### 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]

-(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])

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Rubi [A]  time = 0.145075, antiderivative size = 160, normalized size of antiderivative = 1., 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 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 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 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]

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 = 0.12539, size = 87, normalized size = 0.54 $-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a \left (4 d^2 e^2 x^2+3 d^4+8 e^4 x^4\right )+5 b d^2 x^2 \left (d^2+2 e^2 x^2\right )+15 c d^4 x^4\right )}{15 d^6 x^5}$

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]

-(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)))/(15*d^6*x^5)

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Maple [A]  time = 0.005, size = 82, normalized size = 0.5 \begin{align*} -{\frac{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}}{15\,{x}^{5}{d}^{6}}\sqrt{ex+d}\sqrt{-ex+d}} \end{align*}

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

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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^6/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.41799, size = 173, normalized size = 1.08 \begin{align*} -\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}} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [B]  time = 1.83967, size = 1489, normalized size = 9.31 \begin{align*} \text{result too large to display} \end{align*}

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)