3.2.36 \(\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx\)

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

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Rubi [A]  time = 0.37, antiderivative size = 248, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {520, 1251, 897, 1157, 385, 199, 208} \begin {gather*} -\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}} \end {gather*}

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 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]

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 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 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 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^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*}

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Mathematica [A]  time = 0.19, size = 173, normalized size = 0.82 \begin {gather*} \frac {-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 )-d \left (d^2-e^2 x^2\right ) \left (a \left (8 d^4+10 d^2 e^2 x^2+15 e^4 x^4\right )+6 \left (2 b d^4 x^2+3 b d^2 e^2 x^4+4 c d^4 x^4\right )\right )}{48 d^7 x^6 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.44, size = 397, normalized size = 1.87 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right ) \left (-5 a e^6-6 b d^2 e^4-8 c d^4 e^2\right )}{8 d^7}-\frac {e^2 \sqrt {d+e x} \left (\frac {d+e x}{d-e x}+1\right ) \left (\frac {33 a e^4 (d+e x)^4}{(d-e x)^4}-\frac {28 a e^4 (d+e x)^3}{(d-e x)^3}+\frac {118 a e^4 (d+e x)^2}{(d-e x)^2}-\frac {28 a e^4 (d+e x)}{d-e x}+33 a e^4+\frac {30 b d^2 e^2 (d+e x)^4}{(d-e x)^4}-\frac {72 b d^2 e^2 (d+e x)^3}{(d-e x)^3}+\frac {84 b d^2 e^2 (d+e x)^2}{(d-e x)^2}-\frac {72 b d^2 e^2 (d+e x)}{d-e x}+30 b d^2 e^2+\frac {24 c d^4 (d+e x)^4}{(d-e x)^4}-\frac {96 c d^4 (d+e x)^3}{(d-e x)^3}+\frac {144 c d^4 (d+e x)^2}{(d-e x)^2}-\frac {96 c d^4 (d+e x)}{d-e x}+24 c d^4\right )}{24 d^7 \sqrt {d-e x} \left (\frac {d+e x}{d-e x}-1\right )^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(e^2*Sqrt[d + e*x]*(1 + (d + e*x)/(d - e*x))*(24*c*d^4 + 30*b*d^2*e^2 + 33*a*e^4 - (96*c*d^4*(d + e*x))/
(d - e*x) - (72*b*d^2*e^2*(d + e*x))/(d - e*x) - (28*a*e^4*(d + e*x))/(d - e*x) + (144*c*d^4*(d + e*x)^2)/(d -
e*x)^2 + (84*b*d^2*e^2*(d + e*x)^2)/(d - e*x)^2 + (118*a*e^4*(d + e*x)^2)/(d - e*x)^2 - (96*c*d^4*(d + e*x)^3
)/(d - e*x)^3 - (72*b*d^2*e^2*(d + e*x)^3)/(d - e*x)^3 - (28*a*e^4*(d + e*x)^3)/(d - e*x)^3 + (24*c*d^4*(d + e
*x)^4)/(d - e*x)^4 + (30*b*d^2*e^2*(d + e*x)^4)/(d - e*x)^4 + (33*a*e^4*(d + e*x)^4)/(d - e*x)^4))/(d^7*Sqrt[d
- e*x]*(-1 + (d + e*x)/(d - e*x))^6) + ((-8*c*d^4*e^2 - 6*b*d^2*e^4 - 5*a*e^6)*ArcTanh[Sqrt[d + e*x]/Sqrt[d -
e*x]])/(8*d^7)

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fricas [A]  time = 1.30, size = 137, normalized size = 0.65 \begin {gather*} \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 {gather*}

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

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]

