\(\int \frac {A+B x^2+C x^4+D x^6}{x^9 (a+b x^2)^{3/2}} \, dx\) [231]

Optimal result

Integrand size = 32, antiderivative size = 238 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\frac {b \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )}{a^5 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{8 a^2 x^8}+\frac {(15 A b-8 a B) \sqrt {a+b x^2}}{48 a^3 x^6}-\frac {\left (123 A b^2-88 a b B+48 a^2 C\right ) \sqrt {a+b x^2}}{192 a^4 x^4}+\frac {\left (187 A b^3-8 a \left (19 b^2 B-14 a b C+8 a^2 D\right )\right ) \sqrt {a+b x^2}}{128 a^5 x^2}-\frac {b \left (315 A b^3-8 a \left (35 b^2 B-30 a b C+24 a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{11/2}} \] Output:

b*(A*b^3-a*(B*b^2-C*a*b+D*a^2))/a^5/(b*x^2+a)^(1/2)-1/8*A*(b*x^2+a)^(1/2)/ 
a^2/x^8+1/48*(15*A*b-8*B*a)*(b*x^2+a)^(1/2)/a^3/x^6-1/192*(123*A*b^2-88*B* 
a*b+48*C*a^2)*(b*x^2+a)^(1/2)/a^4/x^4+1/128*(187*A*b^3-8*a*(19*B*b^2-14*C* 
a*b+8*D*a^2))*(b*x^2+a)^(1/2)/a^5/x^2-1/128*b*(315*A*b^3-8*a*(35*B*b^2-30* 
C*a*b+24*D*a^2))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(11/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {a} \left (945 A b^4 x^8+105 a b^3 x^6 \left (3 A-8 B x^2\right )+2 a^2 b^2 x^4 \left (-63 A-140 B x^2+360 C x^4\right )+8 a^3 b x^2 \left (9 A+14 B x^2+30 C x^4-72 D x^6\right )-16 a^4 \left (3 A+4 B x^2+6 C x^4+12 D x^6\right )\right )}{x^8 \sqrt {a+b x^2}}-3 b \left (315 A b^3-8 a \left (35 b^2 B-30 a b C+24 a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{384 a^{11/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^9*(a + b*x^2)^(3/2)),x]
 

Output:

((Sqrt[a]*(945*A*b^4*x^8 + 105*a*b^3*x^6*(3*A - 8*B*x^2) + 2*a^2*b^2*x^4*( 
-63*A - 140*B*x^2 + 360*C*x^4) + 8*a^3*b*x^2*(9*A + 14*B*x^2 + 30*C*x^4 - 
72*D*x^6) - 16*a^4*(3*A + 4*B*x^2 + 6*C*x^4 + 12*D*x^6)))/(x^8*Sqrt[a + b* 
x^2]) - 3*b*(315*A*b^3 - 8*a*(35*b^2*B - 30*a*b*C + 24*a^2*D))*ArcTanh[Sqr 
t[a + b*x^2]/Sqrt[a]])/(384*a^(11/2))
 

Rubi [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2331, 2124, 27, 1192, 1582, 25, 361, 27, 361, 25, 27, 359, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {1}{2} \int \frac {D x^6+C x^4+B x^2+A}{x^{10} \left (b x^2+a\right )^{3/2}}dx^2\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-8 a D x^4-8 a C x^2+9 A b-8 a B}{2 x^8 \left (b x^2+a\right )^{3/2}}dx^2}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-8 a D x^4-8 a C x^2+9 A b-8 a B}{x^8 \left (b x^2+a\right )^{3/2}}dx^2}{8 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {-8 a D x^8-8 a (b C-2 a D) x^4+9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )}{x^4 \left (a-x^4\right )^4}d\sqrt {b x^2+a}}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 1582

