3.2.55 \(\int \frac {x^8 (A+B x^2+C x^4+D x^6)}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=381 \[ -\frac {x^9 \left (2 A b^3-a \left (23 a^2 D-16 a b C+9 b^2 B\right )\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {x \sqrt {a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 a b^7}+\frac {x^3 \sqrt {a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{24 a^2 b^6}-\frac {x^5 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {x^7 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right )}{16 b^{15/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.66, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {1804, 1585, 1263, 1584, 459, 288, 321, 217, 206} \begin {gather*} -\frac {x^9 \left (2 A b^3-a \left (23 a^2 D-16 a b C+9 b^2 B\right )\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {x^9 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {x^7 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}-\frac {x^5 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{30 a^2 b^5 \sqrt {a+b x^2}}+\frac {x^3 \sqrt {a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{24 a^2 b^6}-\frac {x \sqrt {a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 a b^7}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (198 a^2 b C-429 a^3 D-72 a b^2 B+16 A b^3\right )}{16 b^{15/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^9)/(7*a*(a + b*x^2)^(7/2)) - ((2*A*b^3 - a*(9*b^2*B - 16*a*b*C + 23*a
^2*D))*x^9)/(35*a^2*b^3*(a + b*x^2)^(5/2)) - ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*x^7)/(210*a^2
*b^4*(a + b*x^2)^(3/2)) + (D*x^9)/(6*b^3*(a + b*x^2)^(3/2)) - ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*
D))*x^5)/(30*a^2*b^5*Sqrt[a + b*x^2]) - ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*x*Sqrt[a + b*x^2])
/(16*a*b^7) + ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*x^3*Sqrt[a + b*x^2])/(24*a^2*b^6) + ((16*A*b
^3 - 72*a*b^2*B + 198*a^2*b*C - 429*a^3*D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(15/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 288

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

Rule 321

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

Rule 459

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

Rule 1263

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, 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*(f*x)^(m + 1)*(d + e*x^2)^(q + 1))/(2*d*f*(q + 1)), x] + Dist[f/(2*d*(
q + 1)), Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x*Qx + R*(m + 2*q + 3)*x, x], x], x]] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[q, -1] && GtQ[m, 0]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1804

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[((c*x)^m*(a + b*x^2)^(p + 1)*(a*g - b*f*x))/(2*a*b*(p + 1)), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^7 \left (\left (2 A b-\frac {9 a \left (b^2 B-a b C+a^2 D\right )}{b^2}\right ) x-7 a \left (C-\frac {a D}{b}\right ) x^3-7 a D x^5\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^8 \left (2 A b-\frac {9 a \left (b^2 B-a b C+a^2 D\right )}{b^2}-7 a \left (C-\frac {a D}{b}\right ) x^2-7 a D x^4\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^7 \left (\left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}\right ) x+\frac {35 a^2 D x^3}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^8 \left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}+\frac {35 a^2 D x^2}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (\frac {315 a^3 D}{b}-6 b \left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}\right )\right ) \int \frac {x^8}{\left (a+b x^2\right )^{5/2}} \, dx}{210 a^2 b^2}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (\frac {315 a^3 D}{b}-6 b \left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}\right )\right ) \int \frac {x^6}{\left (a+b x^2\right )^{3/2}} \, dx}{90 a^2 b^3}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {\left (\frac {315 a^3 D}{b}-6 b \left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}\right )\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{18 a^2 b^4}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^3 \sqrt {a+b x^2}}{24 a^2 b^6}+\frac {\left (\frac {315 a^3 D}{b}-6 b \left (8 A b-\frac {9 a \left (4 b^2 B-11 a b C+18 a^2 D\right )}{b^2}\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{24 a b^5}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x \sqrt {a+b x^2}}{16 a b^7}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^3 \sqrt {a+b x^2}}{24 a^2 b^6}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^7}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x \sqrt {a+b x^2}}{16 a b^7}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^3 \sqrt {a+b x^2}}{24 a^2 b^6}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^7}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^9}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\left (2 A b^3-a \left (9 b^2 B-16 a b C+23 a^2 D\right )\right ) x^9}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^7}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^5}{30 a^2 b^5 \sqrt {a+b x^2}}-\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x \sqrt {a+b x^2}}{16 a b^7}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) x^3 \sqrt {a+b x^2}}{24 a^2 b^6}+\frac {\left (16 A b^3-72 a b^2 B+198 a^2 b C-429 a^3 D\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{15/2}}\\ \end {align*}

