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

Optimal. Leaf size=279 \[ \frac{x^7 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac{x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{x^5 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{12 a b^5 \sqrt{a+b x^2}}-\frac{x \sqrt{a+b x^2} \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 a b^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^7)/(7*a*(a + b*x^2)^(7/2)) + ((b^2*B - 2*a*b*C + 3*a^2*D)*x^7)/(5*a*b
^3*(a + b*x^2)^(5/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*x^5)/(60*a*b^4*(a + b*x^2)^(3/2)) + (D*x^7)/(4*b^3*(a
+ b*x^2)^(3/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*x^3)/(12*a*b^5*Sqrt[a + b*x^2]) - ((8*b^2*B - 36*a*b*C + 9
9*a^2*D)*x*Sqrt[a + b*x^2])/(8*a*b^6) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])
/(8*b^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.45491, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{x^7 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac{x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{x^5 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{12 a b^5 \sqrt{a+b x^2}}-\frac{x \sqrt{a+b x^2} \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 a b^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(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^7)/(7*a*(a + b*x^2)^(7/2)) + ((b^2*B - 2*a*b*C + 3*a^2*D)*x^7)/(5*a*b
^3*(a + b*x^2)^(5/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*x^5)/(60*a*b^4*(a + b*x^2)^(3/2)) + (D*x^7)/(4*b^3*(a
+ b*x^2)^(3/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*x^3)/(12*a*b^5*Sqrt[a + b*x^2]) - ((8*b^2*B - 36*a*b*C + 9
9*a^2*D)*x*Sqrt[a + b*x^2])/(8*a*b^6) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])
/(8*b^(13/2))

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 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 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 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 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^6 \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^7}{7 a \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^5 \left (-7 a \left (B-\frac{a (b C-a D)}{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^7}{7 a \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^6 \left (-7 a \left (B-\frac{a (b C-a D)}{b^2}\right )-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^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{x^5 \left (-7 a \left (2 B-\frac{a (9 b C-16 a D)}{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^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\int \frac{x^6 \left (-7 a \left (2 B-\frac{a (9 b C-16 a D)}{b^2}\right )+\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^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac{x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{20 a b^3}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac{x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{12 a b^4}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt{a+b x^2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{4 a b^5}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt{a+b x^2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt{a+b x^2}}{8 a b^6}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b^6}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt{a+b x^2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt{a+b x^2}}{8 a b^6}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b^6}\\ &=\frac{\left (A-\frac{a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt{a+b x^2}}-\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt{a+b x^2}}{8 a b^6}+\frac{\left (8 b^2 B-36 a b C+99 a^2 D\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.430943, size = 229, normalized size = 0.82 \[ \frac{\sqrt{a+b x^2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 \sqrt{a} b^{13/2} \sqrt{\frac{b x^2}{a}+1}}-\frac{x \left (42 a^4 b^2 \left (20 B-300 C x^2+957 D x^4\right )+8 a^3 b^3 x^2 \left (350 B-1827 C x^2+2178 D x^4\right )+a^2 b^4 x^4 \left (3248 B-6336 C x^2+1155 D x^4\right )-630 a^5 b \left (6 C-55 D x^2\right )+10395 a^6 D+2 a b^5 x^6 \left (704 B-105 \left (2 C x^2+D x^4\right )\right )-120 A b^6 x^6\right )}{840 a b^6 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(10395*a^6*D - 120*A*b^6*x^6 - 630*a^5*b*(6*C - 55*D*x^2) + 42*a^4*b^2*(20*B - 300*C*x^2 + 957*D*x^4) + a^
2*b^4*x^4*(3248*B - 6336*C*x^2 + 1155*D*x^4) + 8*a^3*b^3*x^2*(350*B - 1827*C*x^2 + 2178*D*x^4) + 2*a*b^5*x^6*(
704*B - 105*(2*C*x^2 + D*x^4))))/(840*a*b^6*(a + b*x^2)^(7/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*Sqrt[a + b*x
^2]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(13/2)*Sqrt[1 + (b*x^2)/a])

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 460, normalized size = 1.7 \begin{align*}{\frac{D{x}^{11}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{11\,aD{x}^{9}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{99\,{a}^{2}D{x}^{7}}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{99\,{a}^{2}D{x}^{5}}{40\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{33\,D{x}^{3}{a}^{2}}{8\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{99\,{a}^{2}Dx}{8\,{b}^{6}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{99\,{a}^{2}D}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{13}{2}}}}+{\frac{C{x}^{9}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,aC{x}^{7}}{14\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,aC{x}^{5}}{10\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{3\,aC{x}^{3}}{2\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,aCx}{2\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{9\,aC}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}}-{\frac{B{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{B{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}-{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Ax}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aAx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{7\,{b}^{3}a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.20426, size = 358, normalized size = 1.28 \begin{align*} \frac{{\left ({\left ({\left ({\left (105 \,{\left (\frac{2 \, D x^{2}}{b} - \frac{11 \, D a^{4} b^{9} - 4 \, C a^{3} b^{10}}{a^{3} b^{11}}\right )} x^{2} - \frac{8 \,{\left (2178 \, D a^{5} b^{8} - 792 \, C a^{4} b^{9} + 176 \, B a^{3} b^{10} - 15 \, A a^{2} b^{11}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{406 \,{\left (99 \, D a^{6} b^{7} - 36 \, C a^{5} b^{8} + 8 \, B a^{4} b^{9}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{350 \,{\left (99 \, D a^{7} b^{6} - 36 \, C a^{6} b^{7} + 8 \, B a^{5} b^{8}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{105 \,{\left (99 \, D a^{8} b^{5} - 36 \, C a^{7} b^{6} + 8 \, B a^{6} b^{7}\right )}}{a^{3} b^{11}}\right )} x}{840 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{{\left (99 \, D a^{2} - 36 \, C a b + 8 \, B b^{2}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{13}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*((((105*(2*D*x^2/b - (11*D*a^4*b^9 - 4*C*a^3*b^10)/(a^3*b^11))*x^2 - 8*(2178*D*a^5*b^8 - 792*C*a^4*b^9 +
176*B*a^3*b^10 - 15*A*a^2*b^11)/(a^3*b^11))*x^2 - 406*(99*D*a^6*b^7 - 36*C*a^5*b^8 + 8*B*a^4*b^9)/(a^3*b^11))
*x^2 - 350*(99*D*a^7*b^6 - 36*C*a^6*b^7 + 8*B*a^5*b^8)/(a^3*b^11))*x^2 - 105*(99*D*a^8*b^5 - 36*C*a^7*b^6 + 8*
B*a^6*b^7)/(a^3*b^11))*x/(b*x^2 + a)^(7/2) - 1/8*(99*D*a^2 - 36*C*a*b + 8*B*b^2)*log(abs(-sqrt(b)*x + sqrt(b*x
^2 + a)))/b^(13/2)