\(\int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [144]

Optimal result

Integrand size = 35, antiderivative size = 226 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {534, 1279, 464, 277, 270} \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}} \]

[In]

Int[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/7*(a*(d^2 - e^2*x^2))/(d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e^2)*(d^2 - e^2*x^2))/(35*d^4
*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d
- e*x]*Sqrt[d + e*x]) - (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[d - e*x]
*Sqrt[d + e*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 464

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

Rule 534

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 1279

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Wit
h[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x,
x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Dist[1/(d*f^2*(m + 1)), Int[(f*x)^(m + 2)
*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q},
x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-7 b d^2-6 a e^2-7 c d^2 x^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{7 d^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (35 c d^4-4 e^2 \left (-7 b d^2-6 a e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{35 d^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (2 e^2 \left (35 c d^4-4 e^2 \left (-7 b d^2-6 a e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{105 d^6 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.55 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (35 c d^4 x^4 \left (d^2+2 e^2 x^2\right )+7 b \left (3 d^6 x^2+4 d^4 e^2 x^4+8 d^2 e^4 x^6\right )+3 a \left (5 d^6+6 d^4 e^2 x^2+8 d^2 e^4 x^4+16 e^6 x^6\right )\right )}{105 d^8 x^7} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.52

method	result	size
gosper	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 x^{7} d^{8}}\)	\(118\)
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 x^{7} d^{8}}\)	\(118\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \operatorname {csgn}\left (e \right )^{2} \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 d^{8} x^{7}}\)	\(122\)

[In]

int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(48*a*e^6*x^6+56*b*d^2*e^4*x^6+70*c*d^4*e^2*x^6+24*a*d^2*e^4*x^4+28*b*d^4*
e^2*x^4+35*c*d^6*x^4+18*a*d^4*e^2*x^2+21*b*d^6*x^2+15*a*d^6)/x^7/d^8

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.49 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (15 \, a d^{6} + 2 \, {\left (35 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 24 \, a e^{6}\right )} x^{6} + {\left (35 \, c d^{6} + 28 \, b d^{4} e^{2} + 24 \, a d^{2} e^{4}\right )} x^{4} + 3 \, {\left (7 \, b d^{6} + 6 \, a d^{4} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{105 \, d^{8} x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**4+b*x**2+a)/x**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} c e^{2}}{3 \, d^{4} x} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{4}}{15 \, d^{6} x} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{6}}{35 \, d^{8} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{3 \, d^{2} x^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{15 \, d^{4} x^{3}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{35 \, d^{6} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{5 \, d^{2} x^{5}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{35 \, d^{4} x^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{7 \, d^{2} x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1451 vs. \(2 (206) = 412\).

Time = 0.69 (sec) , antiderivative size = 1451, normalized size of antiderivative = 6.42 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-4/105*(105*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqr
t(-e*x + d)))^13 + 105*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sq
rt(d) - sqrt(-e*x + d)))^13 + 105*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqr
t(2)*sqrt(d) - sqrt(-e*x + d)))^13 - 1960*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e
*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 - 1400*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x +
d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 - 840*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqr
t(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 + 16240*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-
e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 + 12656*b*d^2*e^6*((sqrt(2)*sqrt
(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 + 14448*a*e^8*((sqrt
(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 - 80640*c*d^
4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7
- 69888*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e
*x + d)))^7 - 40704*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) -
sqrt(-e*x + d)))^7 + 259840*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt
(2)*sqrt(d) - sqrt(-e*x + d)))^5 + 202496*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e
*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^5 + 231168*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d)
- sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^5 - 501760*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/
sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^3 - 358400*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqr
t(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^3 - 215040*a*e^8*((sqrt(2)*sqrt
(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^3 + 430080*c*d^4*e^4*(
(sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))) + 430080*
b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))
) + 430080*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*
x + d))))/((((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d
)))^2 - 4)^7*d^8*e)

Mupad [B] (verification not implemented)

Time = 8.53 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{7\,d}+\frac {x^2\,\left (21\,b\,d^7+18\,a\,d^5\,e^2\right )}{105\,d^8}+\frac {x^4\,\left (35\,c\,d^7+28\,b\,d^5\,e^2+24\,a\,d^3\,e^4\right )}{105\,d^8}+\frac {x^7\,\left (70\,c\,d^4\,e^3+56\,b\,d^2\,e^5+48\,a\,e^7\right )}{105\,d^8}+\frac {x^3\,\left (21\,b\,d^6\,e+18\,a\,d^4\,e^3\right )}{105\,d^8}+\frac {x^5\,\left (35\,c\,d^6\,e+28\,b\,d^4\,e^3+24\,a\,d^2\,e^5\right )}{105\,d^8}+\frac {x^6\,\left (70\,c\,d^5\,e^2+56\,b\,d^3\,e^4+48\,a\,d\,e^6\right )}{105\,d^8}+\frac {a\,e\,x}{7\,d^2}\right )}{x^7\,\sqrt {d+e\,x}} \]

[In]

int((a + b*x^2 + c*x^4)/(x^8*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

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