3.143 \(\int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx\)

Optimal. Leaf size=160 \[ -\frac {\left (d^2-e^2 x^2\right ) \left (8 a e^4+10 b d^2 e^2+15 c d^4\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (4 a e^2+5 b d^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}} \]

[Out]

-1/5*a*(-e^2*x^2+d^2)/d^2/x^5/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-1/15*(4*a*e^2+5*b*d^2)*(-e^2*x^2+d^2)/d^4/x^3/(-e*x
+d)^(1/2)/(e*x+d)^(1/2)-1/15*(8*a*e^4+10*b*d^2*e^2+15*c*d^4)*(-e^2*x^2+d^2)/d^6/x/(-e*x+d)^(1/2)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {520, 1265, 453, 264} \[ -\frac {\left (d^2-e^2 x^2\right ) \left (8 a e^4+10 b d^2 e^2+15 c d^4\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (4 a e^2+5 b d^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^6*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(5*d^2*x^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((5*b*d^2 + 4*a*e^2)*(d^2 - e^2*x^2))/(15*d^4*x
^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((15*c*d^4 + 10*b*d^2*e^2 + 8*a*e^4)*(d^2 - e^2*x^2))/(15*d^6*x*Sqrt[d - e*x
]*Sqrt[d + e*x])

Rule 264

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 453

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 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 1265

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*} \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x^6 \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 )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-5 b d^2-4 a e^2-5 c d^2 x^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 b d^2+4 a e^2\right ) \left (d^2-e^2 x^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (15 c d^4-2 e^2 \left (-5 b d^2-4 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}{15 d^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 b d^2+4 a e^2\right ) \left (d^2-e^2 x^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (15 c d^4+10 b d^2 e^2+8 a e^4\right ) \left (d^2-e^2 x^2\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 87, normalized size = 0.54 \[ -\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a \left (3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right )+5 b d^2 x^2 \left (d^2+2 e^2 x^2\right )+15 c d^4 x^4\right )}{15 d^6 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^6*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/15*(Sqrt[d - e*x]*Sqrt[d + e*x]*(15*c*d^4*x^4 + 5*b*d^2*x^2*(d^2 + 2*e^2*x^2) + a*(3*d^4 + 4*d^2*e^2*x^2 +
8*e^4*x^4)))/(d^6*x^5)

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 76, normalized size = 0.48 \[ -\frac {{\left (3 \, a d^{4} + {\left (15 \, c d^{4} + 10 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} x^{4} + {\left (5 \, b d^{4} + 4 \, a d^{2} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{15 \, d^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/15*(3*a*d^4 + (15*c*d^4 + 10*b*d^2*e^2 + 8*a*e^4)*x^4 + (5*b*d^4 + 4*a*d^2*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*
x + d)/(d^6*x^5)

________________________________________________________________________________________

giac [B]  time = 2.55, size = 1103, normalized size = 6.89 \[ -\frac {4 \, {\left (15 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{2} + 15 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{4} - 240 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{2} + 15 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{6} - 160 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{4} + 1440 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{2} - 80 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{6} + 800 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{4} - 3840 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{2} + 928 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{6} - 2560 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{4} + 3840 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{2} - 1280 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{6} + 3840 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{4} + 3840 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{6}\right )} e^{\left (-1\right )}}{15 \, {\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{2} - 4\right )}^{5} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-4/15*(15*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e
+ d)))^9*e^2 + 15*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) -
sqrt(-x*e + d)))^9*e^4 - 240*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*
sqrt(d) - sqrt(-x*e + d)))^7*e^2 + 15*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqr
t(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^6 - 160*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e
+ d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^7*e^4 + 1440*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d)
- sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^2 - 80*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e
+ d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^7*e^6 + 800*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d)
)/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^4 - 3840*c*d^4*((sqrt(2)*sqrt(d) - sqr
t(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^2 + 928*a*((sqrt(2)*sqrt(d)
- sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^5*e^6 - 2560*b*d^2*((sqrt
(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^3*e^4 + 3840*c
*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))*e^2
- 1280*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d))
)^3*e^6 + 3840*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt
(-x*e + d)))*e^4 + 3840*a*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) -
sqrt(-x*e + d)))*e^6)*e^(-1)/((((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqr
t(d) - sqrt(-x*e + d)))^2 - 4)^5*d^6)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 82, normalized size = 0.51 \[ -\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (8 a \,e^{4} x^{4}+10 b \,d^{2} e^{2} x^{4}+15 c \,d^{4} x^{4}+4 a \,d^{2} e^{2} x^{2}+5 b \,d^{4} x^{2}+3 a \,d^{4}\right )}{15 d^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/15*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(8*a*e^4*x^4+10*b*d^2*e^2*x^4+15*c*d^4*x^4+4*a*d^2*e^2*x^2+5*b*d^4*x^2+3*a*
d^4)/x^5/d^6

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 148, normalized size = 0.92 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{d^{2} x} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{3 \, d^{4} x} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{15 \, d^{6} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{3 \, d^{2} x^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{15 \, d^{4} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{5 \, d^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.73, size = 146, normalized size = 0.91 \[ -\frac {\sqrt {d-e\,x}\,\left (\frac {a}{5\,d}+\frac {x^4\,\left (15\,c\,d^5+10\,b\,d^3\,e^2+8\,a\,d\,e^4\right )}{15\,d^6}+\frac {x^5\,\left (15\,c\,d^4\,e+10\,b\,d^2\,e^3+8\,a\,e^5\right )}{15\,d^6}+\frac {x^2\,\left (5\,b\,d^5+4\,a\,d^3\,e^2\right )}{15\,d^6}+\frac {x^3\,\left (5\,b\,d^4\,e+4\,a\,d^2\,e^3\right )}{15\,d^6}+\frac {a\,e\,x}{5\,d^2}\right )}{x^5\,\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/(x^6*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

-((d - e*x)^(1/2)*(a/(5*d) + (x^4*(15*c*d^5 + 10*b*d^3*e^2 + 8*a*d*e^4))/(15*d^6) + (x^5*(8*a*e^5 + 10*b*d^2*e
^3 + 15*c*d^4*e))/(15*d^6) + (x^2*(5*b*d^5 + 4*a*d^3*e^2))/(15*d^6) + (x^3*(4*a*d^2*e^3 + 5*b*d^4*e))/(15*d^6)
+ (a*e*x)/(5*d^2)))/(x^5*(d + e*x)^(1/2))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**6/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________