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}-\frac {\int -\frac {\left (48 D a^3-40 b^2 B a+45 A b^3\right ) x^4+6 a \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{x^4 \left (a-x^4\right )^3}d\sqrt {b x^2+a}}{6 a^2}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\int \frac {\left (48 D a^3-40 b^2 B a+45 A b^3\right ) x^4+6 a \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{x^4 \left (a-x^4\right )^3}d\sqrt {b x^2+a}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 361

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}-\frac {1}{4} \int -\frac {3 \left (b \left (\frac {99 A b^2}{a}-88 B b+48 a C\right ) x^4+8 \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )\right )}{x^4 \left (a-x^4\right )^2}d\sqrt {b x^2+a}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{4} \int \frac {b \left (\frac {99 A b^2}{a}-88 B b+48 a C\right ) x^4+8 \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{x^4 \left (a-x^4\right )^2}d\sqrt {b x^2+a}+\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 361

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{4} \left (\frac {\sqrt {a+b x^2} \left (171 A b^3-8 a \left (8 a^2 D-14 a b C+19 b^2 B\right )\right )}{2 a^2 \left (a-x^4\right )}-\frac {1}{2} \int -\frac {a \left (\frac {171 A b^3}{a^2}-\frac {152 B b^2}{a}+112 C b-64 a D\right ) x^4+16 \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{a x^4 \left (a-x^4\right )}d\sqrt {b x^2+a}\right )+\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{4} \left (\frac {1}{2} \int \frac {\left (171 A b^3-8 a \left (8 D a^2-14 b C a+19 b^2 B\right )\right ) x^4+16 a \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{a^2 x^4 \left (a-x^4\right )}d\sqrt {b x^2+a}+\frac {\sqrt {a+b x^2} \left (171 A b^3-8 a \left (8 a^2 D-14 a b C+19 b^2 B\right )\right )}{2 a^2 \left (a-x^4\right )}\right )+\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{4} \left (\frac {\int \frac {\left (-64 D a^3+112 b C a^2-152 b^2 B a+171 A b^3\right ) x^4+16 a \left (9 A b^3-8 a \left (D a^2-b C a+b^2 B\right )\right )}{x^4 \left (a-x^4\right )}d\sqrt {b x^2+a}}{2 a^2}+\frac {\sqrt {a+b x^2} \left (171 A b^3-8 a \left (8 a^2 D-14 a b C+19 b^2 B\right )\right )}{2 a^2 \left (a-x^4\right )}\right )+\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{4} \left (\frac {\left (-192 a^3 D+240 a^2 b C-280 a b^2 B+315 A b^3\right ) \int \frac {1}{a-x^4}d\sqrt {b x^2+a}-\frac {16 \left (9 A b^3-8 a \left (a^2 D-a b C+b^2 B\right )\right )}{x^2}}{2 a^2}+\frac {\sqrt {a+b x^2} \left (171 A b^3-8 a \left (8 a^2 D-14 a b C+19 b^2 B\right )\right )}{2 a^2 \left (a-x^4\right )}\right )+\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}}{6 a^2}+\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {b^2 \sqrt {a+b x^2} (9 A b-8 a B)}{6 a^2 \left (a-x^4\right )^3}+\frac {\frac {b \sqrt {a+b x^2} \left (48 a^2 C-88 a b B+99 A b^2\right )}{4 a \left (a-x^4\right )^2}+\frac {3}{4} \left (\frac {\sqrt {a+b x^2} \left (171 A b^3-8 a \left (8 a^2 D-14 a b C+19 b^2 B\right )\right )}{2 a^2 \left (a-x^4\right )}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (-192 a^3 D+240 a^2 b C-280 a b^2 B+315 A b^3\right )}{\sqrt {a}}-\frac {16 \left (9 A b^3-8 a \left (a^2 D-a b C+b^2 B\right )\right )}{x^2}}{2 a^2}\right )}{6 a^2}\right )}{4 a}-\frac {A}{4 a x^8 \sqrt {a+b x^2}}\right )\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^9*(a + b*x^2)^(3/2)),x]
 