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 0.002415226181251/exp(1)*(-1/12*(33*a*(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)))^11*exp(1)^7+30*b*d^2*(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)))^11*exp(1)^5+24*c*d^4*(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)))^11*exp(1)^
3+20*a*(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)))^9*exp(1)^7-168*b*d^2*(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)))^9*exp(1)^5-288*c*d^4*(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)))^9
*exp(1)^3+1440*a*(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)))^7*exp(1)^7+192*b*d^2*(2*sqrt(d+x*exp(1))/(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*e
xp(1)+2*d))-1/2*(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))/sqrt(d+x*exp(1)))^7*exp(1)^5+768*c*d^4*(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)))^7*exp(1)^3+5760*a*(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)))^5*exp(1)^7+768*b*d^2*(2*sqrt(d+x*exp(1))/(2*sqrt(2)*sqrt(d)-2*s
qrt(-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)))^5*exp(1)^5+3072*c*d^4*
(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)))^5*exp(1)^3+1280*a*(2*sqrt(d+x*exp(1))/(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))-1/2*(2*s
qrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))/sqrt(d+x*exp(1)))^3*exp(1)^7-10752*b*d^2*(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)))^3*exp(1)^
5-18432*c*d^4*(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)))^3*exp(1)^3+33792*a*(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)))*exp(1)^7+30720*b*d^2*(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
)))*exp(1)^5+24576*c*d^4*(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)))*exp(1)^3)/d^7/((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)))^2-4)^6-1/16*(5*a*exp(1)^7+6*b*
d^2*exp(1)^5+8*c*d^4*exp(1)^3)*ln(abs(2*sqrt(d+x*exp(1))/(2*sqrt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))-1/2*(2*sq
rt(2)*sqrt(d)-2*sqrt(-d-x*exp(1)+2*d))/sqrt(d+x*exp(1))+2))/d^7+1/16*(5*a*exp(1)^7+6*b*d^2*exp(1)^5+8*c*d^4*ex
p(1)^3)*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^7)

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maple [C]  time = 0.04, size = 306, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (15 a \,e^{6} x^{6} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+18 b \,d^{2} e^{4} x^{6} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+24 c \,d^{4} e^{2} x^{6} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+15 \sqrt {-e^{2} x^{2}+d^{2}}\, a d \,e^{4} x^{4} \mathrm {csgn}\relax (d )+18 \sqrt {-e^{2} x^{2}+d^{2}}\, b \,d^{3} e^{2} x^{4} \mathrm {csgn}\relax (d )+24 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{5} x^{4} \mathrm {csgn}\relax (d )+10 \sqrt {-e^{2} x^{2}+d^{2}}\, a \,d^{3} e^{2} x^{2} \mathrm {csgn}\relax (d )+12 \sqrt {-e^{2} x^{2}+d^{2}}\, b \,d^{5} x^{2} \mathrm {csgn}\relax (d )+8 \sqrt {-e^{2} x^{2}+d^{2}}\, a \,d^{5} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{48 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x^{6}} \end {gather*}

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*(
-e^2*x^2+d^2)^(1/2)*csgn(d)*d*x^4*a*e^4+18*(-e^2*x^2+d^2)^(1/2)*csgn(d)*d^3*x^4*b*e^2+24*(-e^2*x^2+d^2)^(1/2)*
csgn(d)*d^5*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 [A]  time = 1.03, size = 271, normalized size = 1.28 \begin {gather*} -\frac {c e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, b e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {5 \, a e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{7}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{8 \, d^{4} x^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{16 \, d^{6} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{4 \, d^{2} x^{4}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{24 \, d^{4} x^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{6 \, d^{2} x^{6}} \end {gather*}

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]

-1/2*c*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/8*b*e^4*log(2*d^2/abs(x) + 2*sqrt(-e^2*
x^2 + d^2)*d/abs(x))/d^5 - 5/16*a*e^6*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^7 - 1/2*sqrt(-e^2*
x^2 + d^2)*c/(d^2*x^2) - 3/8*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^2) - 5/16*sqrt(-e^2*x^2 + d^2)*a*e^4/(d^6*x^2)
- 1/4*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^4) - 5/24*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^4) - 1/6*sqrt(-e^2*x^2 + d^2)*
a/(d^2*x^6)

