\(\int \frac {a+b x^2+c x^4}{(d+e x^2)^{13/2}} \, dx\) [285]

Optimal result

Integrand size = 24, antiderivative size = 210 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right ) x^{11}}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]

[Out]

a*x/d/(e*x^2+d)^(11/2)+1/3*(10*a*e+b*d)*x^3/d^2/(e*x^2+d)^(11/2)+1/15*(3*c*d^2+8*e*(10*a*e+b*d))*x^5/d^3/(e*x^
2+d)^(11/2)+2/35*e*(3*c*d^2+8*e*(10*a*e+b*d))*x^7/d^4/(e*x^2+d)^(11/2)+8/315*e^2*(3*c*d^2+8*e*(10*a*e+b*d))*x^
9/d^5/(e*x^2+d)^(11/2)+16/3465*e^3*(3*c*d^2+8*e*(10*a*e+b*d))*x^11/d^6/(e*x^2+d)^(11/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1169, 1817, 12, 277, 270} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {a x}{d \left (d+e x^2\right )^{11/2}} \]

[In]

Int[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(a*x)/(d*(d + e*x^2)^(11/2)) + ((b*d + 10*a*e)*x^3)/(3*d^2*(d + e*x^2)^(11/2)) + ((3*c*d^2 + 8*e*(b*d + 10*a*e
))*x^5)/(15*d^3*(d + e*x^2)^(11/2)) + (2*e*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^7)/(35*d^4*(d + e*x^2)^(11/2)) + (
8*e^2*(3*c*d^2 + 8*e*(b*d + 10*a*e))*x^9)/(315*d^5*(d + e*x^2)^(11/2)) + (16*e^3*(3*c*d^2 + 8*e*(b*d + 10*a*e)
)*x^11)/(3465*d^6*(d + e*x^2)^(11/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

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

Rule 277

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

Rule 1169

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[a^p*x*((d + e*x^2
)^(q + 1)/d), x] + Dist[1/d, Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x]
- e*a^p*(2*q + 3)), 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
] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]

Rule 1817

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {\int \frac {x^2 \left (10 a e+d \left (b+c x^2\right )\right )}{\left (d+e x^2\right )^{13/2}} \, dx}{d} \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\int \frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^4}{\left (d+e x^2\right )^{13/2}} \, dx}{3 d^2} \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {1}{3} \left (3 c+\frac {8 e (b d+10 a e)}{d^2}\right ) \int \frac {x^4}{\left (d+e x^2\right )^{13/2}} \, dx \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {\left (2 e \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^6}{\left (d+e x^2\right )^{13/2}} \, dx}{5 d^3} \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {\left (8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^8}{\left (d+e x^2\right )^{13/2}} \, dx}{35 d^4} \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {\left (16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^{10}}{\left (d+e x^2\right )^{13/2}} \, dx}{315 d^5} \\ & = \frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right ) x^{11}}{3465 d^6 \left (d+e x^2\right )^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {5 a \left (693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right )+d x^3 \left (3 c d x^2 \left (231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right )+b \left (1155 d^4+1848 d^3 e x^2+1584 d^2 e^2 x^4+704 d e^3 x^6+128 e^4 x^8\right )\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x]

[Out]