Output:

(-1/4*A/(a*x^8*Sqrt[a + b*x^2]) - (b*((b^2*(9*A*b - 8*a*B)*Sqrt[a + b*x^2] 
)/(6*a^2*(a - x^4)^3) + ((b*(99*A*b^2 - 88*a*b*B + 48*a^2*C)*Sqrt[a + b*x^ 
2])/(4*a*(a - x^4)^2) + (3*(((171*A*b^3 - 8*a*(19*b^2*B - 14*a*b*C + 8*a^2 
*D))*Sqrt[a + b*x^2])/(2*a^2*(a - x^4)) + ((-16*(9*A*b^3 - 8*a*(b^2*B - a* 
b*C + a^2*D)))/x^2 + ((315*A*b^3 - 280*a*b^2*B + 240*a^2*b*C - 192*a^3*D)* 
ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/(2*a^2)))/4)/(6*a^2)))/(4*a))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 361

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] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[x^m*(a + b*x^2)^(p + 1)*E 
xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c 
- a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], 
 x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 
2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1582

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
+ e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ 
(2*p)*(q + 1))   Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e 
*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - 
 b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] 
&& ILtQ[m/2, 0]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(-\frac {315 \left (-\frac {\left (-\frac {8 x^{2} B}{3}+A \right ) x^{6} b^{3} a^{\frac {3}{2}}}{3}+\frac {2 x^{4} b^{2} \left (-\frac {40}{7} C \,x^{4}+\frac {20}{9} x^{2} B +A \right ) a^{\frac {5}{2}}}{15}-\frac {8 \left (-8 D x^{6}+\frac {10}{3} C \,x^{4}+\frac {14}{9} x^{2} B +A \right ) x^{2} b \,a^{\frac {7}{2}}}{105}+\frac {16 \left (4 D x^{6}+2 C \,x^{4}+\frac {4}{3} x^{2} B +A \right ) a^{\frac {9}{2}}}{315}+\left (-A \,b^{3} \sqrt {a}+\sqrt {b \,x^{2}+a}\, \left (b^{3} A -\frac {8}{9} a \,b^{2} B +\frac {16}{21} a^{2} b C -\frac {64}{105} a^{3} D\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )\right ) x^{8} b \right )}{128 a^{\frac {11}{2}} \sqrt {b \,x^{2}+a}\, x^{8}}\)	\(182\)
default	\(A \left (-\frac {1}{8 a \,x^{8} \sqrt {b \,x^{2}+a}}-\frac {9 b \left (-\frac {1}{6 a \,x^{6} \sqrt {b \,x^{2}+a}}-\frac {7 b \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+B \left (-\frac {1}{6 a \,x^{6} \sqrt {b \,x^{2}+a}}-\frac {7 b \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+C \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )+D \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )\)	\(418\)