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Mathematica [A]  time = 0.58, size = 273, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a+b x^2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 \sqrt {a} b^{15/2} \sqrt {\frac {b x^2}{a}+1}}+\frac {x \left (45045 a^6 D-2310 a^5 b \left (9 C-65 D x^2\right )+42 a^4 b^2 \left (180 B-1650 C x^2+4147 D x^4\right )-12 a^3 b^3 \left (140 A-2100 B x^2+6699 C x^4-6292 D x^6\right )+a^2 b^4 x^2 \left (-5600 A+29232 B x^2-34848 C x^4+5005 D x^6\right )-2 a b^5 x^4 \left (3248 A-6336 B x^2+1155 C x^4+455 D x^6\right )+4 b^6 x^6 \left (35 \left (6 B x^2+3 C x^4+2 D x^6\right )-704 A\right )\right )}{1680 b^7 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

(x*(45045*a^6*D - 2310*a^5*b*(9*C - 65*D*x^2) + 42*a^4*b^2*(180*B - 1650*C*x^2 + 4147*D*x^4) - 12*a^3*b^3*(140
*A - 2100*B*x^2 + 6699*C*x^4 - 6292*D*x^6) - 2*a*b^5*x^4*(3248*A - 6336*B*x^2 + 1155*C*x^4 + 455*D*x^6) + a^2*
b^4*x^2*(-5600*A + 29232*B*x^2 - 34848*C*x^4 + 5005*D*x^6) + 4*b^6*x^6*(-704*A + 35*(6*B*x^2 + 3*C*x^4 + 2*D*x
^6))))/(1680*b^7*(a + b*x^2)^(7/2)) + ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*Sqrt[a + b*x^2]*ArcS
inh[(Sqrt[b]*x)/Sqrt[a]])/(16*Sqrt[a]*b^(15/2)*Sqrt[1 + (b*x^2)/a])

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IntegrateAlgebraic [A]  time = 1.40, size = 306, normalized size = 0.80 \begin {gather*} \frac {\log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \left (429 a^3 D-198 a^2 b C+72 a b^2 B-16 A b^3\right )}{16 b^{15/2}}+\frac {45045 a^6 D x-20790 a^5 b C x+150150 a^5 b D x^3+7560 a^4 b^2 B x-69300 a^4 b^2 C x^3+174174 a^4 b^2 D x^5-1680 a^3 A b^3 x+25200 a^3 b^3 B x^3-80388 a^3 b^3 C x^5+75504 a^3 b^3 D x^7-5600 a^2 A b^4 x^3+29232 a^2 b^4 B x^5-34848 a^2 b^4 C x^7+5005 a^2 b^4 D x^9-6496 a A b^5 x^5+12672 a b^5 B x^7-2310 a b^5 C x^9-910 a b^5 D x^{11}-2816 A b^6 x^7+840 b^6 B x^9+420 b^6 C x^{11}+280 b^6 D x^{13}}{1680 b^7 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

(-1680*a^3*A*b^3*x + 7560*a^4*b^2*B*x - 20790*a^5*b*C*x + 45045*a^6*D*x - 5600*a^2*A*b^4*x^3 + 25200*a^3*b^3*B
*x^3 - 69300*a^4*b^2*C*x^3 + 150150*a^5*b*D*x^3 - 6496*a*A*b^5*x^5 + 29232*a^2*b^4*B*x^5 - 80388*a^3*b^3*C*x^5
+ 174174*a^4*b^2*D*x^5 - 2816*A*b^6*x^7 + 12672*a*b^5*B*x^7 - 34848*a^2*b^4*C*x^7 + 75504*a^3*b^3*D*x^7 + 840
*b^6*B*x^9 - 2310*a*b^5*C*x^9 + 5005*a^2*b^4*D*x^9 + 420*b^6*C*x^11 - 910*a*b^5*D*x^11 + 280*b^6*D*x^13)/(1680
*b^7*(a + b*x^2)^(7/2)) + ((-16*A*b^3 + 72*a*b^2*B - 198*a^2*b*C + 429*a^3*D)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^
2]])/(16*b^(15/2))

