normalized size = 1.36 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{6}}{7 \, e^{2}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{4}}{35 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{4}}{5 \, e^{2}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x^{2}}{35 \, e^{6}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x^{2}}{15 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x^{2}}{3 \, e^{2}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{6}}{35 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{4}}{15 \, e^{6}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{2}}{3 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/7*sqrt(-e^2*x^2 + d^2)*c*x^6/e^2 - 6/35*sqrt(-e^2*x^2 + d^2)*c*d^2*x^4/e^4 - 1/5*sqrt(-e^2*x^2 + d^2)*b*x^4
/e^2 - 8/35*sqrt(-e^2*x^2 + d^2)*c*d^4*x^2/e^6 - 4/15*sqrt(-e^2*x^2 + d^2)*b*d^2*x^2/e^4 - 1/3*sqrt(-e^2*x^2 +
d^2)*a*x^2/e^2 - 16/35*sqrt(-e^2*x^2 + d^2)*c*d^6/e^8 - 8/15*sqrt(-e^2*x^2 + d^2)*b*d^4/e^6 - 2/3*sqrt(-e^2*x
^2 + d^2)*a*d^2/e^4

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mupad [B]  time = 1.49, size = 215, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {d-e\,x}\,\left (\frac {48\,c\,d^7+56\,b\,d^5\,e^2+70\,a\,d^3\,e^4}{105\,e^8}+\frac {x^5\,\left (18\,c\,d^2\,e^5+21\,b\,e^7\right )}{105\,e^8}+\frac {c\,x^7}{7\,e}+\frac {x^3\,\left (24\,c\,d^4\,e^3+28\,b\,d^2\,e^5+35\,a\,e^7\right )}{105\,e^8}+\frac {x\,\left (48\,c\,d^6\,e+56\,b\,d^4\,e^3+70\,a\,d^2\,e^5\right )}{105\,e^8}+\frac {x^4\,\left (18\,c\,d^3\,e^4+21\,b\,d\,e^6\right )}{105\,e^8}+\frac {x^2\,\left (24\,c\,d^5\,e^2+28\,b\,d^3\,e^4+35\,a\,d\,e^6\right )}{105\,e^8}+\frac {c\,d\,x^6}{7\,e^2}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*x^2 + c*x^4))/((d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

-((d - e*x)^(1/2)*((48*c*d^7 + 70*a*d^3*e^4 + 56*b*d^5*e^2)/(105*e^8) + (x^5*(21*b*e^7 + 18*c*d^2*e^5))/(105*e
^8) + (c*x^7)/(7*e) + (x^3*(35*a*e^7 + 28*b*d^2*e^5 + 24*c*d^4*e^3))/(105*e^8) + (x*(70*a*d^2*e^5 + 56*b*d^4*e
^3 + 48*c*d^6*e))/(105*e^8) + (x^4*(18*c*d^3*e^4 + 21*b*d*e^6))/(105*e^8) + (x^2*(28*b*d^3*e^4 + 24*c*d^5*e^2
+ 35*a*d*e^6))/(105*e^8) + (c*d*x^6)/(7*e^2)))/(d + e*x)^(1/2)

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sympy [C]  time = 135.14, size = 367, normalized size = 2.31 \begin {gather*} - \frac {i a d^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {a d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {i b d^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {b d^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {i c d^{7} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {13}{4}, - \frac {11}{4} & -3, -3, - \frac {5}{2}, 1 \\- \frac {7}{2}, - \frac {13}{4}, -3, - \frac {11}{4}, - \frac {5}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {c d^{7} {G_{6, 6}^{2, 6}\left (\begin {matrix} -4, - \frac {15}{4}, - \frac {7}{2}, - \frac {13}{4}, -3, 1 & \\- \frac {15}{4}, - \frac {13}{4} & -4, - \frac {7}{2}, - \frac {7}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-I*a*d**3*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2, -5/4, -1, -3/4, -1/2, 0), ()), d**2/(e**2*x**2))/
(4*pi**(3/2)*e**4) - a*d**3*meijerg(((-2, -7/4, -3/2, -5/4, -1, 1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)),
d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**4) - I*b*d**5*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)),
((-5/2, -9/4, -2, -7/4, -3/2, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**6) - b*d**5*meijerg(((-3, -11/4, -5/2
, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e
**6) - I*c*d**7*meijerg(((-13/4, -11/4), (-3, -3, -5/2, 1)), ((-7/2, -13/4, -3, -11/4, -5/2, 0), ()), d**2/(e*
*2*x**2))/(4*pi**(3/2)*e**8) - c*d**7*meijerg(((-4, -15/4, -7/2, -13/4, -3, 1), ()), ((-15/4, -13/4), (-4, -7/
2, -7/2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**8)

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