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

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

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Rubi [A]  time = 0.45, antiderivative size = 279, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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^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*}

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Mathematica [A]  time = 0.48, size = 229, normalized size = 0.82 \begin {gather*} \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 (10395 a^6 D-630 a^5 b \left (6 C-55 D x^2\right )+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 )+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}} \end {gather*}

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]

-1/840*(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))))/(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])

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IntegrateAlgebraic [A]  time = 1.12, size = 241, normalized size = 0.86 \begin {gather*} \frac {\log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \left (-99 a^2 D+36 a b C-8 b^2 B\right )}{8 b^{13/2}}+\frac {-10395 a^6 D x+3780 a^5 b C x-34650 a^5 b D x^3-840 a^4 b^2 B x+12600 a^4 b^2 C x^3-40194 a^4 b^2 D x^5-2800 a^3 b^3 B x^3+14616 a^3 b^3 C x^5-17424 a^3 b^3 D x^7-3248 a^2 b^4 B x^5+6336 a^2 b^4 C x^7-1155 a^2 b^4 D x^9-1408 a b^5 B x^7+420 a b^5 C x^9+210 a b^5 D x^{11}+120 A b^6 x^7}{840 a b^6 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-840*a^4*b^2*B*x + 3780*a^5*b*C*x - 10395*a^6*D*x - 2800*a^3*b^3*B*x^3 + 12600*a^4*b^2*C*x^3 - 34650*a^5*b*D*
x^3 - 3248*a^2*b^4*B*x^5 + 14616*a^3*b^3*C*x^5 - 40194*a^4*b^2*D*x^5 + 120*A*b^6*x^7 - 1408*a*b^5*B*x^7 + 6336
*a^2*b^4*C*x^7 - 17424*a^3*b^3*D*x^7 + 420*a*b^5*C*x^9 - 1155*a^2*b^4*D*x^9 + 210*a*b^5*D*x^11)/(840*a*b^6*(a
+ b*x^2)^(7/2)) + ((-8*b^2*B + 36*a*b*C - 99*a^2*D)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(13/2))

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fricas [A]  time = 1.55, size = 816, normalized size = 2.92 \begin {gather*} \left [\frac {105 \, {\left ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\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 (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, -\frac {105 \, {\left ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{840 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \end {gather*}

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]

[1/1680*(105*((99*D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*b^6)*x^8 + 99*D*a^7 - 36*C*a^6*b + 8*B*a^5*b^2 + 4*(99*D*a^
4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^6 + 6*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^4 + 4*(99*D*a^6*b -
36*C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(210*D*a*b^6*x^11
- 105*(11*D*a^2*b^5 - 4*C*a*b^6)*x^9 - 8*(2178*D*a^3*b^4 - 792*C*a^2*b^5 + 176*B*a*b^6 - 15*A*b^7)*x^7 - 406*
(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^5 - 350*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^3 - 105*(9
9*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x)*sqrt(b*x^2 + a))/(a*b^11*x^8 + 4*a^2*b^10*x^6 + 6*a^3*b^9*x^4 + 4*a
^4*b^8*x^2 + a^5*b^7), -1/840*(105*((99*D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*b^6)*x^8 + 99*D*a^7 - 36*C*a^6*b + 8*
B*a^5*b^2 + 4*(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^6 + 6*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*
x^4 + 4*(99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (210*D*a*
b^6*x^11 - 105*(11*D*a^2*b^5 - 4*C*a*b^6)*x^9 - 8*(2178*D*a^3*b^4 - 792*C*a^2*b^5 + 176*B*a*b^6 - 15*A*b^7)*x^
7 - 406*(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^5 - 350*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^3
- 105*(99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x)*sqrt(b*x^2 + a))/(a*b^11*x^8 + 4*a^2*b^10*x^6 + 6*a^3*b^9*x
^4 + 4*a^4*b^8*x^2 + a^5*b^7)]

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giac [A]  time = 0.59, size = 265, normalized size = 0.95 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 460, normalized size = 1.65 \begin {gather*} \frac {D x^{11}}{4 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {C \,x^{9}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {11 D a \,x^{9}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {B \,x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {9 C a \,x^{7}}{14 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {99 D a^{2} x^{7}}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {A \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {B \,x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {9 C a \,x^{5}}{10 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}-\frac {99 D a^{2} x^{5}}{40 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{4}}-\frac {5 A a \,x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {15 A \,a^{2} x}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {B \,x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {3 C a \,x^{3}}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{4}}-\frac {33 D a^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}+\frac {3 A a x}{56 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}+\frac {A x}{14 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {A x}{7 \sqrt {b \,x^{2}+a}\, a \,b^{3}}-\frac {B x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {9 C a x}{2 \sqrt {b \,x^{2}+a}\, b^{5}}-\frac {99 D a^{2} x}{8 \sqrt {b \,x^{2}+a}\, b^{6}}+\frac {B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {9}{2}}}-\frac {9 C a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}}+\frac {99 D a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {13}{2}}} \end {gather*}

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

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maxima [B]  time = 1.79, size = 986, normalized size = 3.53

result too large to display

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]

1/4*D*x^11/((b*x^2 + a)^(7/2)*b) - 11/8*D*a*x^9/((b*x^2 + a)^(7/2)*b^2) + 1/2*C*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))*B*x - 99/280*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 5
6*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^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
))*C*a*x/b - 33/40*D*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*C*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*B*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*A*x^5/((b*x^2 + a)^(7/2)*b) - 33/8*D*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*C*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*
B*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 - 99/8*D*a^3*x^3/((b*x^2 + a)^(5/2)*b^5) +
9/2*C*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - B*a*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*A*a*x^3/((b*x^2 + a)^(7/2)*b^2)
+ 4587/280*D*a^2*x/(sqrt(b*x^2 + a)*b^6) + 561/280*D*a^3*x/((b*x^2 + a)^(3/2)*b^6) - 2871/280*D*a^4*x/((b*x^2
+ a)^(5/2)*b^6) - 417/70*C*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*C*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*C*a^3*x
/((b*x^2 + a)^(5/2)*b^5) + 139/105*B*x/(sqrt(b*x^2 + a)*b^4) + 17/105*B*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*B*
a^2*x/((b*x^2 + a)^(5/2)*b^4) + 1/14*A*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*A*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*A*a*
x/((b*x^2 + a)^(5/2)*b^3) - 15/56*A*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 99/8*D*a^2*arcsinh(b*x/sqrt(a*b))/b^(13/2)
- 9/2*C*a*arcsinh(b*x/sqrt(a*b))/b^(11/2) + B*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^6\,\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^6*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(9/2),x)

[Out]

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

[Out]

Timed out

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