(5*a*(693*d^5*x + 2310*d^4*e*x^3 + 3696*d^3*e^2*x^5 + 3168*d^2*e^3*x^7 + 1408*d*e^4*x^9 + 256*e^5*x^11) + d*x^
3*(3*c*d*x^2*(231*d^3 + 198*d^2*e*x^2 + 88*d*e^2*x^4 + 16*e^3*x^6) + b*(1155*d^4 + 1848*d^3*e*x^2 + 1584*d^2*e
^2*x^4 + 704*d*e^3*x^6 + 128*e^4*x^8)))/(3465*d^6*(d + e*x^2)^(11/2))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(\frac {\left (\left (\frac {1}{5} c \,x^{4}+\frac {1}{3} b \,x^{2}+a \right ) d^{5}+\frac {10 e \left (\frac {9}{175} c \,x^{4}+\frac {4}{25} b \,x^{2}+a \right ) x^{2} d^{4}}{3}+\frac {16 e^{2} \left (\frac {1}{70} c \,x^{4}+\frac {3}{35} b \,x^{2}+a \right ) x^{4} d^{3}}{3}+\frac {32 e^{3} x^{6} \left (\frac {1}{330} c \,x^{4}+\frac {2}{45} b \,x^{2}+a \right ) d^{2}}{7}+\frac {128 e^{4} x^{8} \left (\frac {b \,x^{2}}{55}+a \right ) d}{63}+\frac {256 a \,e^{5} x^{10}}{693}\right ) x}{\left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(133\)
gosper	\(\frac {x \left (1280 a \,e^{5} x^{10}+128 b d \,e^{4} x^{10}+48 c \,d^{2} e^{3} x^{10}+7040 a d \,e^{4} x^{8}+704 b \,d^{2} e^{3} x^{8}+264 c \,d^{3} e^{2} x^{8}+15840 a \,d^{2} e^{3} x^{6}+1584 b \,d^{3} e^{2} x^{6}+594 c \,d^{4} e \,x^{6}+18480 a \,d^{3} e^{2} x^{4}+1848 b \,d^{4} e \,x^{4}+693 c \,d^{5} x^{4}+11550 a \,d^{4} e \,x^{2}+1155 b \,d^{5} x^{2}+3465 a \,d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(172\)
trager	\(\frac {x \left (1280 a \,e^{5} x^{10}+128 b d \,e^{4} x^{10}+48 c \,d^{2} e^{3} x^{10}+7040 a d \,e^{4} x^{8}+704 b \,d^{2} e^{3} x^{8}+264 c \,d^{3} e^{2} x^{8}+15840 a \,d^{2} e^{3} x^{6}+1584 b \,d^{3} e^{2} x^{6}+594 c \,d^{4} e \,x^{6}+18480 a \,d^{3} e^{2} x^{4}+1848 b \,d^{4} e \,x^{4}+693 c \,d^{5} x^{4}+11550 a \,d^{4} e \,x^{2}+1155 b \,d^{5} x^{2}+3465 a \,d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(172\)
default	\(a \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )+c \left (-\frac {x^{3}}{8 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {3 d \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )}{8 e}\right )+b \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )\)	\(421\)

[In]

int((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x,method=_RETURNVERBOSE)

[Out]

((1/5*c*x^4+1/3*b*x^2+a)*d^5+10/3*e*(9/175*c*x^4+4/25*b*x^2+a)*x^2*d^4+16/3*e^2*(1/70*c*x^4+3/35*b*x^2+a)*x^4*
d^3+32/7*e^3*x^6*(1/330*c*x^4+2/45*b*x^2+a)*d^2+128/63*e^4*x^8*(1/55*b*x^2+a)*d+256/693*a*e^5*x^10)/(e*x^2+d)^
(11/2)*x/d^6

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left (16 \, {\left (3 \, c d^{2} e^{3} + 8 \, b d e^{4} + 80 \, a e^{5}\right )} x^{11} + 88 \, {\left (3 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} + 80 \, a d e^{4}\right )} x^{9} + 198 \, {\left (3 \, c d^{4} e + 8 \, b d^{3} e^{2} + 80 \, a d^{2} e^{3}\right )} x^{7} + 3465 \, a d^{5} x + 231 \, {\left (3 \, c d^{5} + 8 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} x^{5} + 1155 \, {\left (b d^{5} + 10 \, a d^{4} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{3465 \, {\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \]

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="fricas")

[Out]

1/3465*(16*(3*c*d^2*e^3 + 8*b*d*e^4 + 80*a*e^5)*x^11 + 88*(3*c*d^3*e^2 + 8*b*d^2*e^3 + 80*a*d*e^4)*x^9 + 198*(
3*c*d^4*e + 8*b*d^3*e^2 + 80*a*d^2*e^3)*x^7 + 3465*a*d^5*x + 231*(3*c*d^5 + 8*b*d^4*e + 80*a*d^3*e^2)*x^5 + 11
55*(b*d^5 + 10*a*d^4*e)*x^3)*sqrt(e*x^2 + d)/(d^6*e^6*x^12 + 6*d^7*e^5*x^10 + 15*d^8*e^4*x^8 + 20*d^9*e^3*x^6
+ 15*d^10*e^2*x^4 + 6*d^11*e*x^2 + d^12)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11602 vs. \(2 (206) = 412\).

