3.2.42 \(\int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx\)

Optimal. Leaf size=226 \[ -\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}} \]

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Rubi [A]  time = 0.18, antiderivative size = 226, normalized size of antiderivative = 1.00, 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, 453, 271, 264} \begin {gather*} -\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (6 a e^2+7 b d^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-(a*(d^2 - e^2*x^2))/(7*d^2*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((7*b*d^2 + 6*a*e^2)*(d^2 - e^2*x^2))/(35*d^4*x
^5*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^3*Sqrt[d -
e*x]*Sqrt[d + e*x]) - (2*e^2*(35*c*d^4 + 28*b*d^2*e^2 + 24*a*e^4)*(d^2 - e^2*x^2))/(105*d^8*x*Sqrt[d - e*x]*S
qrt[d + e*x])

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]

Rule 271

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

Rubi steps

\begin {align*} \int \frac {a+b x^2+c x^4}{x^8 \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^8 \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 )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-7 b d^2-6 a e^2-7 c d^2 x^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{7 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (35 c d^4-4 e^2 \left (-7 b d^2-6 a e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{35 d^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (2 e^2 \left (35 c d^4-4 e^2 \left (-7 b d^2-6 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}{105 d^6 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{7 d^2 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (7 b d^2+6 a e^2\right ) \left (d^2-e^2 x^2\right )}{35 d^4 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^8 x \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 124, normalized size = 0.55 \begin {gather*} -\frac {\sqrt {d-e x} \sqrt {d+e x} \left (3 a \left (5 d^6+6 d^4 e^2 x^2+8 d^2 e^4 x^4+16 e^6 x^6\right )+7 b \left (3 d^6 x^2+4 d^4 e^2 x^4+8 d^2 e^4 x^6\right )+35 c d^4 x^4 \left (d^2+2 e^2 x^2\right )\right )}{105 d^8 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/105*(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*c*d^4*x^4*(d^2 + 2*e^2*x^2) + 7*b*(3*d^6*x^2 + 4*d^4*e^2*x^4 + 8*d^2*e
^4*x^6) + 3*a*(5*d^6 + 6*d^4*e^2*x^2 + 8*d^2*e^4*x^4 + 16*e^6*x^6)))/(d^8*x^7)

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IntegrateAlgebraic [B]  time = 0.22, size = 477, normalized size = 2.11 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (\frac {105 a e^7 (d+e x)^6}{(d-e x)^6}-\frac {210 a e^7 (d+e x)^5}{(d-e x)^5}+\frac {903 a e^7 (d+e x)^4}{(d-e x)^4}-\frac {636 a e^7 (d+e x)^3}{(d-e x)^3}+\frac {903 a e^7 (d+e x)^2}{(d-e x)^2}-\frac {210 a e^7 (d+e x)}{d-e x}+105 a e^7+\frac {105 b d^2 e^5 (d+e x)^6}{(d-e x)^6}-\frac {350 b d^2 e^5 (d+e x)^5}{(d-e x)^5}+\frac {791 b d^2 e^5 (d+e x)^4}{(d-e x)^4}-\frac {1092 b d^2 e^5 (d+e x)^3}{(d-e x)^3}+\frac {791 b d^2 e^5 (d+e x)^2}{(d-e x)^2}-\frac {350 b d^2 e^5 (d+e x)}{d-e x}+105 b d^2 e^5+\frac {105 c d^4 e^3 (d+e x)^6}{(d-e x)^6}-\frac {490 c d^4 e^3 (d+e x)^5}{(d-e x)^5}+\frac {1015 c d^4 e^3 (d+e x)^4}{(d-e x)^4}-\frac {1260 c d^4 e^3 (d+e x)^3}{(d-e x)^3}+\frac {1015 c d^4 e^3 (d+e x)^2}{(d-e x)^2}-\frac {490 c d^4 e^3 (d+e x)}{d-e x}+105 c d^4 e^3\right )}{105 d^8 \sqrt {d-e x} \left (\frac {d+e x}{d-e x}-1\right )^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-2*Sqrt[d + e*x]*(105*c*d^4*e^3 + 105*b*d^2*e^5 + 105*a*e^7 - (490*c*d^4*e^3*(d + e*x))/(d - e*x) - (350*b*d^
2*e^5*(d + e*x))/(d - e*x) - (210*a*e^7*(d + e*x))/(d - e*x) + (1015*c*d^4*e^3*(d + e*x)^2)/(d - e*x)^2 + (791
*b*d^2*e^5*(d + e*x)^2)/(d - e*x)^2 + (903*a*e^7*(d + e*x)^2)/(d - e*x)^2 - (1260*c*d^4*e^3*(d + e*x)^3)/(d -
e*x)^3 - (1092*b*d^2*e^5*(d + e*x)^3)/(d - e*x)^3 - (636*a*e^7*(d + e*x)^3)/(d - e*x)^3 + (1015*c*d^4*e^3*(d +
e*x)^4)/(d - e*x)^4 + (791*b*d^2*e^5*(d + e*x)^4)/(d - e*x)^4 + (903*a*e^7*(d + e*x)^4)/(d - e*x)^4 - (490*c*
d^4*e^3*(d + e*x)^5)/(d - e*x)^5 - (350*b*d^2*e^5*(d + e*x)^5)/(d - e*x)^5 - (210*a*e^7*(d + e*x)^5)/(d - e*x)
^5 + (105*c*d^4*e^3*(d + e*x)^6)/(d - e*x)^6 + (105*b*d^2*e^5*(d + e*x)^6)/(d - e*x)^6 + (105*a*e^7*(d + e*x)^
6)/(d - e*x)^6))/(105*d^8*Sqrt[d - e*x]*(-1 + (d + e*x)/(d - e*x))^7)

