\(\int \frac {A+B x^2+C x^4+D x^6+F x^8}{(a+b x^2)^{9/2}} \, dx\) [267]

Optimal result

Integrand size = 34, antiderivative size = 251 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b^3 B-a b^2 C+a^2 b D-a^3 F}{b^4}\right ) x}{7 \left (a+b x^2\right )^{7/2}}+\frac {\left (\frac {6 A}{a}+\frac {b^3 B-8 a b^2 C+15 a^2 b D-22 a^3 F}{b^4}\right ) x}{35 a \left (a+b x^2\right )^{5/2}}+\frac {\left (24 A b^4+a \left (4 b^3 B+3 a b^2 C-45 a^2 b D+122 a^3 F\right )\right ) x}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac {\left (48 A b^4+a \left (8 b^3 B+6 a b^2 C+15 a^2 b D-176 a^3 F\right )\right ) x}{105 a^4 b^4 \sqrt {a+b x^2}}+\frac {F \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \] Output:

1/7*(A/a-(B*b^3-C*a*b^2+D*a^2*b-F*a^3)/b^4)*x/(b*x^2+a)^(7/2)+1/35*(6*A/a+ 
(B*b^3-8*C*a*b^2+15*D*a^2*b-22*F*a^3)/b^4)*x/a/(b*x^2+a)^(5/2)+1/105*(24*A 
*b^4+a*(4*B*b^3+3*C*a*b^2-45*D*a^2*b+122*F*a^3))*x/a^3/b^4/(b*x^2+a)^(3/2) 
+1/105*(48*A*b^4+a*(8*B*b^3+6*C*a*b^2+15*D*a^2*b-176*F*a^3))*x/a^4/b^4/(b* 
x^2+a)^(1/2)+F*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (-105 a^7 F-350 a^6 b F x^2-406 a^5 b^2 F x^4+48 A b^7 x^6-176 a^4 b^3 F x^6+8 a b^6 x^4 \left (21 A+B x^2\right )+2 a^2 b^5 x^2 \left (105 A+14 B x^2+3 C x^4\right )+a^3 b^4 \left (105 A+35 B x^2+21 C x^4+15 D x^6\right )\right )}{105 a^4 b^4 \left (a+b x^2\right )^{7/2}}-\frac {F \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{9/2}} \] Input:

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

Output:

(x*(-105*a^7*F - 350*a^6*b*F*x^2 - 406*a^5*b^2*F*x^4 + 48*A*b^7*x^6 - 176* 
a^4*b^3*F*x^6 + 8*a*b^6*x^4*(21*A + B*x^2) + 2*a^2*b^5*x^2*(105*A + 14*B*x 
^2 + 3*C*x^4) + a^3*b^4*(105*A + 35*B*x^2 + 21*C*x^4 + 15*D*x^6)))/(105*a^ 
4*b^4*(a + b*x^2)^(7/2)) - (F*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(9/2)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2345, 25, 2345, 25, 1471, 25, 27, 298, 224, 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.

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {\frac {7 a F x^6}{b}+\frac {7 a (b D-a F) x^4}{b^2}+\frac {7 a \left (F a^2-b D a+b^2 C\right ) x^2}{b^3}+6 A+\frac {a \left (-F a^3+b D a^2-b^2 C a+b^3 B\right )}{b^4}}{\left (b x^2+a\right )^{7/2}}dx}{7 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {7 a F x^6}{b}+\frac {7 a (b D-a F) x^4}{b^2}+\frac {7 a \left (F a^2-b D a+b^2 C\right ) x^2}{b^3}+6 A+\frac {a \left (-F a^3+b D a^2-b^2 C a+b^3 B\right )}{b^4}}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {\frac {35 a^2 F x^4}{b^2}+\frac {35 a^2 (b D-2 a F) x^2}{b^3}+24 A+\frac {a \left (17 F a^3-10 b D a^2+3 b^2 C a+4 b^3 B\right )}{b^4}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {35 a^2 F x^4}{b^2}+\frac {35 a^2 (b D-2 a F) x^2}{b^3}+24 A+\frac {a \left (17 F a^3-10 b D a^2+3 b^2 C a+4 b^3 B\right )}{b^4}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {\frac {\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {-71 F a^4+105 b F x^2 a^3+15 b D a^3+6 b^2 C a^2+8 b^3 B a+48 A b^4}{b^4 \left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {48 A b^4+105 a^3 F x^2 b+a \left (-71 F a^3+15 b D a^2+6 b^2 C a+8 b^3 B\right )}{b^4 \left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {48 A b^4+105 a^3 F x^2 b+a \left (-71 F a^3+15 b D a^2+6 b^2 C a+8 b^3 B\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b^4}+\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {\frac {105 a^3 F \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {x \left (-176 a^4 F+15 a^3 b D+6 a^2 b^2 C+8 a b^3 B+48 A b^4\right )}{a \sqrt {a+b x^2}}}{3 a b^4}+\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {105 a^3 F \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {x \left (-176 a^4 F+15 a^3 b D+6 a^2 b^2 C+8 a b^3 B+48 A b^4\right )}{a \sqrt {a+b x^2}}}{3 a b^4}+\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x \left (\frac {A}{a}-\frac {a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}+\frac {\frac {x \left (\frac {-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B}{b^4}+\frac {6 A}{a}\right )}{5 \left (a+b x^2\right )^{5/2}}+\frac {\frac {x \left (\frac {122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B}{b^4}+\frac {24 A}{a}\right )}{3 \left (a+b x^2\right )^{3/2}}+\frac {\frac {105 a^3 F \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {x \left (-176 a^4 F+15 a^3 b D+6 a^2 b^2 C+8 a b^3 B+48 A b^4\right )}{a \sqrt {a+b x^2}}}{3 a b^4}}{5 a}}{7 a}\)

