### 3.138 $$\int \frac{a+b x^2+c x^4}{x^7 \sqrt{d-e x} \sqrt{d+e x}} \, dx$$

Optimal. Leaf size=212 $-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2}-\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{6 d^2 x^6}$

[Out]

-(a*Sqrt[d - e*x]*Sqrt[d + e*x])/(6*d^2*x^6) - ((6*b*d^2 + 5*a*e^2)*Sqrt[d - e*x]*Sqrt[d + e*x])/(24*d^4*x^4)
- ((8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d - e*x]*Sqrt[d + e*x])/(16*d^6*x^2) - (e^2*(8*c*d^4 + 6*b*d^2*e^2 +
5*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(16*d^7*Sqrt[d - e*x]*Sqrt[d + e*x])

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Rubi [A]  time = 0.372665, antiderivative size = 248, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {520, 1251, 897, 1157, 385, 199, 208} $-\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(6*d^2*x^6*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((6*b*d^2 + 5*a*e^2)*(d^2 - e^2*x^2))/(24*d^4*x
^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*(d^2 - e^2*x^2))/(16*d^6*x^2*Sqrt[d - e*x
]*Sqrt[d + e*x]) - (e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/
(16*d^7*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 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]

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 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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]

Rubi steps

\begin{align*} \int \frac{a+b x^2+c x^4}{x^7 \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^7 \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^4 \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}}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^4} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{-5 a-\frac{6 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac{6 c d^2 x^2}{e^4}}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^3} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{6 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (\left (6 b+\frac{8 c d^2}{e^2}+\frac{5 a e^2}{d^2}\right ) \sqrt{d^2-e^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^2} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{8 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (e^2 \left (6 b+\frac{8 c d^2}{e^2}+\frac{5 a e^2}{d^2}\right ) \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 )}{16 d^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.20004, size = 173, normalized size = 0.82 $\frac{-d \left (d^2-e^2 x^2\right ) \left (a \left (10 d^2 e^2 x^2+8 d^4+15 e^4 x^4\right )+6 \left (3 b d^2 e^2 x^4+2 b d^4 x^2+4 c d^4 x^4\right )\right )-3 e^2 x^6 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{48 d^7 x^6 \sqrt{d-e x} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^7*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-(d*(d^2 - e^2*x^2)*(6*(2*b*d^4*x^2 + 4*c*d^4*x^4 + 3*b*d^2*e^2*x^4) + a*(8*d^4 + 10*d^2*e^2*x^2 + 15*e^4*x^4
))) - 3*e^2*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*x^6*Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(48*d^7*
x^6*Sqrt[d - e*x]*Sqrt[d + e*x])

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Maple [C]  time = 0.033, size = 306, normalized size = 1.4 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{48\,{d}^{7}{x}^{6}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 15\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}a{e}^{6}+18\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}b{d}^{2}{e}^{4}+24\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}c{d}^{4}{e}^{2}+15\,{\it csgn} \left ( d \right ) d\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}a{e}^{4}+18\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}b{e}^{2}+24\,{\it csgn} \left ( d \right ){d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}c+10\,{\it csgn} \left ( d \right ){x}^{2}a{d}^{3}{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( d \right ){x}^{2}b{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+8\,{\it csgn} \left ( d \right ) a{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \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^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/48*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^7*(15*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^6*a*e^6+18*ln(2*d*((-e
^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^6*b*d^2*e^4+24*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^6*c*d^4*e^2+15*c
sgn(d)*d*(-e^2*x^2+d^2)^(1/2)*x^4*a*e^4+18*csgn(d)*d^3*(-e^2*x^2+d^2)^(1/2)*x^4*b*e^2+24*csgn(d)*d^5*(-e^2*x^2
+d^2)^(1/2)*x^4*c+10*csgn(d)*x^2*a*d^3*e^2*(-e^2*x^2+d^2)^(1/2)+12*csgn(d)*x^2*b*d^5*(-e^2*x^2+d^2)^(1/2)+8*cs
gn(d)*a*d^5*(-e^2*x^2+d^2)^(1/2))*csgn(d)/(-e^2*x^2+d^2)^(1/2)/x^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^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44854, size = 298, normalized size = 1.41 \begin{align*} \frac{3 \,{\left (8 \, c d^{4} e^{2} + 6 \, b d^{2} e^{4} + 5 \, a e^{6}\right )} x^{6} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) -{\left (8 \, a d^{5} + 3 \,{\left (8 \, c d^{5} + 6 \, b d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 2 \,{\left (6 \, b d^{5} + 5 \, a d^{3} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{48 \, d^{7} x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/48*(3*(8*c*d^4*e^2 + 6*b*d^2*e^4 + 5*a*e^6)*x^6*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x) - (8*a*d^5 + 3*(8*
c*d^5 + 6*b*d^3*e^2 + 5*a*d*e^4)*x^4 + 2*(6*b*d^5 + 5*a*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(d^7*x^6)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**7/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: MellinTransformStripError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

sage0*x