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

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

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Rubi [A]  time = 0.72, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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 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 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 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^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 (\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) \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*}

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Mathematica [A]  time = 0.59, size = 221, normalized size = 0.85 \begin {gather*} \frac {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 )+\sqrt {b} x \left (945 a^7 F-210 a^6 b \left (D-15 F x^2\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 )+2 a^2 b^5 x^2 \left (35 A+21 B x^2+15 C x^4\right )+4 a b^6 x^4 \left (14 A+3 B x^2\right )+16 A b^7 x^6\right )}{210 a^3 b^{11/2} \left (a+b x^2\right )^{7/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.84, size = 224, normalized size = 0.86 \begin {gather*} \frac {945 a^7 F x-210 a^6 b D x+3150 a^6 b F x^3-700 a^5 b^2 D x^3+3654 a^5 b^2 F x^5-812 a^4 b^3 D x^5+1584 a^4 b^3 F x^7-352 a^3 b^4 D x^7+105 a^3 b^4 F x^9+70 a^2 A b^5 x^3+42 a^2 b^5 B x^5+30 a^2 b^5 C x^7+56 a A b^6 x^5+12 a b^6 B x^7+16 A b^7 x^7}{210 a^3 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(9 a F-2 b D) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{2 b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-210*a^6*b*D*x + 945*a^7*F*x + 70*a^2*A*b^5*x^3 - 700*a^5*b^2*D*x^3 + 3150*a^6*b*F*x^3 + 56*a*A*b^6*x^5 + 42*
a^2*b^5*B*x^5 - 812*a^4*b^3*D*x^5 + 3654*a^5*b^2*F*x^5 + 16*A*b^7*x^7 + 12*a*b^6*B*x^7 + 30*a^2*b^5*C*x^7 - 35
2*a^3*b^4*D*x^7 + 1584*a^4*b^3*F*x^7 + 105*a^3*b^4*F*x^9)/(210*a^3*b^5*(a + b*x^2)^(7/2)) + ((-2*b*D + 9*a*F)*
Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^(11/2))

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fricas [A]  time = 1.52, size = 705, normalized size = 2.70 \begin {gather*} \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 ] \end {gather*}

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]

[-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)]

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giac [A]  time = 0.56, size = 224, normalized size = 0.86 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.01, size = 478, normalized size = 1.83 \begin {gather*} \frac {F \,x^{9}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {D x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {9 F a \,x^{7}}{14 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {C \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {D x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {9 F a \,x^{5}}{10 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}-\frac {B \,x^{3}}{4 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {5 C a \,x^{3}}{8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {A x}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {3 B a x}{28 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {15 C \,a^{2} x}{56 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {D x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {3 F a \,x^{3}}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{4}}+\frac {A x}{35 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a b}+\frac {3 B x}{140 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {3 C a x}{56 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{3}}+\frac {4 A x}{105 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b}+\frac {B x}{35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,b^{2}}+\frac {C x}{14 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {8 A x}{105 \sqrt {b \,x^{2}+a}\, a^{3} b}+\frac {2 B x}{35 \sqrt {b \,x^{2}+a}\, a^{2} b^{2}}+\frac {C x}{7 \sqrt {b \,x^{2}+a}\, a \,b^{3}}-\frac {D x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {9 F a x}{2 \sqrt {b \,x^{2}+a}\, b^{5}}+\frac {D \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {9}{2}}}-\frac {9 F a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}} \end {gather*}

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]

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

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maxima [B]  time = 1.75, size = 826, normalized size = 3.16

result too large to display

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]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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**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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