3.2.45 \(\int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [145]

Optimal. Leaf size=292 \[ -\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^8 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {8 e^4 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^{10} x \sqrt {d-e x} \sqrt {d+e x}} \]

[Out]

-1/9*a*(-e^2*x^2+d^2)/d^2/x^9/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-1/63*(8*a*e^2+9*b*d^2)*(-e^2*x^2+d^2)/d^4/x^7/(-e*x
+d)^(1/2)/(e*x+d)^(1/2)-1/105*(16*a*e^4+18*b*d^2*e^2+21*c*d^4)*(-e^2*x^2+d^2)/d^6/x^5/(-e*x+d)^(1/2)/(e*x+d)^(
1/2)-4/315*e^2*(16*a*e^4+18*b*d^2*e^2+21*c*d^4)*(-e^2*x^2+d^2)/d^8/x^3/(-e*x+d)^(1/2)/(e*x+d)^(1/2)-8/315*e^4*
(16*a*e^4+18*b*d^2*e^2+21*c*d^4)*(-e^2*x^2+d^2)/d^10/x/(-e*x+d)^(1/2)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {534, 1279, 464, 277, 270} \begin {gather*} -\frac {8 e^4 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^{10} x \sqrt {d-e x} \sqrt {d+e x}}-\frac {4 e^2 \left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{315 d^8 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{105 d^6 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (8 a e^2+9 b d^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/9*(a*(d^2 - e^2*x^2))/(d^2*x^9*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((9*b*d^2 + 8*a*e^2)*(d^2 - e^2*x^2))/(63*d^4
*x^7*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(105*d^6*x^5*Sqrt[d
- e*x]*Sqrt[d + e*x]) - (4*e^2*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^8*x^3*Sqrt[d - e*
x]*Sqrt[d + e*x]) - (8*e^4*(21*c*d^4 + 18*b*d^2*e^2 + 16*a*e^4)*(d^2 - e^2*x^2))/(315*d^10*x*Sqrt[d - e*x]*Sqr
t[d + e*x])

Rule 270

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 277

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 464

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 534

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 1279

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^{10} \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^{10} \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 )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-9 b d^2-8 a e^2-9 c d^2 x^2}{x^8 \sqrt {d^2-e^2 x^2}} \, dx}{9 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (63 c d^4-6 e^2 \left (-9 b d^2-8 a e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{63 d^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (4 e^2 \left (63 c d^4-6 e^2 \left (-9 b d^2-8 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}{315 d^6 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^8 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (8 e^4 \left (63 c d^4-6 e^2 \left (-9 b d^2-8 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}{945 d^8 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{9 d^2 x^9 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (9 b d^2+8 a e^2\right ) \left (d^2-e^2 x^2\right )}{63 d^4 x^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{105 d^6 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^8 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {8 e^4 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \left (d^2-e^2 x^2\right )}{315 d^{10} x \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 158, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {d-e x} \sqrt {d+e x} \left (21 c d^4 x^4 \left (3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right )+9 b \left (5 d^8 x^2+6 d^6 e^2 x^4+8 d^4 e^4 x^6+16 d^2 e^6 x^8\right )+a \left (35 d^8+40 d^6 e^2 x^2+48 d^4 e^4 x^4+64 d^2 e^6 x^6+128 e^8 x^8\right )\right )}{315 d^{10} x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/315*(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*c*d^4*x^4*(3*d^4 + 4*d^2*e^2*x^2 + 8*e^4*x^4) + 9*b*(5*d^8*x^2 + 6*d^6
*e^2*x^4 + 8*d^4*e^4*x^6 + 16*d^2*e^6*x^8) + a*(35*d^8 + 40*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 64*d^2*e^6*x^6 + 12
8*e^8*x^8)))/(d^10*x^9)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 2.
time = 0.16, size = 158, normalized size = 0.54

method	result	size
gosper	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 x^{9} d^{10}}\)	\(154\)
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 x^{9} d^{10}}\)	\(154\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \mathrm {csgn}\left (e \right )^{2} \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 d^{10} x^{9}}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/315*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*csgn(e)^2/d^10*(128*a*e^8*x^8+144*b*d^2*e^6*x^8+168*c*d^4*e^4*x^8+64*a*d^2
*e^6*x^6+72*b*d^4*e^4*x^6+84*c*d^6*e^2*x^6+48*a*d^4*e^4*x^4+54*b*d^6*e^2*x^4+63*c*d^8*x^4+40*a*d^6*e^2*x^2+45*
b*d^8*x^2+35*a*d^8)/x^9

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Maxima [A]
time = 0.49, size = 283, normalized size = 0.97 \begin {gather*} -\frac {8 \, \sqrt {-x^{2} e^{2} + d^{2}} c e^{4}}{15 \, d^{6} x} - \frac {4 \, \sqrt {-x^{2} e^{2} + d^{2}} c e^{2}}{15 \, d^{4} x^{3}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} c}{5 \, d^{2} x^{5}} - \frac {16 \, \sqrt {-x^{2} e^{2} + d^{2}} b e^{6}}{35 \, d^{8} x} - \frac {8 \, \sqrt {-x^{2} e^{2} + d^{2}} b e^{4}}{35 \, d^{6} x^{3}} - \frac {6 \, \sqrt {-x^{2} e^{2} + d^{2}} b e^{2}}{35 \, d^{4} x^{5}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} b}{7 \, d^{2} x^{7}} - \frac {128 \, \sqrt {-x^{2} e^{2} + d^{2}} a e^{8}}{315 \, d^{10} x} - \frac {64 \, \sqrt {-x^{2} e^{2} + d^{2}} a e^{6}}{315 \, d^{8} x^{3}} - \frac {16 \, \sqrt {-x^{2} e^{2} + d^{2}} a e^{4}}{105 \, d^{6} x^{5}} - \frac {8 \, \sqrt {-x^{2} e^{2} + d^{2}} a e^{2}}{63 \, d^{4} x^{7}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} a}{9 \, d^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-8/15*sqrt(-x^2*e^2 + d^2)*c*e^4/(d^6*x) - 4/15*sqrt(-x^2*e^2 + d^2)*c*e^2/(d^4*x^3) - 1/5*sqrt(-x^2*e^2 + d^2
)*c/(d^2*x^5) - 16/35*sqrt(-x^2*e^2 + d^2)*b*e^6/(d^8*x) - 8/35*sqrt(-x^2*e^2 + d^2)*b*e^4/(d^6*x^3) - 6/35*sq
rt(-x^2*e^2 + d^2)*b*e^2/(d^4*x^5) - 1/7*sqrt(-x^2*e^2 + d^2)*b/(d^2*x^7) - 128/315*sqrt(-x^2*e^2 + d^2)*a*e^8
/(d^10*x) - 64/315*sqrt(-x^2*e^2 + d^2)*a*e^6/(d^8*x^3) - 16/105*sqrt(-x^2*e^2 + d^2)*a*e^4/(d^6*x^5) - 8/63*s
qrt(-x^2*e^2 + d^2)*a*e^2/(d^4*x^7) - 1/9*sqrt(-x^2*e^2 + d^2)*a/(d^2*x^9)

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Fricas [A]
time = 0.44, size = 144, normalized size = 0.49 \begin {gather*} -\frac {{\left (35 \, a d^{8} + 8 \, {\left (21 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 16 \, a e^{8}\right )} x^{8} + 4 \, {\left (21 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 16 \, a d^{2} e^{6}\right )} x^{6} + 3 \, {\left (21 \, c d^{8} + 18 \, b d^{6} e^{2} + 16 \, a d^{4} e^{4}\right )} x^{4} + 5 \, {\left (9 \, b d^{8} + 8 \, a d^{6} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, d^{10} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(35*a*d^8 + 8*(21*c*d^4*e^4 + 18*b*d^2*e^6 + 16*a*e^8)*x^8 + 4*(21*c*d^6*e^2 + 18*b*d^4*e^4 + 16*a*d^2*
e^6)*x^6 + 3*(21*c*d^8 + 18*b*d^6*e^2 + 16*a*d^4*e^4)*x^4 + 5*(9*b*d^8 + 8*a*d^6*e^2)*x^2)*sqrt(e*x + d)*sqrt(
-e*x + d)/(d^10*x^9)