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^9/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-315/128/a^(11/2)/(b*x^2+a)^(1/2)*(-1/3*(-8/3*x^2*B+A)*x^6*b^3*a^(3/2)+2/1 
5*x^4*b^2*(-40/7*C*x^4+20/9*x^2*B+A)*a^(5/2)-8/105*(-8*D*x^6+10/3*C*x^4+14 
/9*x^2*B+A)*x^2*b*a^(7/2)+16/315*(4*D*x^6+2*C*x^4+4/3*x^2*B+A)*a^(9/2)+(-A 
*b^3*a^(1/2)+(b*x^2+a)^(1/2)*(b^3*A-8/9*a*b^2*B+16/21*a^2*b*C-64/105*a^3*D 
)*arctanh((b*x^2+a)^(1/2)/a^(1/2)))*x^8*b)/x^8
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.22 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (192 \, D a^{3} b^{2} - 240 \, C a^{2} b^{3} + 280 \, B a b^{4} - 315 \, A b^{5}\right )} x^{10} + {\left (192 \, D a^{4} b - 240 \, C a^{3} b^{2} + 280 \, B a^{2} b^{3} - 315 \, A a b^{4}\right )} x^{8}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (192 \, D a^{4} b - 240 \, C a^{3} b^{2} + 280 \, B a^{2} b^{3} - 315 \, A a b^{4}\right )} x^{8} + {\left (192 \, D a^{5} - 240 \, C a^{4} b + 280 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3}\right )} x^{6} + 48 \, A a^{5} + 2 \, {\left (48 \, C a^{5} - 56 \, B a^{4} b + 63 \, A a^{3} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{768 \, {\left (a^{6} b x^{10} + a^{7} x^{8}\right )}}, -\frac {3 \, {\left ({\left (192 \, D a^{3} b^{2} - 240 \, C a^{2} b^{3} + 280 \, B a b^{4} - 315 \, A b^{5}\right )} x^{10} + {\left (192 \, D a^{4} b - 240 \, C a^{3} b^{2} + 280 \, B a^{2} b^{3} - 315 \, A a b^{4}\right )} x^{8}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, {\left (192 \, D a^{4} b - 240 \, C a^{3} b^{2} + 280 \, B a^{2} b^{3} - 315 \, A a b^{4}\right )} x^{8} + {\left (192 \, D a^{5} - 240 \, C a^{4} b + 280 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3}\right )} x^{6} + 48 \, A a^{5} + 2 \, {\left (48 \, C a^{5} - 56 \, B a^{4} b + 63 \, A a^{3} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{384 \, {\left (a^{6} b x^{10} + a^{7} x^{8}\right )}}\right ] \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^9/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/768*(3*((192*D*a^3*b^2 - 240*C*a^2*b^3 + 280*B*a*b^4 - 315*A*b^5)*x^10 
 + (192*D*a^4*b - 240*C*a^3*b^2 + 280*B*a^2*b^3 - 315*A*a*b^4)*x^8)*sqrt(a 
)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(3*(192*D*a^4*b 
- 240*C*a^3*b^2 + 280*B*a^2*b^3 - 315*A*a*b^4)*x^8 + (192*D*a^5 - 240*C*a^ 
4*b + 280*B*a^3*b^2 - 315*A*a^2*b^3)*x^6 + 48*A*a^5 + 2*(48*C*a^5 - 56*B*a 
^4*b + 63*A*a^3*b^2)*x^4 + 8*(8*B*a^5 - 9*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/( 
a^6*b*x^10 + a^7*x^8), -1/384*(3*((192*D*a^3*b^2 - 240*C*a^2*b^3 + 280*B*a 
*b^4 - 315*A*b^5)*x^10 + (192*D*a^4*b - 240*C*a^3*b^2 + 280*B*a^2*b^3 - 31 
5*A*a*b^4)*x^8)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (3*(192*D*a^ 
4*b - 240*C*a^3*b^2 + 280*B*a^2*b^3 - 315*A*a*b^4)*x^8 + (192*D*a^5 - 240* 
C*a^4*b + 280*B*a^3*b^2 - 315*A*a^2*b^3)*x^6 + 48*A*a^5 + 2*(48*C*a^5 - 56 
*B*a^4*b + 63*A*a^3*b^2)*x^4 + 8*(8*B*a^5 - 9*A*a^4*b)*x^2)*sqrt(b*x^2 + a 
))/(a^6*b*x^10 + a^7*x^8)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**9/(b*x**2+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 \, D b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {15 \, C b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {7}{2}}} + \frac {35 \, B b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {9}{2}}} - \frac {315 \, A b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {11}{2}}} - \frac {3 \, D b}{2 \, \sqrt {b x^{2} + a} a^{2}} + \frac {15 \, C b^{2}}{8 \, \sqrt {b x^{2} + a} a^{3}} - \frac {35 \, B b^{3}}{16 \, \sqrt {b x^{2} + a} a^{4}} + \frac {315 \, A b^{4}}{128 \, \sqrt {b x^{2} + a} a^{5}} - \frac {D}{2 \, \sqrt {b x^{2} + a} a x^{2}} + \frac {5 \, C b}{8 \, \sqrt {b x^{2} + a} a^{2} x^{2}} - \frac {35 \, B b^{2}}{48 \, \sqrt {b x^{2} + a} a^{3} x^{2}} + \frac {105 \, A b^{3}}{128 \, \sqrt {b x^{2} + a} a^{4} x^{2}} - \frac {C}{4 \, \sqrt {b x^{2} + a} a x^{4}} + \frac {7 \, B b}{24 \, \sqrt {b x^{2} + a} a^{2} x^{4}} - \frac {21 \, A b^{2}}{64 \, \sqrt {b x^{2} + a} a^{3} x^{4}} - \frac {B}{6 \, \sqrt {b x^{2} + a} a x^{6}} + \frac {3 \, A b}{16 \, \sqrt {b x^{2} + a} a^{2} x^{6}} - \frac {A}{8 \, \sqrt {b x^{2} + a} a x^{8}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^9/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