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fricas [A]  time = 1.77, size = 987, normalized size = 2.59 \begin {gather*} \left [\frac {105 \, {\left ({\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{8} + 429 \, D a^{7} - 198 \, C a^{6} b + 72 \, B a^{5} b^{2} - 16 \, A a^{4} b^{3} + 4 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{6} + 6 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{4} + 4 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (280 \, D b^{7} x^{13} - 70 \, {\left (13 \, D a b^{6} - 6 \, C b^{7}\right )} x^{11} + 35 \, {\left (143 \, D a^{2} b^{5} - 66 \, C a b^{6} + 24 \, B b^{7}\right )} x^{9} + 176 \, {\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{7} + 406 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{5} + 350 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{3} + 105 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{3360 \, {\left (b^{12} x^{8} + 4 \, a b^{11} x^{6} + 6 \, a^{2} b^{10} x^{4} + 4 \, a^{3} b^{9} x^{2} + a^{4} b^{8}\right )}}, \frac {105 \, {\left ({\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{8} + 429 \, D a^{7} - 198 \, C a^{6} b + 72 \, B a^{5} b^{2} - 16 \, A a^{4} b^{3} + 4 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{6} + 6 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{4} + 4 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (280 \, D b^{7} x^{13} - 70 \, {\left (13 \, D a b^{6} - 6 \, C b^{7}\right )} x^{11} + 35 \, {\left (143 \, D a^{2} b^{5} - 66 \, C a b^{6} + 24 \, B b^{7}\right )} x^{9} + 176 \, {\left (429 \, D a^{3} b^{4} - 198 \, C a^{2} b^{5} + 72 \, B a b^{6} - 16 \, A b^{7}\right )} x^{7} + 406 \, {\left (429 \, D a^{4} b^{3} - 198 \, C a^{3} b^{4} + 72 \, B a^{2} b^{5} - 16 \, A a b^{6}\right )} x^{5} + 350 \, {\left (429 \, D a^{5} b^{2} - 198 \, C a^{4} b^{3} + 72 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5}\right )} x^{3} + 105 \, {\left (429 \, D a^{6} b - 198 \, C a^{5} b^{2} + 72 \, B a^{4} b^{3} - 16 \, A a^{3} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, {\left (b^{12} x^{8} + 4 \, a b^{11} x^{6} + 6 \, a^{2} b^{10} x^{4} + 4 \, a^{3} b^{9} x^{2} + a^{4} b^{8}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[1/3360*(105*((429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a^7 - 198*C*a^6*b + 72*B*a^5
*b^2 - 16*A*a^4*b^3 + 4*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 1
98*C*a^4*b^3 + 72*B*a^3*b^4 - 16*A*a^2*b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4
)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(280*D*b^7*x^13 - 70*(13*D*a*b^6 - 6*C*b^7)
*x^11 + 35*(143*D*a^2*b^5 - 66*C*a*b^6 + 24*B*b^7)*x^9 + 176*(429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*
A*b^7)*x^7 + 406*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^5 + 350*(429*D*a^5*b^2 - 198*C*
a^4*b^3 + 72*B*a^3*b^4 - 16*A*a^2*b^5)*x^3 + 105*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x
)*sqrt(b*x^2 + a))/(b^12*x^8 + 4*a*b^11*x^6 + 6*a^2*b^10*x^4 + 4*a^3*b^9*x^2 + a^4*b^8), 1/1680*(105*((429*D*a
^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a^7 - 198*C*a^6*b + 72*B*a^5*b^2 - 16*A*a^4*b^3 +
4*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 198*C*a^4*b^3 + 72*B*a^
3*b^4 - 16*A*a^2*b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x^2)*sqrt(-b)*arctan
(sqrt(-b)*x/sqrt(b*x^2 + a)) + (280*D*b^7*x^13 - 70*(13*D*a*b^6 - 6*C*b^7)*x^11 + 35*(143*D*a^2*b^5 - 66*C*a*b
^6 + 24*B*b^7)*x^9 + 176*(429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^7 + 406*(429*D*a^4*b^3 - 19
8*C*a^3*b^4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^5 + 350*(429*D*a^5*b^2 - 198*C*a^4*b^3 + 72*B*a^3*b^4 - 16*A*a^2*b^
5)*x^3 + 105*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x)*sqrt(b*x^2 + a))/(b^12*x^8 + 4*a*b
^11*x^6 + 6*a^2*b^10*x^4 + 4*a^3*b^9*x^2 + a^4*b^8)]

