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

Optimal. Leaf size=261 \[ \frac{x^3 \left (a \left (-71 a^2 b D+162 a^3 F+15 a b^2 C+6 b^3 B\right )+8 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (a \left (17 a^2 b D-24 a^3 F-10 a b^2 C+3 b^3 B\right )+4 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac{x^3 \left (A b^4-a \left (a^2 b D+a^3 (-F)-a b^2 C+b^3 B\right )\right )}{7 a b^4 \left (a+b x^2\right )^{7/2}}-\frac{x (b D-4 a F)}{b^5 \sqrt{a+b x^2}}+\frac{(2 b D-9 a F) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}+\frac{F x \sqrt{a+b x^2}}{2 b^5} \]

[Out]

((A*b^4 - a*(b^3*B - a*b^2*C + a^2*b*D - a^3*F))*x^3)/(7*a*b^4*(a + b*x^2)^(7/2)) + ((4*A*b^4 + a*(3*b^3*B - 1
0*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*Sqrt[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))

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Rubi [A]  time = 0.716416, antiderivative size = 257, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {1804, 1800, 1585, 1263, 1584, 455, 388, 217, 206} \[ \frac{x^3 \left (a \left (-71 a^2 b D+162 a^3 F+15 a b^2 C+6 b^3 B\right )+8 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (a \left (17 a^2 b D-24 a^3 F-10 a b^2 C+3 b^3 B\right )+4 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac{x^3 \left (\frac{A}{a}-\frac{a^2 b D+a^3 (-F)-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}-\frac{x (b D-4 a F)}{b^5 \sqrt{a+b x^2}}+\frac{(2 b D-9 a F) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}+\frac{F x \sqrt{a+b x^2}}{2 b^5} \]

Antiderivative was successfully verified.

[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 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]

Rule 1800

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 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 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 455

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 388

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 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 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])

Rubi steps

\begin{align*} \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 (\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 (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-\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) \operatorname{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]  time = 0.504825, size = 221, normalized size = 0.85 \[ \frac{\sqrt{b} x \left (2 a^2 b^5 x^2 \left (35 A+21 B x^2+15 C x^4\right )+14 a^5 b^2 x^2 \left (261 F x^2-50 D\right )+4 a^4 b^3 x^4 \left (396 F x^2-203 D\right )+a^3 b^4 x^6 \left (105 F x^2-352 D\right )-210 a^6 b \left (D-15 F x^2\right )+945 a^7 F+4 a b^6 x^4 \left (14 A+3 B x^2\right )+16 A b^7 x^6\right )+105 a^{7/2} \left (a+b x^2\right )^3 \sqrt{\frac{b x^2}{a}+1} (2 b D-9 a F) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{210 a^3 b^{11/2} \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[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]

(Sqrt[b]*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)) + 105*a^(7/2)*(2*b*D - 9*a*F)*(a + b*x^2)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]
*x)/Sqrt[a]])/(210*a^3*b^(11/2)*(a + b*x^2)^(7/2))

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Maple [B]  time = 0.01, size = 478, normalized size = 1.8 \begin{align*} -{\frac{5\,aC{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Cx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Fa{x}^{3}}{2\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,Fax}{2\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{Ax}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aCx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{35\,ab} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,Ax}{105\,b{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,Ax}{105\,b{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{Cx}{7\,{b}^{3}a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Bax}{28\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Bx}{35\,{b}^{2}a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{D\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{2\,Bx}{35\,{b}^{2}{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{F{x}^{9}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Fa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}}-{\frac{C{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Cx}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{B{x}^{3}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Bx}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{D{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{D{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{D{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,Fa{x}^{7}}{14\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,Fa{x}^{5}}{10\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{Dx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*C/b^2*a*x^3/(b*x^2+a)^(7/2)-15/56*C/b^3*a^2*x/(b*x^2+a)^(7/2)+3/2*F/b^4*a*x^3/(b*x^2+a)^(3/2)+9/2*F/b^5*a
*x/(b*x^2+a)^(1/2)-1/7*A/b*x/(b*x^2+a)^(7/2)+3/56*C/b^3*a*x/(b*x^2+a)^(5/2)+1/35*A/b/a*x/(b*x^2+a)^(5/2)+4/105
*A/b/a^2*x/(b*x^2+a)^(3/2)+8/105*A/b/a^3*x/(b*x^2+a)^(1/2)+1/7*C/b^3/a*x/(b*x^2+a)^(1/2)-3/28*B/b^2*a*x/(b*x^2
+a)^(7/2)+1/35*B/b^2/a*x/(b*x^2+a)^(3/2)+D/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+2/35*B*x/a^2/b^2/(b*x^2+a)^(1
/2)+1/2*F*x^9/b/(b*x^2+a)^(7/2)-9/2*F/b^(11/2)*a*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-1/2*C*x^5/b/(b*x^2+a)^(7/2)+1/1
4*C/b^3*x/(b*x^2+a)^(3/2)-1/4*B*x^3/b/(b*x^2+a)^(7/2)+3/140*B/b^2*x/(b*x^2+a)^(5/2)-1/7*D*x^7/b/(b*x^2+a)^(7/2
)-1/5*D/b^2*x^5/(b*x^2+a)^(5/2)-1/3*D/b^3*x^3/(b*x^2+a)^(3/2)+9/14*F/b^2*a*x^7/(b*x^2+a)^(7/2)+9/10*F/b^3*a*x^
5/(b*x^2+a)^(5/2)-D*x/b^4/(b*x^2+a)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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Giac [A]  time = 1.4274, size = 302, normalized size = 1.16 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)