3/2*D*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 15/8*C*b^2*arcsinh(a/(sqrt 
(a*b)*abs(x)))/a^(7/2) + 35/16*B*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) 
 - 315/128*A*b^4*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(11/2) - 3/2*D*b/(sqrt(b* 
x^2 + a)*a^2) + 15/8*C*b^2/(sqrt(b*x^2 + a)*a^3) - 35/16*B*b^3/(sqrt(b*x^2 
 + a)*a^4) + 315/128*A*b^4/(sqrt(b*x^2 + a)*a^5) - 1/2*D/(sqrt(b*x^2 + a)* 
a*x^2) + 5/8*C*b/(sqrt(b*x^2 + a)*a^2*x^2) - 35/48*B*b^2/(sqrt(b*x^2 + a)* 
a^3*x^2) + 105/128*A*b^3/(sqrt(b*x^2 + a)*a^4*x^2) - 1/4*C/(sqrt(b*x^2 + a 
)*a*x^4) + 7/24*B*b/(sqrt(b*x^2 + a)*a^2*x^4) - 21/64*A*b^2/(sqrt(b*x^2 + 
a)*a^3*x^4) - 1/6*B/(sqrt(b*x^2 + a)*a*x^6) + 3/16*A*b/(sqrt(b*x^2 + a)*a^ 
2*x^6) - 1/8*A/(sqrt(b*x^2 + a)*a*x^8)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.61 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left (192 \, D a^{3} b - 240 \, C a^{2} b^{2} + 280 \, B a b^{3} - 315 \, A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{128 \, \sqrt {-a} a^{5}} - \frac {D a^{3} b - C a^{2} b^{2} + B a b^{3} - A b^{4}}{\sqrt {b x^{2} + a} a^{5}} - \frac {192 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} D a^{3} b - 576 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{4} b + 576 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D a^{5} b - 192 \, \sqrt {b x^{2} + a} D a^{6} b - 336 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} C a^{2} b^{2} + 1104 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{3} b^{2} - 1200 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C a^{4} b^{2} + 432 \, \sqrt {b x^{2} + a} C a^{5} b^{2} + 456 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a b^{3} - 1544 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} b^{3} + 1784 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} b^{3} - 696 \, \sqrt {b x^{2} + a} B a^{4} b^{3} - 561 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{4} + 1929 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b^{4} - 2295 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} b^{4} + 975 \, \sqrt {b x^{2} + a} A a^{3} b^{4}}{384 \, a^{5} b^{4} x^{8}} \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^9/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