Time = 154.15 (sec) , antiderivative size = 11602, normalized size of antiderivative = 55.25 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(13/2),x)

[Out]

a*(693*d**55*x/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119
/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**
8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1
+ e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**
2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) +
945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**
(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x
**30*sqrt(1 + e*x**2/d)) + 9240*d**54*e*x**3/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt
(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d
) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 346846
5*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(10
7/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**
10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*
sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x
**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 57981*d**53*e**2*x**5/(693*d**(123/2)*sqrt(1 + e*x**2
/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(
117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5
*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*s
qrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 +
e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/
d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*
d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 227238*d**52*e**3*x
**7/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x*
*4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 +
e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d
) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468
465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(
101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**1
3*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1
+ e*x**2/d)) + 622138*d**51*e**4*x**9/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e
*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 94
5945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(
111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e
**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**
20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1
+ e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d)
+ 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 1260152*d**50*e**5*x**11/(693*d**(123/2)*sqrt(1 + e*x**2/d)
+ 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117
/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x*
*10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt
(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x
**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d)
+ 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**
(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 1949890*d**49*e**6*x**
13/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**
4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e
*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d)
+ 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 34684
65*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(1
01/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13
*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1
+ e*x**2/d)) + 2345412*d**48*e**7*x**15/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 +
e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 9
45945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**
(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*
e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x*
*20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(
1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d
) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 2209281*d**47*e**8*x**17/(693*d**(123/2)*sqrt(1 + e*x**2/d
) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(11
7/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x
**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqr
t(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*
x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d)
+ 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d*
*(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 1628256*d**46*e**9*x*
*19/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x*
*4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 +
e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d
) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468
465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(
101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**1
3*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1
+ e*x**2/d)) + 930465*d**45*e**10*x**21/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 +
e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) +
945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d*
*(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)
*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x
**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt
(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/
d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 404550*d**44*e**11*x**23/(693*d**(123/2)*sqrt(1 + e*x**2/
d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(1
17/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*
x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sq
rt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e
*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d
) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d
**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 129456*d**43*e**12*x
**25/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x
**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 +
e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/
d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 346
8465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**
(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**
13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(
1 + e*x**2/d)) + 28768*d**42*e**13*x**27/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 +
e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) +
945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d*
*(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)
*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x
**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt
(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/
d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 3968*d**41*e**14*x**29/(693*d**(123/2)*sqrt(1 + e*x**2/d)
+ 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117
/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x*
*10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt
(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x
**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d)
+ 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**
(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 256*d**40*e**15*x**31/
(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*s
qrt(1 + e*x**2/d) + 315315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x*
*2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) +
4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*
d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/
2)*e**11*x**22*sqrt(1 + e*x**2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x*
*26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x**2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e
*x**2/d))) + b*(1155*d**30*x**3/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d)
+ 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**
(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6
*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt
(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/
d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 8778*d**29*e*x**5/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 3
