\(\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [138]
Optimal result
Integrand size = 35, antiderivative size = 212 \[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{6 d^2 x^6}-\frac {\left (6 b d^2+5 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{24 d^4 x^4}-\frac {\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{16 d^6 x^2}-\frac {e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt {d^2-e^2 x^2} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^7 \sqrt {d-e x} \sqrt {d+e x}}
\]
[Out]
-1/6*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^6-1/24*(5*a*e^2+6*b*d^2)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^4/x^4-1/16*(
5*a*e^4+6*b*d^2*e^2+8*c*d^4)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^6/x^2-1/16*e^2*(5*a*e^4+6*b*d^2*e^2+8*c*d^4)*arcta
nh((-e^2*x^2+d^2)^(1/2)/d)*(-e^2*x^2+d^2)^(1/2)/d^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2)
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.17, number of
steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {534, 1265, 911, 1171, 393, 205,
214} \[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {e^2 \sqrt {d^2-e^2 x^2} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt {d-e x} \sqrt {d+e x}}
\]
[In]
Int[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
-1/6*(a*(d^2 - e^2*x^2))/(d^2*x^6*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((6*b*d^2 + 5*a*e^2)*(d^2 - e^2*x^2))/(24*d^4
*x^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*(d^2 - e^2*x^2))/(16*d^6*x^2*Sqrt[d - e
*x]*Sqrt[d + e*x]) - (e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]
)/(16*d^7*Sqrt[d - e*x]*Sqrt[d + e*x])
Rule 205
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 393
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
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 911
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]
Rule 1171
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
Rule 1265
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*}
\text {integral}& = \frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {a+b x+c x^2}{x^4 \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} \text {Subst}\left (\int \frac {\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^4} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {-5 a-\frac {6 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac {6 c d^2 x^2}{e^4}}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^3} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{6 d^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (\left (6 b+\frac {8 c d^2}{e^2}+\frac {5 a e^2}{d^2}\right ) \sqrt {d^2-e^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^2} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{8 d^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (e^2 \left (6 b+\frac {8 c d^2}{e^2}+\frac {5 a e^2}{d^2}\right ) \sqrt {d^2-e^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{16 d^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^7 \sqrt {d-e x} \sqrt {d+e x}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.39 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.67
\[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {d \sqrt {d-e x} \sqrt {d+e x} \left (6 \left (2 b d^4 x^2+4 c d^4 x^4+3 b d^2 e^2 x^4\right )+a \left (8 d^4+10 d^2 e^2 x^2+15 e^4 x^4\right )\right )}{x^6}+6 e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{48 d^7}
\]
[In]
Integrate[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
[Out]
-1/48*((d*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(2*b*d^4*x^2 + 4*c*d^4*x^4 + 3*b*d^2*e^2*x^4) + a*(8*d^4 + 10*d^2*e^2
*x^2 + 15*e^4*x^4)))/x^6 + 6*e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e*x]])/d^7
Maple [A] (verified)
Time = 0.44 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.84
| | |
| method | result | size |
| | |
| risch |
\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (15 a \,e^{4} x^{4}+18 b \,d^{2} e^{2} x^{4}+24 c \,d^{4} x^{4}+10 a \,d^{2} e^{2} x^{2}+12 b \,d^{4} x^{2}+8 a \,d^{4}\right )}{48 d^{6} x^{6}}-\frac {e^{2} \left (5 e^{4} a +6 e^{2} d^{2} b +8 d^{4} c \right ) \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{16 d^{6} \sqrt {d^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\) |
\(179\) |
| default |