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mupad [B]  time = 20.05, size = 1621, normalized size = 7.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*e^4)/4 + (6*b*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (53*b*e^4*((d + e*x)^(1/2
) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - d^(1/2))^4) - (87*b*e^4*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2)
- d^(1/2))^6 + (657*b*e^4*((d + e*x)^(1/2) - d^(1/2))^8)/(4*((d - e*x)^(1/2) - d^(1/2))^8) - (121*b*e^4*((d +
e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10)/((256*d^5*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^
(1/2) - d^(1/2))^4 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 + (1536*d^5*((d +
e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) - d^(1/2))^8 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^
(1/2) - d^(1/2))^10 + (256*d^5*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12) - ((c*e^2*((d +
e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (c*e^2)/2 + (15*c*e^2*((d + e*x)^(1/2) - d^(1/2))^4)
/(2*((d - e*x)^(1/2) - d^(1/2))^4))/((16*d^3*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2))^2 - (3
2*d^3*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2) - d^(1/2))^4 + (16*d^3*((d + e*x)^(1/2) - d^(1/2))^6)/((
d - e*x)^(1/2) - d^(1/2))^6) + ((a*e^6)/6 + (4*a*e^6*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^(1/2)
)^2 + (71*a*e^6*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2) - d^(1/2))^4 - (1558*a*e^6*((d + e*x)^(1/2) -
d^(1/2))^6)/(3*((d - e*x)^(1/2) - d^(1/2))^6) - (540*a*e^6*((d + e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) - d
^(1/2))^8 + (4248*a*e^6*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10 - (7683*a*e^6*((d + e*x
)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12 + (5558*a*e^6*((d + e*x)^(1/2) - d^(1/2))^14)/((d - e*x)
^(1/2) - d^(1/2))^14 - (3643*a*e^6*((d + e*x)^(1/2) - d^(1/2))^16)/(2*((d - e*x)^(1/2) - d^(1/2))^16))/((4096*
d^7*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - d^(1/2))^6 - (24576*d^7*((d + e*x)^(1/2) - d^(1/2))^8)/(
(d - e*x)^(1/2) - d^(1/2))^8 + (61440*d^7*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) - d^(1/2))^10 - (81
920*d^7*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) - d^(1/2))^12 + (61440*d^7*((d + e*x)^(1/2) - d^(1/2)
)^14)/((d - e*x)^(1/2) - d^(1/2))^14 - (24576*d^7*((d + e*x)^(1/2) - d^(1/2))^16)/((d - e*x)^(1/2) - d^(1/2))^
16 + (4096*d^7*((d + e*x)^(1/2) - d^(1/2))^18)/((d - e*x)^(1/2) - d^(1/2))^18) - (5*a*e^6*log(((d + e*x)^(1/2)
- d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(16*d^7) - (3*b*e^4*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2)
- d^(1/2))))/(8*d^5) - (c*e^2*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(2*d^3) + (5*a*e^
6*log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 - 1))/(16*d^7) + (3*b*e^4*log(((d + e*x)^(1/
2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 - 1))/(8*d^5) + (c*e^2*log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e
*x)^(1/2) - d^(1/2))^2 - 1))/(2*d^3) + (197*a*e^6*((d + e*x)^(1/2) - d^(1/2))^2)/(8192*d^7*((d - e*x)^(1/2) -
d^(1/2))^2) + (5*a*e^6*((d + e*x)^(1/2) - d^(1/2))^4)/(4096*d^7*((d - e*x)^(1/2) - d^(1/2))^4) + (a*e^6*((d +
e*x)^(1/2) - d^(1/2))^6)/(24576*d^7*((d - e*x)^(1/2) - d^(1/2))^6) + (7*b*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/(
256*d^5*((d - e*x)^(1/2) - d^(1/2))^2) + (b*e^4*((d + e*x)^(1/2) - d^(1/2))^4)/(1024*d^5*((d - e*x)^(1/2) - d^
(1/2))^4) + (c*e^2*((d + e*x)^(1/2) - d^(1/2))^2)/(32*d^3*((d - e*x)^(1/2) - d^(1/2))^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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