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giac [A]  time = 0.64, size = 342, normalized size = 0.90 \begin {gather*} \frac {{\left ({\left ({\left ({\left (35 \, {\left (2 \, {\left (\frac {4 \, D x^{2}}{b} - \frac {13 \, D a^{4} b^{11} - 6 \, C a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac {143 \, D a^{5} b^{10} - 66 \, C a^{4} b^{11} + 24 \, B a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac {176 \, {\left (429 \, D a^{6} b^{9} - 198 \, C a^{5} b^{10} + 72 \, B a^{4} b^{11} - 16 \, A a^{3} b^{12}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {406 \, {\left (429 \, D a^{7} b^{8} - 198 \, C a^{6} b^{9} + 72 \, B a^{5} b^{10} - 16 \, A a^{4} b^{11}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {350 \, {\left (429 \, D a^{8} b^{7} - 198 \, C a^{7} b^{8} + 72 \, B a^{6} b^{9} - 16 \, A a^{5} b^{10}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac {105 \, {\left (429 \, D a^{9} b^{6} - 198 \, C a^{8} b^{7} + 72 \, B a^{7} b^{8} - 16 \, A a^{6} b^{9}\right )}}{a^{3} b^{13}}\right )} x}{1680 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (429 \, D a^{3} - 198 \, C a^{2} b + 72 \, B a b^{2} - 16 \, A b^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/1680*((((35*(2*(4*D*x^2/b - (13*D*a^4*b^11 - 6*C*a^3*b^12)/(a^3*b^13))*x^2 + (143*D*a^5*b^10 - 66*C*a^4*b^11
+ 24*B*a^3*b^12)/(a^3*b^13))*x^2 + 176*(429*D*a^6*b^9 - 198*C*a^5*b^10 + 72*B*a^4*b^11 - 16*A*a^3*b^12)/(a^3*
b^13))*x^2 + 406*(429*D*a^7*b^8 - 198*C*a^6*b^9 + 72*B*a^5*b^10 - 16*A*a^4*b^11)/(a^3*b^13))*x^2 + 350*(429*D*
a^8*b^7 - 198*C*a^7*b^8 + 72*B*a^6*b^9 - 16*A*a^5*b^10)/(a^3*b^13))*x^2 + 105*(429*D*a^9*b^6 - 198*C*a^8*b^7 +
72*B*a^7*b^8 - 16*A*a^6*b^9)/(a^3*b^13))*x/(b*x^2 + a)^(7/2) + 1/16*(429*D*a^3 - 198*C*a^2*b + 72*B*a*b^2 - 1
6*A*b^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(15/2)

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maple [A]  time = 0.30, size = 517, normalized size = 1.36 \begin {gather*} \frac {D x^{13}}{6 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {C \,x^{11}}{4 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {13 D a \,x^{11}}{24 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}+\frac {B \,x^{9}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {11 C a \,x^{9}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}+\frac {143 D a^{2} x^{9}}{48 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {A \,x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {9 B a \,x^{7}}{14 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {99 C \,a^{2} x^{7}}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}+\frac {429 D a^{3} x^{7}}{112 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4}}-\frac {A \,x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {9 B a \,x^{5}}{10 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}-\frac {99 C \,a^{2} x^{5}}{40 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{4}}+\frac {429 D a^{3} x^{5}}{80 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{5}}-\frac {A \,x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {3 B a \,x^{3}}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{4}}-\frac {33 C \,a^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}+\frac {143 D a^{3} x^{3}}{16 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{6}}-\frac {A x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {9 B a x}{2 \sqrt {b \,x^{2}+a}\, b^{5}}-\frac {99 C \,a^{2} x}{8 \sqrt {b \,x^{2}+a}\, b^{6}}+\frac {429 D a^{3} x}{16 \sqrt {b \,x^{2}+a}\, b^{7}}+\frac {A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {9}{2}}}-\frac {9 B a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}}+\frac {99 C \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {13}{2}}}-\frac {429 D a^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)

[Out]

