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

Optimal. Leaf size=102 \[ -\frac {a \sqrt {d-e x} \sqrt {d+e x}}{d^2 x}-\frac {\left (2 b e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )}{e^3}+\frac {c x (e x-d) \sqrt {d+e x}}{2 e^2 \sqrt {d-e x}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 155, normalized size of antiderivative = 1.52, 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, 388, 217, 203} \[ -\frac {a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}+\frac {\sqrt {d^2-e^2 x^2} \left (2 b e^2+c d^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(d^2 - e^2*x^2))/(d^2*x*Sqrt[d - e*x]*Sqrt[d + e*x])) - (c*x*(d^2 - e^2*x^2))/(2*e^2*Sqrt[d - e*x]*Sqrt[d
+ e*x]) + ((c*d^2 + 2*b*e^2)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^3*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 388

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

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^2 \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^2 \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 )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-b d^2-c d^2 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (2 b+\frac {c d^2}{e^2}\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (2 b+\frac {c d^2}{e^2}\right ) \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 )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (c d^2+2 b e^2\right ) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 135, normalized size = 1.32 \[ -\frac {\frac {e \sqrt {d-e x} \sqrt {d+e x} \left (2 a e^2+c d^2 x^2\right )}{d^2 x}+4 \left (b e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )-\frac {2 c d^{5/2} \sqrt {\frac {e x}{d}+1} \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right )}{\sqrt {d+e x}}}{2 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((e*Sqrt[d - e*x]*Sqrt[d + e*x]*(2*a*e^2 + c*d^2*x^2))/(d^2*x) - (2*c*d^(5/2)*Sqrt[1 + (e*x)/d]*ArcSin[Sq
rt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[d + e*x] + 4*(c*d^2 + b*e^2)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^3

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fricas [A]  time = 0.96, size = 90, normalized size = 0.88 \[ -\frac {2 \, {\left (c d^{4} + 2 \, b d^{2} e^{2}\right )} x \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right ) + {\left (c d^{2} e x^{2} + 2 \, a e^{3}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{2 \, d^{2} e^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.05, size = 257, normalized size = 2.52 \[ \frac {1}{2} \, {\left ({\left (\pi + 2 \, \arctan \left (\frac {\sqrt {x e + d} {\left (\frac {{\left (\sqrt {2} \sqrt {d} - \sqrt {-x e + d}\right )}^{2}}{x e + d} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {-x e + d}\right )}}\right )\right )} {\left (c d^{2} + 2 \, b e^{2}\right )} e^{\left (-2\right )} - {\left ({\left (x e + d\right )} c e^{\left (-2\right )} - c d e^{\left (-2\right )}\right )} \sqrt {x e + d} \sqrt {-x e + d} - \frac {8 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{2}}{{\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{2} - 4\right )} d^{2}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 148, normalized size = 1.45 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (-2 b \,d^{2} e^{2} x \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )-c \,d^{4} x \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )+\sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{2} e \,x^{2} \mathrm {csgn}\relax (e )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, a \,e^{3} \mathrm {csgn}\relax (e )\right ) \mathrm {csgn}\relax (e )}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 73, normalized size = 0.72 \[ \frac {c d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{3}} + \frac {b \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c x}{2 \, e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 7.00, size = 306, normalized size = 3.00 \[ \frac {\frac {14\,c\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^3}-\frac {14\,c\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^5}+\frac {2\,c\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^7}-\frac {2\,c\,d^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{\sqrt {d-e\,x}-\sqrt {d}}}{e^3\,{\left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+1\right )}^4}-\frac {4\,b\,\mathrm {atan}\left (\frac {e\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}{\sqrt {e^2}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )}{\sqrt {e^2}}+\frac {2\,c\,d^2\,\mathrm {atan}\left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{e^3}-\frac {\left (\frac {a}{d}+\frac {a\,e\,x}{d^2}\right )\,\sqrt {d-e\,x}}{x\,\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((14*c*d^2*((d + e*x)^(1/2) - d^(1/2))^3)/((d - e*x)^(1/2) - d^(1/2))^3 - (14*c*d^2*((d + e*x)^(1/2) - d^(1/2)
)^5)/((d - e*x)^(1/2) - d^(1/2))^5 + (2*c*d^2*((d + e*x)^(1/2) - d^(1/2))^7)/((d - e*x)^(1/2) - d^(1/2))^7 - (
2*c*d^2*((d + e*x)^(1/2) - d^(1/2)))/((d - e*x)^(1/2) - d^(1/2)))/(e^3*(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*
x)^(1/2) - d^(1/2))^2 + 1)^4) - (4*b*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) + (2*c*d^2*atan(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/e^3 - ((a/d + (a
*e*x)/d^2)*(d - e*x)^(1/2))/(x*(d + e*x)^(1/2))

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sympy [C]  time = 104.02, size = 287, normalized size = 2.81 \[ \frac {i a e {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {a e {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {i b {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} + \frac {b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} - \frac {i c d^{2} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{3}} + \frac {c d^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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