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

Optimal. Leaf size=157 \[ -\frac {\left (d^2-e^2 x^2\right ) \left (2 a e^2+3 b d^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e \sqrt {d-e x} \sqrt {d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {520, 1265, 451, 217, 203} \[ -\frac {\left (d^2-e^2 x^2\right ) \left (2 a e^2+3 b d^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e \sqrt {d-e x} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 451

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[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -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^4 \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^4 \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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-3 b d^2-2 a e^2-3 c d^2 x^2}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (3 b d^2+2 a e^2\right ) \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (c \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{\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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (3 b d^2+2 a e^2\right ) \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (c \sqrt {d^2-e^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (3 b d^2+2 a e^2\right ) \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 81, normalized size = 0.52 \[ -\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a \left (d^2+2 e^2 x^2\right )+3 b d^2 x^2\right )}{3 d^4 x^3}-\frac {2 c \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Sqrt[d - e*x]*Sqrt[d + e*x]*(3*b*d^2*x^2 + a*(d^2 + 2*e^2*x^2)))/(d^4*x^3) - (2*c*ArcTan[Sqrt[d - e*x]/S
qrt[d + e*x]])/e

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fricas [A]  time = 0.83, size = 90, normalized size = 0.57 \[ -\frac {6 \, c d^{4} x^{3} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right ) + {\left (a d^{2} e + {\left (3 \, b d^{2} e + 2 \, a e^{3}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{3 \, d^{4} e x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*c*d^4*x^3*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)) + (a*d^2*e + (3*b*d^2*e + 2*a*e^3)*x^2)*sqr
t(e*x + d)*sqrt(-e*x + d))/(d^4*e*x^3)

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giac [B]  time = 1.47, size = 555, normalized size = 3.54 \[ \frac {1}{3} \, {\left (3 \, {\left (\pi + 2 \, \arctan \left (\frac {\sqrt {x e + d} {\left (\frac {{\left (\sqrt {2} \sqrt {d} - \sqrt {-x e + d}\right )}^{2}}{x e + d} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {-x e + d}\right )}}\right )\right )} c - \frac {4 \, {\left (3 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{2} + 3 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{4} - 24 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{2} - 8 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{4} + 48 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{2} + 48 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{4}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{2} - 4\right )}^{3} d^{4}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(pi + 2*arctan(1/2*sqrt(x*e + d)*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))^2/(x*e + d) - 1)/(sqrt(2)*sqrt(d)
- sqrt(-x*e + d))))*c - 4*(3*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^2 + 3*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^4 - 24*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^2 - 8*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^4 + 48*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^2 + 48*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^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)))^2 - 4)^3*d^4))*e^(-1)

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maple [C]  time = 0.03, size = 146, normalized size = 0.93 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (-3 c \,d^{4} x^{3} \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, a \,e^{3} x^{2} \mathrm {csgn}\relax (e )+3 \sqrt {-e^{2} x^{2}+d^{2}}\, b \,d^{2} e \,x^{2} \mathrm {csgn}\relax (e )+\sqrt {-e^{2} x^{2}+d^{2}}\, a \,d^{2} e \,\mathrm {csgn}\relax (e )\right ) \mathrm {csgn}\relax (e )}{3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^4*(-3*arctan(1/(-e^2*x^2+d^2)^(1/2)*e*x*csgn(e))*x^3*c*d^4+2*(-e^2*x^2+d^2
)^(1/2)*csgn(e)*e^3*x^2*a+3*(-e^2*x^2+d^2)^(1/2)*csgn(e)*e*x^2*b*d^2+a*(-e^2*x^2+d^2)^(1/2)*d^2*csgn(e)*e)*csg
n(e)/(-e^2*x^2+d^2)^(1/2)/x^3/e

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maxima [A]  time = 1.00, size = 85, normalized size = 0.54 \[ \frac {c \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{d^{2} x} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{3 \, d^{4} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{3 \, d^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.27, size = 138, normalized size = 0.88 \[ -\frac {4\,c\,\mathrm {atan}\left (\frac {e\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}{\sqrt {e^2}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )}{\sqrt {e^2}}-\frac {\left (\frac {b}{d}+\frac {b\,e\,x}{d^2}\right )\,\sqrt {d-e\,x}}{x\,\sqrt {d+e\,x}}-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{3\,d}+\frac {2\,a\,e^2\,x^2}{3\,d^3}+\frac {2\,a\,e^3\,x^3}{3\,d^4}+\frac {a\,e\,x}{3\,d^2}\right )}{x^3\,\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 116.43, size = 257, normalized size = 1.64 \[ \frac {i a e^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {5}{2}, \frac {5}{2}, 3 \\2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 3 & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {a e^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 1 & \\\frac {7}{4}, \frac {9}{4} & \frac {3}{2}, 2, 2, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {i b e {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {b e {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {i c {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} + \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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