1/4*C*x^11/b/(b*x^2+a)^(7/2)+99/8*C*a^2/b^(13/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+1/2*B*x^9/b/(b*x^2+a)^(7/2)-9/2
*B*a/b^(11/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+1/6*D*x^13/b/(b*x^2+a)^(7/2)-429/16*D*a^3/b^(15/2)*ln(b^(1/2)*x+(b
*x^2+a)^(1/2))-1/7*A*x^7/b/(b*x^2+a)^(7/2)-1/5*A/b^2*x^5/(b*x^2+a)^(5/2)-1/3*A/b^3*x^3/(b*x^2+a)^(3/2)-A/b^4*x
/(b*x^2+a)^(1/2)+9/10*B*a/b^3*x^5/(b*x^2+a)^(5/2)+3/2*B*a/b^4*x^3/(b*x^2+a)^(3/2)+9/2*B*a/b^5*x/(b*x^2+a)^(1/2
)-13/24*D*a/b^2*x^11/(b*x^2+a)^(7/2)+143/48*D*a^2/b^3*x^9/(b*x^2+a)^(7/2)+429/112*D*a^3/b^4*x^7/(b*x^2+a)^(7/2
)+429/80*D*a^3/b^5*x^5/(b*x^2+a)^(5/2)+143/16*D*a^3/b^6*x^3/(b*x^2+a)^(3/2)+429/16*D*a^3/b^7*x/(b*x^2+a)^(1/2)
-11/8*C*a/b^2*x^9/(b*x^2+a)^(7/2)-99/56*C*a^2/b^3*x^7/(b*x^2+a)^(7/2)-99/40*C*a^2/b^4*x^5/(b*x^2+a)^(5/2)-33/8
*C*a^2/b^5*x^3/(b*x^2+a)^(3/2)+A/b^(9/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))-99/8*C*a^2/b^6*x/(b*x^2+a)^(1/2)+9/14*B
*a/b^2*x^7/(b*x^2+a)^(7/2)

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maxima [B]  time = 1.89, size = 1221, normalized size = 3.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

1/6*D*x^13/((b*x^2 + a)^(7/2)*b) - 13/24*D*a*x^11/((b*x^2 + a)^(7/2)*b^2) + 1/4*C*x^11/((b*x^2 + a)^(7/2)*b) +
143/48*D*a^2*x^9/((b*x^2 + a)^(7/2)*b^3) - 11/8*C*a*x^9/((b*x^2 + a)^(7/2)*b^2) + 1/2*B*x^9/((b*x^2 + a)^(7/2
)*b) - 1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b
^3) + 16*a^3/((b*x^2 + a)^(7/2)*b^4))*A*x + 429/560*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2
)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/((b*x^2 + a)^(7/2)*b^4))*D*a^3*x/b^3 - 99/280*(35*x^6/((b
*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/((b*x^2 +
a)^(7/2)*b^4))*C*a^2*x/b^2 + 9/70*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^
2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/((b*x^2 + a)^(7/2)*b^4))*B*a*x/b + 143/80*D*a^3*x*(15*x^4/((b*x^2 + a)^(5/2
)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^4 - 33/40*C*a^2*x*(15*x^4/((b*x^2 +
a)^(5/2)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^3 + 3/10*B*a*x*(15*x^4/((b*
x^2 + a)^(5/2)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^2 - 1/15*A*x*(15*x^4/(
(b*x^2 + a)^(5/2)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b + 143/16*D*a^3*x*(3
*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^5 - 33/8*C*a^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*
a/((b*x^2 + a)^(3/2)*b^2))/b^4 + 3/2*B*a*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^3 - 1
/3*A*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 429/16*D*a^4*x^3/((b*x^2 + a)^(5/2)*b
^6) - 99/8*C*a^3*x^3/((b*x^2 + a)^(5/2)*b^5) + 9/2*B*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - A*a*x^3/((b*x^2 + a)^(5
/2)*b^3) - 19877/560*D*a^3*x/(sqrt(b*x^2 + a)*b^7) - 2431/560*D*a^4*x/((b*x^2 + a)^(3/2)*b^7) + 12441/560*D*a^
5*x/((b*x^2 + a)^(5/2)*b^7) + 4587/280*C*a^2*x/(sqrt(b*x^2 + a)*b^6) + 561/280*C*a^3*x/((b*x^2 + a)^(3/2)*b^6)
- 2871/280*C*a^4*x/((b*x^2 + a)^(5/2)*b^6) - 417/70*B*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*B*a^2*x/((b*x^2 + a)^
(3/2)*b^5) + 261/70*B*a^3*x/((b*x^2 + a)^(5/2)*b^5) + 139/105*A*x/(sqrt(b*x^2 + a)*b^4) + 17/105*A*a*x/((b*x^2
+ a)^(3/2)*b^4) - 29/35*A*a^2*x/((b*x^2 + a)^(5/2)*b^4) - 429/16*D*a^3*arcsinh(b*x/sqrt(a*b))/b^(15/2) + 99/8
*C*a^2*arcsinh(b*x/sqrt(a*b))/b^(13/2) - 9/2*B*a*arcsinh(b*x/sqrt(a*b))/b^(11/2) + A*arcsinh(b*x/sqrt(a*b))/b^
(9/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^8\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(9/2),x)

[Out]

int((x^8*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(9/2), x)

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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(x**8*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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