-1/128*(192*D*a^3*b - 240*C*a^2*b^2 + 280*B*a*b^3 - 315*A*b^4)*arctan(sqrt 
(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^5) - (D*a^3*b - C*a^2*b^2 + B*a*b^3 - A* 
b^4)/(sqrt(b*x^2 + a)*a^5) - 1/384*(192*(b*x^2 + a)^(7/2)*D*a^3*b - 576*(b 
*x^2 + a)^(5/2)*D*a^4*b + 576*(b*x^2 + a)^(3/2)*D*a^5*b - 192*sqrt(b*x^2 + 
 a)*D*a^6*b - 336*(b*x^2 + a)^(7/2)*C*a^2*b^2 + 1104*(b*x^2 + a)^(5/2)*C*a 
^3*b^2 - 1200*(b*x^2 + a)^(3/2)*C*a^4*b^2 + 432*sqrt(b*x^2 + a)*C*a^5*b^2 
+ 456*(b*x^2 + a)^(7/2)*B*a*b^3 - 1544*(b*x^2 + a)^(5/2)*B*a^2*b^3 + 1784* 
(b*x^2 + a)^(3/2)*B*a^3*b^3 - 696*sqrt(b*x^2 + a)*B*a^4*b^3 - 561*(b*x^2 + 
 a)^(7/2)*A*b^4 + 1929*(b*x^2 + a)^(5/2)*A*a*b^4 - 2295*(b*x^2 + a)^(3/2)* 
A*a^2*b^4 + 975*sqrt(b*x^2 + a)*A*a^3*b^4)/(a^5*b^4*x^8)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^9\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^9*(a + b*x^2)^(3/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^9*(a + b*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.53 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^9 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-48 \sqrt {b \,x^{2}+a}\, a^{5}+8 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-96 \sqrt {b \,x^{2}+a}\, a^{4} c \,x^{4}-192 \sqrt {b \,x^{2}+a}\, a^{4} d \,x^{6}-14 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}+240 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{6}-576 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{8}+35 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}+720 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{8}+105 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-576 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d \,x^{8}+720 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c \,x^{8}-576 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,x^{10}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}+720 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,x^{10}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}+576 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d \,x^{8}-720 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c \,x^{8}+576 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,x^{10}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}-720 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,x^{10}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}}{384 a^{5} x^{8} \left (b \,x^{2}+a \right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^9/(b*x^2+a)^(3/2),x)
 

Output:

( - 48*sqrt(a + b*x**2)*a**5 + 8*sqrt(a + b*x**2)*a**4*b*x**2 - 96*sqrt(a 
+ b*x**2)*a**4*c*x**4 - 192*sqrt(a + b*x**2)*a**4*d*x**6 - 14*sqrt(a + b*x 
**2)*a**3*b**2*x**4 + 240*sqrt(a + b*x**2)*a**3*b*c*x**6 - 576*sqrt(a + b* 
x**2)*a**3*b*d*x**8 + 35*sqrt(a + b*x**2)*a**2*b**3*x**6 + 720*sqrt(a + b* 
x**2)*a**2*b**2*c*x**8 + 105*sqrt(a + b*x**2)*a*b**4*x**8 - 576*sqrt(a)*lo 
g((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b*d*x**8 + 720*sq 
rt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*x* 
*8 - 576*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a** 
2*b**2*d*x**10 + 105*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/ 
sqrt(a))*a*b**4*x**8 + 720*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt( 
b)*x)/sqrt(a))*a*b**3*c*x**10 + 105*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a 
) + sqrt(b)*x)/sqrt(a))*b**5*x**10 + 576*sqrt(a)*log((sqrt(a + b*x**2) + s 
qrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b*d*x**8 - 720*sqrt(a)*log((sqrt(a + b*x 
**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*x**8 + 576*sqrt(a)*log((s 
qrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d*x**10 - 105*sq 
rt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**4*x**8 - 
720*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*c 
*x**10 - 105*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a)) 
*b**5*x**10)/(384*a**5*x**8*(a + b*x**2))