8115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e*
*3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqr
t(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x*
*2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 3811
5*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 29997*d**28*e**2
*x**7/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x
**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 +
e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d)
+ 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d*
*(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*
x**22*sqrt(1 + e*x**2/d)) + 61028*d**27*e**3*x**9/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*
sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**
2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600
830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/
2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**2
0*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 82397*d**26*e**4*x**11/(3465*d**(73/2)
*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d
) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d
**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e
**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sq
rt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2
/d)) + 77658*d**25*e**5*x**13/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) +
190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(6
5/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x
**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1
+ e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d)
+ 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 52003*d**24*e**6*x**15/(3465*d**(73/2)*sqrt(1 + e*x**2/d)
+ 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)
*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*
sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e
*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 3
8115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 24472*d**23*e
**7*x**17/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e*
*2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(
1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2
/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 19057
5*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e*
*11*x**22*sqrt(1 + e*x**2/d)) + 7728*d**22*e**8*x**19/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x
**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e
*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) +
1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**
(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*
x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d)) + 1472*d**21*e**9*x**21/(3465*d**(73
/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d) + 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**
2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 160083
0*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2
)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18
*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x
**2/d)) + 128*d**20*e**10*x**23/(3465*d**(73/2)*sqrt(1 + e*x**2/d) + 38115*d**(71/2)*e*x**2*sqrt(1 + e*x**2/d)
+ 190575*d**(69/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 571725*d**(67/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 1143450*d**
(65/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 1600830*d**(63/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 1600830*d**(61/2)*e**6
*x**12*sqrt(1 + e*x**2/d) + 1143450*d**(59/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 571725*d**(57/2)*e**8*x**16*sqrt
(1 + e*x**2/d) + 190575*d**(55/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 38115*d**(53/2)*e**10*x**20*sqrt(1 + e*x**2/
d) + 3465*d**(51/2)*e**11*x**22*sqrt(1 + e*x**2/d))) + c*(231*d**14*x**5/(1155*d**(41/2)*sqrt(1 + e*x**2/d) +
9240*d**(39/2)*e*x**2*sqrt(1 + e*x**2/d) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3
*x**6*sqrt(1 + e*x**2/d) + 80850*d**(33/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 +
e*x**2/d) + 32340*d**(29/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 115
5*d**(25/2)*e**8*x**16*sqrt(1 + e*x**2/d)) + 891*d**13*e*x**7/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39
/2)*e*x**2*sqrt(1 + e*x**2/d) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(
1 + e*x**2/d) + 80850*d**(33/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) +
32340*d**(29/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)
*e**8*x**16*sqrt(1 + e*x**2/d)) + 1375*d**12*e**2*x**9/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39/2)*e*x
**2*sqrt(1 + e*x**2/d) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(1 + e*x
**2/d) + 80850*d**(33/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 32340*
d**(29/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)*e**8*x
**16*sqrt(1 + e*x**2/d)) + 1105*d**11*e**3*x**11/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39/2)*e*x**2*sq
rt(1 + e*x**2/d) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(1 + e*x**2/d)
+ 80850*d**(33/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 32340*d**(29
/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)*e**8*x**16*s
qrt(1 + e*x**2/d)) + 510*d**10*e**4*x**13/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39/2)*e*x**2*sqrt(1 +
e*x**2/d) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 8085
0*d**(33/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 32340*d**(29/2)*e**
6*x**12*sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)*e**8*x**16*sqrt(1 +
e*x**2/d)) + 136*d**9*e**5*x**15/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39/2)*e*x**2*sqrt(1 + e*x**2/d
) + 32340*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 80850*d**(33
/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 32340*d**(29/2)*e**6*x**12*
sqrt(1 + e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)*e**8*x**16*sqrt(1 + e*x**2/
d)) + 16*d**8*e**6*x**17/(1155*d**(41/2)*sqrt(1 + e*x**2/d) + 9240*d**(39/2)*e*x**2*sqrt(1 + e*x**2/d) + 32340
*d**(37/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 64680*d**(35/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 80850*d**(33/2)*e**4*
x**8*sqrt(1 + e*x**2/d) + 64680*d**(31/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 32340*d**(29/2)*e**6*x**12*sqrt(1 +
e*x**2/d) + 9240*d**(27/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 1155*d**(25/2)*e**8*x**16*sqrt(1 + e*x**2/d)))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.60 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=-\frac {c x^{3}}{8 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {256 \, a x}{693 \, \sqrt {e x^{2} + d} d^{6}} + \frac {128 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{5}} + \frac {32 \, a x}{231 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{4}} + \frac {80 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{3}} + \frac {10 \, a x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d^{2}} + \frac {a x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} d} + \frac {c x}{264 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} + \frac {16 \, c x}{1155 \, \sqrt {e x^{2} + d} d^{4} e^{2}} + \frac {8 \, c x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e^{2}} + \frac {2 \, c x}{385 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e^{2}} + \frac {c x}{231 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e^{2}} - \frac {3 \, c d x}{88 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{2}} - \frac {b x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {128 \, b x}{3465 \, \sqrt {e x^{2} + d} d^{5} e} + \frac {64 \, b x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4} e} + \frac {16 \, b x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3} e} + \frac {8 \, b x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2} e} + \frac {b x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d e} \]