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Sympy [F(-1)] Timed out
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**10/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1931 vs. \(2 (263) = 526\).
time = 7.39, size = 1931, normalized size = 6.61 \begin {gather*} -\frac {4 \, {\left (315 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{17} e^{6} + 315 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{17} e^{8} - 6720 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{15} e^{6} + 315 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{17} e^{10} - 5040 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{15} e^{8} + 76608 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{13} e^{6} - 3360 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{15} e^{10} + 68544 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{13} e^{8} - 580608 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{11} e^{6} + 76608 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{13} e^{10} - 509184 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{11} e^{8} + 2892288 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{6} - 327168 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{11} e^{10} + 2363904 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{8} - 9289728 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{6} + 2728448 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{9} e^{10} - 8146944 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{8} + 19611648 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{6} - 5234688 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{7} e^{10} + 17547264 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{8} - 27525120 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{6} + 19611648 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{5} e^{10} - 20643840 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{8} + 20643840 \, c d^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{6} - 13762560 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{3} e^{10} + 20643840 \, b d^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{8} + 20643840 \, a {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )} e^{10}\right )} e^{\left (-1\right )}}{315 \, {\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}{\sqrt {x e + d}} - \frac {\sqrt {x e + d}}{\sqrt {2} \sqrt {d} - \sqrt {-x e + d}}\right )}^{2} - 4\right )}^{9} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-4/315*(315*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)))^17*e^6 + 315*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)))^17*e^8 - 6720*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)))^15*e^6 + 315*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)))^17*e^10 - 5040*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)))^15*e^8 + 76608*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^6 - 3360*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)))^15*e^10 + 68544*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^8 - 580608*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)))^11*e^6 +
76608*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^10 - 509184*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - s
qrt(-x*e + d)))^11*e^8 + 2892288*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^6 - 327168*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^10 + 2363904*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^8 - 9289728*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^6 + 2728448*a*((sqrt(2)*sqrt(d) - sq
rt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqrt(-x*e + d)))^9*e^10 - 8146944*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^8 + 1961164
8*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^6 - 5234688*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^10 + 17547264*b*d^2*((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*s
qrt(d) - sqrt(-x*e + d)))^5*e^8 - 27525120*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^6 + 19611648*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^10 - 20643840*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^8 + 20643840*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^6 - 13762560*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^10 + 20643840*
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^
8 + 20643840*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^10)*e^(-1)/((((sqrt(2)*sqrt(d) - sqrt(-x*e + d))/sqrt(x*e + d) - sqrt(x*e + d)/(sqrt(2)*sqrt(d) - sqr
t(-x*e + d)))^2 - 4)^9*d^10)

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Mupad [B]
time = 1.87, size = 290, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {d-e\,x}\,\left (\frac {a}{9\,d}+\frac {x^2\,\left (45\,b\,d^9+40\,a\,d^7\,e^2\right )}{315\,d^{10}}+\frac {x^6\,\left (84\,c\,d^7\,e^2+72\,b\,d^5\,e^4+64\,a\,d^3\,e^6\right )}{315\,d^{10}}+\frac {x^7\,\left (84\,c\,d^6\,e^3+72\,b\,d^4\,e^5+64\,a\,d^2\,e^7\right )}{315\,d^{10}}+\frac {x^4\,\left (63\,c\,d^9+54\,b\,d^7\,e^2+48\,a\,d^5\,e^4\right )}{315\,d^{10}}+\frac {x^9\,\left (168\,c\,d^4\,e^5+144\,b\,d^2\,e^7+128\,a\,e^9\right )}{315\,d^{10}}+\frac {x^3\,\left (45\,b\,d^8\,e+40\,a\,d^6\,e^3\right )}{315\,d^{10}}+\frac {x^5\,\left (63\,c\,d^8\,e+54\,b\,d^6\,e^3+48\,a\,d^4\,e^5\right )}{315\,d^{10}}+\frac {x^8\,\left (168\,c\,d^5\,e^4+144\,b\,d^3\,e^6+128\,a\,d\,e^8\right )}{315\,d^{10}}+\frac {a\,e\,x}{9\,d^2}\right )}{x^9\,\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^10*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

-((d - e*x)^(1/2)*(a/(9*d) + (x^2*(45*b*d^9 + 40*a*d^7*e^2))/(315*d^10) + (x^6*(64*a*d^3*e^6 + 72*b*d^5*e^4 +
84*c*d^7*e^2))/(315*d^10) + (x^7*(64*a*d^2*e^7 + 72*b*d^4*e^5 + 84*c*d^6*e^3))/(315*d^10) + (x^4*(63*c*d^9 + 4
8*a*d^5*e^4 + 54*b*d^7*e^2))/(315*d^10) + (x^9*(128*a*e^9 + 144*b*d^2*e^7 + 168*c*d^4*e^5))/(315*d^10) + (x^3*
(40*a*d^6*e^3 + 45*b*d^8*e))/(315*d^10) + (x^5*(48*a*d^4*e^5 + 54*b*d^6*e^3 + 63*c*d^8*e))/(315*d^10) + (x^8*(
144*b*d^3*e^6 + 168*c*d^5*e^4 + 128*a*d*e^8))/(315*d^10) + (a*e*x)/(9*d^2)))/(x^9*(d + e*x)^(1/2))

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