\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (15 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{6} x^{6}+18 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{4} x^{6}+24 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) c \,d^{4} e^{2} x^{6}+15 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d a \,e^{4} x^{4}+18 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} b \,e^{2} x^{4}+24 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d^{5} c \,x^{4}+10 \,\operatorname {csgn}\left (d \right ) a \,d^{3} e^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+12 \,\operatorname {csgn}\left (d \right ) b \,d^{5} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+8 \,\operatorname {csgn}\left (d \right ) a \,d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \operatorname {csgn}\left (d \right )}{48 d^{7} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{6}}\) |
\(306\) |
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[In]
int((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-1/48*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(15*a*e^4*x^4+18*b*d^2*e^2*x^4+24*c*d^4*x^4+10*a*d^2*e^2*x^2+12*b*d^4*x^2+8
*a*d^4)/d^6/x^6-1/16*e^2*(5*a*e^4+6*b*d^2*e^2+8*c*d^4)/d^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^
(1/2))/x)*((e*x+d)*(-e*x+d))^(1/2)/(e*x+d)^(1/2)/(-e*x+d)^(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65
\[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {3 \, {\left (8 \, c d^{4} e^{2} + 6 \, b d^{2} e^{4} + 5 \, a e^{6}\right )} x^{6} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (8 \, a d^{5} + 3 \, {\left (8 \, c d^{5} + 6 \, b d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 2 \, {\left (6 \, b d^{5} + 5 \, a d^{3} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{48 \, d^{7} x^{6}}
\]
[In]
integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")
[Out]
1/48*(3*(8*c*d^4*e^2 + 6*b*d^2*e^4 + 5*a*e^6)*x^6*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x) - (8*a*d^5 + 3*(8*
c*d^5 + 6*b*d^3*e^2 + 5*a*d*e^4)*x^4 + 2*(6*b*d^5 + 5*a*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(d^7*x^6)
Sympy [F(-1)]
Timed out. \[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out}
\]
[In]
integrate((c*x**4+b*x**2+a)/x**7/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
[Out]
Timed out
Maxima [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.28
\[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {c e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, b e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {5 \, a e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{7}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{8 \, d^{4} x^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{16 \, d^{6} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{4 \, d^{2} x^{4}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{24 \, d^{4} x^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{6 \, d^{2} x^{6}}
\]
[In]
integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")
[Out]
-1/2*c*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/8*b*e^4*log(2*d^2/abs(x) + 2*sqrt(-e^2*
x^2 + d^2)*d/abs(x))/d^5 - 5/16*a*e^6*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^7 - 1/2*sqrt(-e^2*
x^2 + d^2)*c/(d^2*x^2) - 3/8*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^2) - 5/16*sqrt(-e^2*x^2 + d^2)*a*e^4/(d^6*x^2)
- 1/4*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^4) - 5/24*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^4) - 1/6*sqrt(-e^2*x^2 + d^2)*
a/(d^2*x^6)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1434 vs. \(2 (184) = 368\).
Time = 0.77 (sec) , antiderivative size = 1434, normalized size of antiderivative = 6.76
\[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display}
\]
[In]
integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")
[Out]
-1/48*(3*(8*c*d^4*e^3 + 6*b*d^2*e^5 + 5*a*e^7)*log(abs(-(sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) + sqr
t(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)) + 2))/d^7 - 3*(8*c*d^4*e^3 + 6*b*d^2*e^5 + 5*a*e^7)*log(abs(-(sq
rt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) + sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)) - 2))/d^7 - 4
*(24*c*d^4*e^3*((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 + 30*b*d^2*e^5*((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 + 33*a*e^7*((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 - 288*c*d^4*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(s
qrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 - 168*b*d^2*e^5*((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 + 20*a*e^7*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - s
qrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 + 768*c*d^4*e^3*((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 + 192*b*d^2*e^5*((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 + 1440*a*e^7*((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 + 3072*c*d^4*e^3*((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 + 768*b*d^2*e^5*((sqr
t(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^5 + 5760*a*e^
7*((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 - 18
432*c*d^4*e^3*((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 - 10752*b*d^2*e^5*((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 + 1280*a*e^7*((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)))^3 + 24576*c*d^4*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)
/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))) + 30720*b*d^2*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqr
t(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))) + 33792*a*e^7*((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) - sq
rt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^2 - 4)^6*d^7))/e
Mupad [B] (verification not implemented)
Time = 24.81 (sec) , antiderivative size = 1621, normalized size of antiderivative = 7.65
\[
\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display}
\]
[In]
int((a + b*x^2 + c*x^4)/(x^7*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
[Out]
((b*e^4)/4 + (6*b*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (53*b*e^4*((d + e*x)^(1/2
) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - d^(1/2))^4) - (87*b*e^4*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2)
- d^(1/2))^6 + (657*b*e^4*((d + e*x)^(1/2) - d^(1/2))^8)/(4*((d - e*x)^(1/2) - d^(1/2))^8) - (121*b*e^4*((d +
e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10)/((256*d^5*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^
(1/2) - d^(1/2))^4 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 + (1536*d^5*((d +
e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) - d^(1/2))^8 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^
(1/2) - d^(1/2))^10 + (256*d^5*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12) - ((c*e^2*((d +
e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (c*e^2)/2 + (15*c*e^2*((d + e*x)^(1/2) - d^(1/2))^4)
/(2*((d - e*x)^(1/2) - d^(1/2))^4))/((16*d^3*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (3
2*d^3*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2) - d^(1/2))^4 + (16*d^3*((d + e*x)^(1/2) - d^(1/2))^6)/((
d - e*x)^(1/2) - d^(1/2))^6) + ((a*e^6)/6 + (4*a*e^6*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2)
)^2 + (71*a*e^6*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2) - d^(1/2))^4 - (1558*a*e^6*((d + e*x)^(1/2) -
d^(1/2))^6)/(3*((d - e*x)^(1/2) - d^(1/2))^6) - (540*a*e^6*((d + e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) - d
^(1/2))^8 + (4248*a*e^6*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10 - (7683*a*e^6*((d + e*x
)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12 + (5558*a*e^6*((d + e*x)^(1/2) - d^(1/2))^14)/((d - e*x)
^(1/2) - d^(1/2))^14 - (3643*a*e^6*((d + e*x)^(1/2) - d^(1/2))^16)/(2*((d - e*x)^(1/2) - d^(1/2))^16))/((4096*
d^7*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 - (24576*d^7*((d + e*x)^(1/2) - d^(1/2))^8)/(
(d - e*x)^(1/2) - d^(1/2))^8 + (61440*d^7*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10 - (81
920*d^7*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12 + (61440*d^7*((d + e*x)^(1/2) - d^(1/2)
)^14)/((d - e*x)^(1/2) - d^(1/2))^14 - (24576*d^7*((d + e*x)^(1/2) - d^(1/2))^16)/((d - e*x)^(1/2) - d^(1/2))^
16 + (4096*d^7*((d + e*x)^(1/2) - d^(1/2))^18)/((d - e*x)^(1/2) - d^(1/2))^18) - (5*a*e^6*log(((d + e*x)^(1/2)
- d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(16*d^7) - (3*b*e^4*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2)
- d^(1/2))))/(8*d^5) - (c*e^2*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(2*d^3) + (5*a*e^
6*log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 - 1))/(16*d^7) + (3*b*e^4*log(((d + e*x)^(1/
2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 - 1))/(8*d^5) + (c*e^2*log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e
*x)^(1/2) - d^(1/2))^2 - 1))/(2*d^3) + (197*a*e^6*((d + e*x)^(1/2) - d^(1/2))^2)/(8192*d^7*((d - e*x)^(1/2) -
d^(1/2))^2) + (5*a*e^6*((d + e*x)^(1/2) - d^(1/2))^4)/(4096*d^7*((d - e*x)^(1/2) - d^(1/2))^4) + (a*e^6*((d +
e*x)^(1/2) - d^(1/2))^6)/(24576*d^7*((d - e*x)^(1/2) - d^(1/2))^6) + (7*b*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/(
256*d^5*((d - e*x)^(1/2) - d^(1/2))^2) + (b*e^4*((d + e*x)^(1/2) - d^(1/2))^4)/(1024*d^5*((d - e*x)^(1/2) - d^
(1/2))^4) + (c*e^2*((d + e*x)^(1/2) - d^(1/2))^2)/(32*d^3*((d - e*x)^(1/2) - d^(1/2))^2)