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="maxima")

[Out]

-1/8*c*x^3/((e*x^2 + d)^(11/2)*e) + 256/693*a*x/(sqrt(e*x^2 + d)*d^6) + 128/693*a*x/((e*x^2 + d)^(3/2)*d^5) +
32/231*a*x/((e*x^2 + d)^(5/2)*d^4) + 80/693*a*x/((e*x^2 + d)^(7/2)*d^3) + 10/99*a*x/((e*x^2 + d)^(9/2)*d^2) +
1/11*a*x/((e*x^2 + d)^(11/2)*d) + 1/264*c*x/((e*x^2 + d)^(9/2)*e^2) + 16/1155*c*x/(sqrt(e*x^2 + d)*d^4*e^2) +
8/1155*c*x/((e*x^2 + d)^(3/2)*d^3*e^2) + 2/385*c*x/((e*x^2 + d)^(5/2)*d^2*e^2) + 1/231*c*x/((e*x^2 + d)^(7/2)*
d*e^2) - 3/88*c*d*x/((e*x^2 + d)^(11/2)*e^2) - 1/11*b*x/((e*x^2 + d)^(11/2)*e) + 128/3465*b*x/(sqrt(e*x^2 + d)
*d^5*e) + 64/3465*b*x/((e*x^2 + d)^(3/2)*d^4*e) + 16/1155*b*x/((e*x^2 + d)^(5/2)*d^3*e) + 8/693*b*x/((e*x^2 +
d)^(7/2)*d^2*e) + 1/99*b*x/((e*x^2 + d)^(9/2)*d*e)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, {\left (3 \, c d^{2} e^{8} + 8 \, b d e^{9} + 80 \, a e^{10}\right )} x^{2}}{d^{6} e^{5}} + \frac {11 \, {\left (3 \, c d^{3} e^{7} + 8 \, b d^{2} e^{8} + 80 \, a d e^{9}\right )}}{d^{6} e^{5}}\right )} + \frac {99 \, {\left (3 \, c d^{4} e^{6} + 8 \, b d^{3} e^{7} + 80 \, a d^{2} e^{8}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {231 \, {\left (3 \, c d^{5} e^{5} + 8 \, b d^{4} e^{6} + 80 \, a d^{3} e^{7}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {1155 \, {\left (b d^{5} e^{5} + 10 \, a d^{4} e^{6}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {3465 \, a}{d}\right )} x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}}} \]

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(13/2),x, algorithm="giac")

[Out]

1/3465*(((2*(4*x^2*(2*(3*c*d^2*e^8 + 8*b*d*e^9 + 80*a*e^10)*x^2/(d^6*e^5) + 11*(3*c*d^3*e^7 + 8*b*d^2*e^8 + 80
*a*d*e^9)/(d^6*e^5)) + 99*(3*c*d^4*e^6 + 8*b*d^3*e^7 + 80*a*d^2*e^8)/(d^6*e^5))*x^2 + 231*(3*c*d^5*e^5 + 8*b*d
^4*e^6 + 80*a*d^3*e^7)/(d^6*e^5))*x^2 + 1155*(b*d^5*e^5 + 10*a*d^4*e^6)/(d^6*e^5))*x^2 + 3465*a/d)*x/(e*x^2 +
d)^(11/2)

Mupad [B] (verification not implemented)

Time = 7.91 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {x\,\left (\frac {a}{11\,d}-\frac {d\,\left (\frac {b}{11\,d}-\frac {c}{11\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{11/2}}-\frac {x\,\left (\frac {c}{9\,e^2}-\frac {-c\,d^2+b\,d\,e+10\,a\,e^2}{99\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{9/2}}+\frac {x\,\left (3\,c\,d^2+8\,b\,d\,e+80\,a\,e^2\right )}{693\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (6\,c\,d^2+16\,b\,d\,e+160\,a\,e^2\right )}{1155\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (24\,c\,d^2+64\,b\,d\,e+640\,a\,e^2\right )}{3465\,d^5\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (48\,c\,d^2+128\,b\,d\,e+1280\,a\,e^2\right )}{3465\,d^6\,e^2\,\sqrt {e\,x^2+d}} \]

[In]

int((a + b*x^2 + c*x^4)/(d + e*x^2)^(13/2),x)

[Out]

(x*(a/(11*d) - (d*(b/(11*d) - c/(11*e)))/e))/(d + e*x^2)^(11/2) - (x*(c/(9*e^2) - (10*a*e^2 - c*d^2 + b*d*e)/(
99*d^2*e^2)))/(d + e*x^2)^(9/2) + (x*(80*a*e^2 + 3*c*d^2 + 8*b*d*e))/(693*d^3*e^2*(d + e*x^2)^(7/2)) + (x*(160
*a*e^2 + 6*c*d^2 + 16*b*d*e))/(1155*d^4*e^2*(d + e*x^2)^(5/2)) + (x*(640*a*e^2 + 24*c*d^2 + 64*b*d*e))/(3465*d
^5*e^2*(d + e*x^2)^(3/2)) + (x*(1280*a*e^2 + 48*c*d^2 + 128*b*d*e))/(3465*d^6*e^2*(d + e*x^2)^(1/2))