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fricas [A]  time = 1.52, size = 110, normalized size = 0.49 \begin {gather*} -\frac {{\left (15 \, a d^{6} + 2 \, {\left (35 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 24 \, a e^{6}\right )} x^{6} + {\left (35 \, c d^{6} + 28 \, b d^{4} e^{2} + 24 \, a d^{2} e^{4}\right )} x^{4} + 3 \, {\left (7 \, b d^{6} + 6 \, a d^{4} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{105 \, d^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(15*a*d^6 + 2*(35*c*d^4*e^2 + 28*b*d^2*e^4 + 24*a*e^6)*x^6 + (35*c*d^6 + 28*b*d^4*e^2 + 24*a*d^2*e^4)*x
^4 + 3*(7*b*d^6 + 6*a*d^4*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d)/(d^8*x^7)

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giac [B]  time = 4.73, size = 1517, normalized size = 6.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-4/105*(105*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)))^13*e^4 + 105*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)))^13*e^6 - 1960*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sq
rt(2)*sqrt(d) - sqrt(-x*e + d)))^11*e^4 + 105*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)))^13*e^8 - 1400*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)))^11*e^6 + 16240*c*d^4*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/s
qrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^4 - 840*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)))^11*e^8 + 12656*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^6 - 80640*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^4 + 14448*
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)))^9*e^8
- 69888*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^6 + 259840*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^4 - 40704*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^8 + 202496*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^6 - 501760*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)))^3*e^4 + 231168*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^8 - 358400*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^6 + 430080*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^4 - 215040*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^8 + 430080*
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^
6 + 430080*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^8)*e^(-1)/((((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-
x*e + d)))^2 - 4)^7*d^8)

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maple [A]  time = 0.01, size = 118, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 d^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(48*a*e^6*x^6+56*b*d^2*e^4*x^6+70*c*d^4*e^2*x^6+24*a*d^2*e^4*x^4+28*b*d^4*
e^2*x^4+35*c*d^6*x^4+18*a*d^4*e^2*x^2+21*b*d^6*x^2+15*a*d^6)/x^7/d^8

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maxima [A]  time = 1.01, size = 226, normalized size = 1.00 \begin {gather*} -\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} c e^{2}}{3 \, d^{4} x} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{4}}{15 \, d^{6} x} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{6}}{35 \, d^{8} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{3 \, d^{2} x^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{15 \, d^{4} x^{3}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{35 \, d^{6} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{5 \, d^{2} x^{5}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{35 \, d^{4} x^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{7 \, d^{2} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-2/3*sqrt(-e^2*x^2 + d^2)*c*e^2/(d^4*x) - 8/15*sqrt(-e^2*x^2 + d^2)*b*e^4/(d^6*x) - 16/35*sqrt(-e^2*x^2 + d^2)
*a*e^6/(d^8*x) - 1/3*sqrt(-e^2*x^2 + d^2)*c/(d^2*x^3) - 4/15*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^3) - 8/35*sqrt(
-e^2*x^2 + d^2)*a*e^4/(d^6*x^3) - 1/5*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^5) - 6/35*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*
x^5) - 1/7*sqrt(-e^2*x^2 + d^2)*a/(d^2*x^7)

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mupad [B]  time = 1.82, size = 218, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {d-e\,x}\,\left (\frac {a}{7\,d}+\frac {x^2\,\left (21\,b\,d^7+18\,a\,d^5\,e^2\right )}{105\,d^8}+\frac {x^4\,\left (35\,c\,d^7+28\,b\,d^5\,e^2+24\,a\,d^3\,e^4\right )}{105\,d^8}+\frac {x^7\,\left (70\,c\,d^4\,e^3+56\,b\,d^2\,e^5+48\,a\,e^7\right )}{105\,d^8}+\frac {x^3\,\left (21\,b\,d^6\,e+18\,a\,d^4\,e^3\right )}{105\,d^8}+\frac {x^5\,\left (35\,c\,d^6\,e+28\,b\,d^4\,e^3+24\,a\,d^2\,e^5\right )}{105\,d^8}+\frac {x^6\,\left (70\,c\,d^5\,e^2+56\,b\,d^3\,e^4+48\,a\,d\,e^6\right )}{105\,d^8}+\frac {a\,e\,x}{7\,d^2}\right )}{x^7\,\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/(x^8*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

-((d - e*x)^(1/2)*(a/(7*d) + (x^2*(21*b*d^7 + 18*a*d^5*e^2))/(105*d^8) + (x^4*(35*c*d^7 + 24*a*d^3*e^4 + 28*b*
d^5*e^2))/(105*d^8) + (x^7*(48*a*e^7 + 56*b*d^2*e^5 + 70*c*d^4*e^3))/(105*d^8) + (x^3*(18*a*d^4*e^3 + 21*b*d^6
*e))/(105*d^8) + (x^5*(24*a*d^2*e^5 + 28*b*d^4*e^3 + 35*c*d^6*e))/(105*d^8) + (x^6*(56*b*d^3*e^4 + 70*c*d^5*e^
2 + 48*a*d*e^6))/(105*d^8) + (a*e*x)/(7*d^2)))/(x^7*(d + e*x)^(1/2))

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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**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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