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]

-(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])

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Rubi [A]  time = 0.124514, antiderivative size = 157, normalized size of antiderivative = 1., 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 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]

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 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 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])

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*}

Mathematica [A]  time = 0.128342, 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]

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

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Maple [C]  time = 0.022, size = 146, normalized size = 0.9 \begin{align*} -{\frac{{\it csgn} \left ( e \right ) }{3\,{d}^{4}{x}^{3}e}\sqrt{-ex+d}\sqrt{ex+d} \left ( -3\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ){x}^{3}c{d}^{4}+2\,{\it csgn} \left ( e \right ){e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{2}a+3\,{\it csgn} \left ( e \right ) e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{2}b{d}^{2}+a\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{d}^{2}{\it csgn} \left ( e \right ) e \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.38233, size = 203, normalized size = 1.29 \begin{align*} -\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}} \end{align*}

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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Sympy [C]  time = 77.4416, size = 257, normalized size = 1.64 \begin{align*} \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} \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError