\(\int \frac {x^2 (A+B x^2+C x^4+D x^6+F x^8)}{(a+b x^2)^{9/2}} \, dx\) [172]

Optimal result

Integrand size = 37, antiderivative size = 261 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (A b^4-a \left (b^3 B-a b^2 C+a^2 b D-a^3 F\right )\right ) x^3}{7 a b^4 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b D-9 a F) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \]

[Out]

1/7*(A*b^4-a*(B*b^3-C*a*b^2+D*a^2*b-F*a^3))*x^3/a/b^4/(b*x^2+a)^(7/2)+1/35*(4*A*b^4+a*(3*B*b^3-10*C*a*b^2+17*D
*a^2*b-24*F*a^3))*x^3/a^2/b^4/(b*x^2+a)^(5/2)+1/105*(8*A*b^4+a*(6*B*b^3+15*C*a*b^2-71*D*a^2*b+162*F*a^3))*x^3/
a^3/b^4/(b*x^2+a)^(3/2)+1/2*(2*D*b-9*F*a)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(11/2)-(D*b-4*F*a)*x/b^5/(b*x^2
+a)^(1/2)+1/2*F*x*(b*x^2+a)^(1/2)/b^5

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {1818, 1814, 1599, 1277, 1598, 466, 396, 223, 212} \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x^3 \left (a \left (162 a^3 F-71 a^2 b D+15 a b^2 C+6 b^3 B\right )+8 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac {x^3 \left (a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )+4 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {x^3 \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b D-9 a F)}{2 b^{11/2}}-\frac {x (b D-4 a F)}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5} \]

[In]

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

[Out]

((A/a - (b^3*B - a*b^2*C + a^2*b*D - a^3*F)/b^4)*x^3)/(7*(a + b*x^2)^(7/2)) + ((4*A*b^4 + a*(3*b^3*B - 10*a*b^
2*C + 17*a^2*b*D - 24*a^3*F))*x^3)/(35*a^2*b^4*(a + b*x^2)^(5/2)) + ((8*A*b^4 + a*(6*b^3*B + 15*a*b^2*C - 71*a
^2*b*D + 162*a^3*F))*x^3)/(105*a^3*b^4*(a + b*x^2)^(3/2)) - ((b*D - 4*a*F)*x)/(b^5*Sqrt[a + b*x^2]) + (F*x*Sqr
t[a + b*x^2])/(2*b^5) + ((2*b*D - 9*a*F)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(11/2))

Rule 212

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

Rule 223

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 396

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

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1277

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 1598

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 1599

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 1814

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c, Int[(c*x)^(m + 1)*PolynomialQ
uotient[Pq, x, x]*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0
]

Rule 1818

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*} \text {integral}& = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x \left (-\left (\left (4 A b+\frac {3 a \left (b^3 B-a b^2 C+a^2 b D-a^3 F\right )}{b^3}\right ) x\right )-\frac {7 a \left (b^2 C-a b D+a^2 F\right ) x^3}{b^2}-7 a \left (D-\frac {a F}{b}\right ) x^5-7 a F x^7\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^2 \left (-4 A b-\frac {3 a \left (b^3 B-a b^2 C+a^2 b D-a^3 F\right )}{b^3}-\frac {7 a \left (b^2 C-a b D+a^2 F\right ) x^2}{b^2}-7 a \left (D-\frac {a F}{b}\right ) x^4-7 a F x^6\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x \left (\left (8 A b^2+3 a \left (2 b B+5 a C-\frac {12 a^2 D}{b}+\frac {19 a^3 F}{b^2}\right )\right ) x+35 a^2 \left (D-\frac {2 a F}{b}\right ) x^3+35 a^2 F x^5\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^2 \left (8 A b^2+3 a \left (2 b B+5 a C-\frac {12 a^2 D}{b}+\frac {19 a^3 F}{b^2}\right )+35 a^2 \left (D-\frac {2 a F}{b}\right ) x^2+35 a^2 F x^4\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {x \left (-\frac {105 a^3 (b D-3 a F) x}{b^2}-\frac {105 a^3 F x^3}{b}\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^2} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (-\frac {105 a^3 (b D-3 a F)}{b^2}-\frac {105 a^3 F x^2}{b}\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^2} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {\int \frac {\frac {105 a^3 (b D-4 a F)}{b}+105 a^3 F x^2}{\sqrt {a+b x^2}} \, dx}{105 a^3 b^4} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b D-9 a F) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^5} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b D-9 a F) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^5} \\ & = \frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x^3}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (4 A b^4+a \left (3 b^3 B-10 a b^2 C+17 a^2 b D-24 a^3 F\right )\right ) x^3}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 A b^4+a \left (6 b^3 B+15 a b^2 C-71 a^2 b D+162 a^3 F\right )\right ) x^3}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {(b D-4 a F) x}{b^5 \sqrt {a+b x^2}}+\frac {F x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b D-9 a F) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (945 a^7 F+16 A b^7 x^6+4 a b^6 x^4 \left (14 A+3 B x^2\right )-210 a^6 b \left (D-15 F x^2\right )+a^3 b^4 x^6 \left (-352 D+105 F x^2\right )+14 a^5 b^2 x^2 \left (-50 D+261 F x^2\right )+4 a^4 b^3 x^4 \left (-203 D+396 F x^2\right )+2 a^2 b^5 x^2 \left (35 A+21 B x^2+15 C x^4\right )\right )}{210 a^3 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(2 b D-9 a F) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{11/2}} \]

[In]

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

[Out]

(x*(945*a^7*F + 16*A*b^7*x^6 + 4*a*b^6*x^4*(14*A + 3*B*x^2) - 210*a^6*b*(D - 15*F*x^2) + a^3*b^4*x^6*(-352*D +
105*F*x^2) + 14*a^5*b^2*x^2*(-50*D + 261*F*x^2) + 4*a^4*b^3*x^4*(-203*D + 396*F*x^2) + 2*a^2*b^5*x^2*(35*A +
21*B*x^2 + 15*C*x^4)))/(210*a^3*b^5*(a + b*x^2)^(7/2)) + ((2*b*D - 9*a*F)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt
[a + b*x^2])])/b^(11/2)

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} \left (D b -\frac {9 F a}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (x^{2} a^{2} \left (\frac {3}{7} C \,x^{4}+\frac {3}{5} x^{2} B +A \right ) b^{\frac {11}{2}}+\frac {4 a \left (\frac {3 x^{2} B}{14}+A \right ) x^{4} b^{\frac {13}{2}}}{5}-3 a^{6} \left (-15 F \,x^{2}+D\right ) b^{\frac {3}{2}}-10 a^{5} \left (-\frac {261 F \,x^{2}}{50}+D\right ) x^{2} b^{\frac {5}{2}}-\frac {58 a^{4} \left (-\frac {396 F \,x^{2}}{203}+D\right ) x^{4} b^{\frac {7}{2}}}{5}-\frac {176 \left (-\frac {105 F \,x^{2}}{352}+D\right ) a^{3} x^{6} b^{\frac {9}{2}}}{35}+\frac {8 A \,b^{\frac {15}{2}} x^{6}}{35}+\frac {27 F \sqrt {b}\, a^{7}}{2}\right ) x}{3 b^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3}}\)	\(193\)
default	\(D \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+C \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+A \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )+F \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )\)	\(597\)

[In]

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

[Out]

1/3/b^(11/2)*(3*(b*x^2+a)^(7/2)*a^3*(D*b-9/2*F*a)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(x^2*a^2*(3/7*C*x^4+3/5*x
^2*B+A)*b^(11/2)+4/5*a*(3/14*x^2*B+A)*x^4*b^(13/2)-3*a^6*(-15*F*x^2+D)*b^(3/2)-10*a^5*(-261/50*F*x^2+D)*x^2*b^
(5/2)-58/5*a^4*(-396/203*F*x^2+D)*x^4*b^(7/2)-176/35*(-105/352*F*x^2+D)*a^3*x^6*b^(9/2)+8/35*A*b^(15/2)*x^6+27
/2*F*b^(1/2)*a^7)*x)/(b*x^2+a)^(7/2)/a^3

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.70 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [-\frac {105 \, {\left (9 \, F a^{8} - 2 \, D a^{7} b + {\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \, {\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \, {\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\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 (105 \, F a^{3} b^{5} x^{9} + 2 \, {\left (792 \, F a^{4} b^{4} - 176 \, D a^{3} b^{5} + 15 \, C a^{2} b^{6} + 6 \, B a b^{7} + 8 \, A b^{8}\right )} x^{7} + 14 \, {\left (261 \, F a^{5} b^{3} - 58 \, D a^{4} b^{4} + 3 \, B a^{2} b^{6} + 4 \, A a b^{7}\right )} x^{5} + 70 \, {\left (45 \, F a^{6} b^{2} - 10 \, D a^{5} b^{3} + A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{420 \, {\left (a^{3} b^{10} x^{8} + 4 \, a^{4} b^{9} x^{6} + 6 \, a^{5} b^{8} x^{4} + 4 \, a^{6} b^{7} x^{2} + a^{7} b^{6}\right )}}, \frac {105 \, {\left (9 \, F a^{8} - 2 \, D a^{7} b + {\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \, {\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \, {\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, F a^{3} b^{5} x^{9} + 2 \, {\left (792 \, F a^{4} b^{4} - 176 \, D a^{3} b^{5} + 15 \, C a^{2} b^{6} + 6 \, B a b^{7} + 8 \, A b^{8}\right )} x^{7} + 14 \, {\left (261 \, F a^{5} b^{3} - 58 \, D a^{4} b^{4} + 3 \, B a^{2} b^{6} + 4 \, A a b^{7}\right )} x^{5} + 70 \, {\left (45 \, F a^{6} b^{2} - 10 \, D a^{5} b^{3} + A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{3} b^{10} x^{8} + 4 \, a^{4} b^{9} x^{6} + 6 \, a^{5} b^{8} x^{4} + 4 \, a^{6} b^{7} x^{2} + a^{7} b^{6}\right )}}\right ] \]

[In]

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

[Out]

[-1/420*(105*(9*F*a^8 - 2*D*a^7*b + (9*F*a^4*b^4 - 2*D*a^3*b^5)*x^8 + 4*(9*F*a^5*b^3 - 2*D*a^4*b^4)*x^6 + 6*(9
*F*a^6*b^2 - 2*D*a^5*b^3)*x^4 + 4*(9*F*a^7*b - 2*D*a^6*b^2)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt
(b)*x - a) - 2*(105*F*a^3*b^5*x^9 + 2*(792*F*a^4*b^4 - 176*D*a^3*b^5 + 15*C*a^2*b^6 + 6*B*a*b^7 + 8*A*b^8)*x^7
+ 14*(261*F*a^5*b^3 - 58*D*a^4*b^4 + 3*B*a^2*b^6 + 4*A*a*b^7)*x^5 + 70*(45*F*a^6*b^2 - 10*D*a^5*b^3 + A*a^2*b
^6)*x^3 + 105*(9*F*a^7*b - 2*D*a^6*b^2)*x)*sqrt(b*x^2 + a))/(a^3*b^10*x^8 + 4*a^4*b^9*x^6 + 6*a^5*b^8*x^4 + 4*
a^6*b^7*x^2 + a^7*b^6), 1/210*(105*(9*F*a^8 - 2*D*a^7*b + (9*F*a^4*b^4 - 2*D*a^3*b^5)*x^8 + 4*(9*F*a^5*b^3 - 2
*D*a^4*b^4)*x^6 + 6*(9*F*a^6*b^2 - 2*D*a^5*b^3)*x^4 + 4*(9*F*a^7*b - 2*D*a^6*b^2)*x^2)*sqrt(-b)*arctan(sqrt(-b
)*x/sqrt(b*x^2 + a)) + (105*F*a^3*b^5*x^9 + 2*(792*F*a^4*b^4 - 176*D*a^3*b^5 + 15*C*a^2*b^6 + 6*B*a*b^7 + 8*A*
b^8)*x^7 + 14*(261*F*a^5*b^3 - 58*D*a^4*b^4 + 3*B*a^2*b^6 + 4*A*a*b^7)*x^5 + 70*(45*F*a^6*b^2 - 10*D*a^5*b^3 +
A*a^2*b^6)*x^3 + 105*(9*F*a^7*b - 2*D*a^6*b^2)*x)*sqrt(b*x^2 + a))/(a^3*b^10*x^8 + 4*a^4*b^9*x^6 + 6*a^5*b^8*
x^4 + 4*a^6*b^7*x^2 + a^7*b^6)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6987 vs. \(2 (253) = 506\).

Time = 108.77 (sec) , antiderivative size = 6987, normalized size of antiderivative = 26.77 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

A*(35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b*
*2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x
**2/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**
(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqr
t(1 + b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/
a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b
**4*x**8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(
1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*
a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a))) + B*(7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*
sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + 2
*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqr
t(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a))) + C*x**7/(7*a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(
7/2)*b*x**2*sqrt(1 + b*x**2/a) + 21*a**(5/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt(1 + b*x*
*2/a)) + D*(105*a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1
+ b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x
**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/
a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a))
+ 630*a**(203/2)*b**46*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x
**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a
) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) +
630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 157
5*a**(201/2)*b**47*x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/
a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) +
2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*
a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 2100*a*
*(199/2)*b**48*x**6*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) +
630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100
*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(
195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(19
7/2)*b**49*x**8*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630
*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**
(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/
2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 630*a**(195/2)*
b**50*x**10*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**
(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199
/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b
**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 105*a**(193/2)*b**5
1*x**12*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203
/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*
b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(1
09/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 105*a**102*b**(91/2)*x/
(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(20
1/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)
*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(
111/2)*x**12*sqrt(1 + b*x**2/a)) - 665*a**101*b**(93/2)*x**3/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 63
0*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a*
*(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195
/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 1771*a**100*b*
*(95/2)*x**5/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a)
+ 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 15
75*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a*
*(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 2549*a**99*b**(97/2)*x**7/(105*a**(205/2)*b**(99/2)*sqrt(1 + b
*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2
/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a)
+ 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 2
096*a**98*b**(99/2)*x**9/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1
+ b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*
x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2
/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 934*a**97*b**(101/2)*x**11/(105*a**(205/2)*b**(99/
2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sq
rt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1
+ b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b
*x**2/a)) - 176*a**96*b**(103/2)*x**13/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2
)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x*
*6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*s
qrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a))) + F*(-315*a**(311/2)*b**66*sqrt(1 + b
*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**
2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sq
rt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1
+ b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 1890*a**(309/2)*b**67*x**2*sqrt(1 + b*x**2
/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqr
t(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1
+ b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*
x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 4725*a**(307/2)*b**68*x**4*sqrt(1 + b*x**2/a)*a
sinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 +
b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x
**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/
a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 6300*a**(305/2)*b**69*x**6*sqrt(1 + b*x**2/a)*asinh(
sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x*
*2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a
) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) +
70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 4725*a**(303/2)*b**70*x**8*sqrt(1 + b*x**2/a)*asinh(sqrt(
b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a)
+ 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1
050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a*
*(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 1890*a**(301/2)*b**71*x**10*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x
/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1
050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*
a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(29
7/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 315*a**(299/2)*b**72*x**12*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt
(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a
**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(3
01/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*
b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 315*a**155*b**(133/2)*x/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) +
420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*
a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(2
99/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 1995*a**154*b
**(135/2)*x**3/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a
) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) +
1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a
**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 5313*a**153*b**(137/2)*x**5/(70*a**(309/2)*b**(143/2)*sqrt(1
+ b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x
**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/
a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) +
7647*a**152*b**(139/2)*x**7/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqr
t(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1
+ b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*
x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 6323*a**151*b**(141/2)*x**9/(70*a**(309/2)*b**(
143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**
4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sq
rt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1
+ b*x**2/a)) + 2907*a**150*b**(143/2)*x**11/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(
145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/
2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x*
*10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 633*a**149*b**(145/2)*x**13/(70*
a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)
*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**
(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2
)*x**12*sqrt(1 + b*x**2/a)) + 35*a**148*b**(147/2)*x**15/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a*
*(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(30
3/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*
b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (235) = 470\).

