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

Optimal. Leaf size=93 \[ -\frac {a \tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{d}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (b e^2+c d^2\right )}{e^4}+\frac {c (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^4} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 151, normalized size of antiderivative = 1.62, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {520, 1251, 897, 1153, 208} \[ -\frac {a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (b e^2+c d^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

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 897

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 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]

Rule 1251

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*} \int \frac {a+b x^2+c x^4}{x \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 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \frac {a+b x+c x^2}{x \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} \operatorname {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}}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \left (b+\frac {c d^2}{e^2}-\frac {c x^2}{e^2}+\frac {a}{\frac {d^2}{e^2}-\frac {x^2}{e^2}}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (a \sqrt {d^2-e^2 x^2}\right ) \operatorname {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 )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [B]  time = 0.88, size = 217, normalized size = 2.33 \[ \frac {-\frac {3 a \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d \sqrt {d-e x}}+\frac {\left (e^2 x^2-d^2\right ) \left (3 b e^2+2 c d^2+c e^2 x^2\right )}{e^4 \sqrt {d-e x}}+\frac {6 d \sqrt {d+e x} \left (b e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )}{e^4}-\frac {6 d^{3/2} \sqrt {\frac {e x}{d}+1} \left (b e^2+c d^2\right ) \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right )}{e^4}}{3 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-d^2 + e^2*x^2)*(2*c*d^2 + 3*b*e^2 + c*e^2*x^2))/(e^4*Sqrt[d - e*x]) - (6*d^(3/2)*(c*d^2 + b*e^2)*Sqrt[1 +
(e*x)/d]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/e^4 + (6*d*(c*d^2 + b*e^2)*Sqrt[d + e*x]*ArcTan[Sqrt[d - e*x
]/Sqrt[d + e*x]])/e^4 - (3*a*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(d*Sqrt[d - e*x]))/(3*Sqrt[d
+ e*x])

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fricas [A]  time = 0.96, size = 80, normalized size = 0.86 \[ \frac {3 \, a e^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (c d e^{2} x^{2} + 2 \, c d^{3} + 3 \, b d e^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{3 \, d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: schur row 1 1.55494e-10Francis algorithm
not precise enough for[1.0,-220.862474643,10162.5484803,-174574.213802,1032773.91614]schur row 1 3.66198e-10F
rancis algorithm not precise enough for[1.0,-467.909596927,45612.3731035,-1659969.6644,20804885.8013]Bad condi
tionned root j= 2 value 38.9905751966 ratio 0.000133135092941 mindist 0.00241522618125-a*ln(abs(2*sqrt(d+x*exp
(1))/(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))-1/2*(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))/sqrt(d+x*exp(
1))+2))/d+a*ln(abs(2*sqrt(d+x*exp(1))/(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))-1/2*(2*sqrt(2)*sqrt(d)-2*sqr
t(-d-x*exp(1)+2*d))/sqrt(d+x*exp(1))-2))/d+2*((-24*c*exp(1)^12/144/exp(1)^16*sqrt(d+x*exp(1))*sqrt(d+x*exp(1))
+48*c*exp(1)^12*d/144/exp(1)^16)*sqrt(d+x*exp(1))*sqrt(d+x*exp(1))+(-72*c*exp(1)^12*d^2-72*exp(1)^14*b)/144/ex
p(1)^16)*sqrt(d+x*exp(1))*sqrt(-d-x*exp(1)+2*d)

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maple [C]  time = 0.04, size = 143, normalized size = 1.54 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (\sqrt {-e^{2} x^{2}+d^{2}}\, c d \,e^{2} x^{2} \mathrm {csgn}\relax (d )+3 a \,e^{4} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+3 \sqrt {-e^{2} x^{2}+d^{2}}\, b d \,e^{2} \mathrm {csgn}\relax (d )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{3} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{3 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 1.00, size = 105, normalized size = 1.13 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{2}}{3 \, e^{2}} - \frac {a \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2}}{3 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.95, size = 161, normalized size = 1.73 \[ \frac {a\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {\sqrt {d-e\,x}\,\left (\frac {2\,c\,d^3}{3\,e^4}+\frac {c\,x^3}{3\,e}+\frac {c\,d\,x^2}{3\,e^2}+\frac {2\,c\,d^2\,x}{3\,e^3}\right )}{\sqrt {d+e\,x}}-\frac {\left (\frac {b\,d}{e^2}+\frac {b\,x}{e}\right )\,\sqrt {d-e\,x}}{\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 91.28, size = 304, normalized size = 3.27 \[ \frac {i a {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i b d {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {b d {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {i c d^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {c d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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