Time = 0.23 (sec) , antiderivative size = 826, normalized size of antiderivative = 3.16 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*F*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))*D*x + 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))*F*a*x/b
+ 3/10*F*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*D*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 - 1/2*C*x^5/((b*x^2 + a)^(7/2)*b) + 3/2*F*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*D*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 9/2*F*a^2*x^3/((b*x^2 + a)^(
5/2)*b^4) - D*a*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*C*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1/4*B*x^3/((b*x^2 + a)^(7/
2)*b) - 417/70*F*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*F*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*F*a^3*x/((b*x^2 +
a)^(5/2)*b^5) + 139/105*D*x/(sqrt(b*x^2 + a)*b^4) + 17/105*D*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*D*a^2*x/((b*x
^2 + a)^(5/2)*b^4) + 1/14*C*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*C*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*C*a*x/((b*x^2 +
a)^(5/2)*b^3) - 15/56*C*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 3/140*B*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*B*x/(sqrt(b*
x^2 + a)*a^2*b^2) + 1/35*B*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*B*a*x/((b*x^2 + a)^(7/2)*b^2) - 1/7*A*x/((b*x^2
+ a)^(7/2)*b) + 8/105*A*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*A*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*A*x/((b*x^2 + a
)^(5/2)*a*b) - 9/2*F*a*arcsinh(b*x/sqrt(a*b))/b^(11/2) + D*arcsinh(b*x/sqrt(a*b))/b^(9/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (\frac {105 \, F x^{2}}{b} + \frac {2 \, {\left (792 \, F a^{4} b^{7} - 176 \, D a^{3} b^{8} + 15 \, C a^{2} b^{9} + 6 \, B a b^{10} + 8 \, A b^{11}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {14 \, {\left (261 \, F a^{5} b^{6} - 58 \, D a^{4} b^{7} + 3 \, B a^{2} b^{9} + 4 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {70 \, {\left (45 \, F a^{6} b^{5} - 10 \, D a^{5} b^{6} + A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {105 \, {\left (9 \, F a^{7} b^{4} - 2 \, D a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (9 \, F a - 2 \, D b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \]

[In]

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

[Out]

1/210*((((105*F*x^2/b + 2*(792*F*a^4*b^7 - 176*D*a^3*b^8 + 15*C*a^2*b^9 + 6*B*a*b^10 + 8*A*b^11)/(a^3*b^9))*x^
2 + 14*(261*F*a^5*b^6 - 58*D*a^4*b^7 + 3*B*a^2*b^9 + 4*A*a*b^10)/(a^3*b^9))*x^2 + 70*(45*F*a^6*b^5 - 10*D*a^5*
b^6 + A*a^2*b^9)/(a^3*b^9))*x^2 + 105*(9*F*a^7*b^4 - 2*D*a^6*b^5)/(a^3*b^9))*x/(b*x^2 + a)^(7/2) + 1/2*(9*F*a
- 2*D*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^2\,\left (A+B\,x^2+C\,x^4+F\,x^8+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]

[In]

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

[Out]

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