Input:

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

Output:

((A/a - (b^3*B - a*b^2*C + a^2*b*D - a^3*F)/b^4)*x)/(7*(a + b*x^2)^(7/2)) 
+ ((((6*A)/a + (b^3*B - 8*a*b^2*C + 15*a^2*b*D - 22*a^3*F)/b^4)*x)/(5*(a + 
 b*x^2)^(5/2)) + ((((24*A)/a + (4*b^3*B + 3*a*b^2*C - 45*a^2*b*D + 122*a^3 
*F)/b^4)*x)/(3*(a + b*x^2)^(3/2)) + (((48*A*b^4 + 8*a*b^3*B + 6*a^2*b^2*C 
+ 15*a^3*b*D - 176*a^4*F)*x)/(a*Sqrt[a + b*x^2]) + (105*a^3*F*ArcTanh[(Sqr 
t[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b])/(3*a*b^4))/(5*a))/(7*a)
 

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 224

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 298

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

rule 1471

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

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[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[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(\frac {F \,a^{4} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) b^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}+\frac {16 b^{\frac {9}{2}} \left (A \,b^{7} x^{6}+\frac {7 \left (\frac {x^{2} B}{21}+A \right ) x^{4} a \,b^{6}}{2}+\frac {35 \left (\frac {1}{35} C \,x^{4}+\frac {2}{15} x^{2} B +A \right ) x^{2} a^{2} b^{5}}{8}+\frac {35 \left (\frac {1}{7} D x^{6}+\frac {1}{5} C \,x^{4}+\frac {1}{3} x^{2} B +A \right ) a^{3} b^{4}}{16}-\frac {11 F \,a^{4} b^{3} x^{6}}{3}-\frac {203 F \,a^{5} b^{2} x^{4}}{24}-\frac {175 F \,a^{6} b \,x^{2}}{24}-\frac {35 F \,a^{7}}{16}\right ) x}{35}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {17}{2}} a^{4}}\)	\(177\)
default	\(A \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+C \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+D \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+F \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+B \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )\)	\(544\)

Input:

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

Output:

(F*a^4*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))*b^4*(b*x^2+a)^(7/2)+16/35*b^(9/2 
)*(A*b^7*x^6+7/2*(1/21*x^2*B+A)*x^4*a*b^6+35/8*(1/35*C*x^4+2/15*x^2*B+A)*x 
^2*a^2*b^5+35/16*(1/7*D*x^6+1/5*C*x^4+1/3*x^2*B+A)*a^3*b^4-11/3*F*a^4*b^3* 
x^6-203/24*F*a^5*b^2*x^4-175/24*F*a^6*b*x^2-35/16*F*a^7)*x)/(b*x^2+a)^(7/2 
)/b^(17/2)/a^4
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.26 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (F a^{4} b^{4} x^{8} + 4 \, F a^{5} b^{3} x^{6} + 6 \, F a^{6} b^{2} x^{4} + 4 \, F a^{7} b x^{2} + F a^{8}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left ({\left (176 \, F a^{4} b^{4} - 15 \, D a^{3} b^{5} - 6 \, C a^{2} b^{6} - 8 \, B a b^{7} - 48 \, A b^{8}\right )} x^{7} + 7 \, {\left (58 \, F a^{5} b^{3} - 3 \, C a^{3} b^{5} - 4 \, B a^{2} b^{6} - 24 \, A a b^{7}\right )} x^{5} + 35 \, {\left (10 \, F a^{6} b^{2} - B a^{3} b^{5} - 6 \, A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (F a^{7} b - A a^{3} b^{5}\right )} x\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{4} b^{9} x^{8} + 4 \, a^{5} b^{8} x^{6} + 6 \, a^{6} b^{7} x^{4} + 4 \, a^{7} b^{6} x^{2} + a^{8} b^{5}\right )}}, -\frac {105 \, {\left (F a^{4} b^{4} x^{8} + 4 \, F a^{5} b^{3} x^{6} + 6 \, F a^{6} b^{2} x^{4} + 4 \, F a^{7} b x^{2} + F a^{8}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left ({\left (176 \, F a^{4} b^{4} - 15 \, D a^{3} b^{5} - 6 \, C a^{2} b^{6} - 8 \, B a b^{7} - 48 \, A b^{8}\right )} x^{7} + 7 \, {\left (58 \, F a^{5} b^{3} - 3 \, C a^{3} b^{5} - 4 \, B a^{2} b^{6} - 24 \, A a b^{7}\right )} x^{5} + 35 \, {\left (10 \, F a^{6} b^{2} - B a^{3} b^{5} - 6 \, A a^{2} b^{6}\right )} x^{3} + 105 \, {\left (F a^{7} b - A a^{3} b^{5}\right )} x\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{9} x^{8} + 4 \, a^{5} b^{8} x^{6} + 6 \, a^{6} b^{7} x^{4} + 4 \, a^{7} b^{6} x^{2} + a^{8} b^{5}\right )}}\right ] \] Input:

integrate((F*x^8+D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas 
")
 

Output:

[1/210*(105*(F*a^4*b^4*x^8 + 4*F*a^5*b^3*x^6 + 6*F*a^6*b^2*x^4 + 4*F*a^7*b 
*x^2 + F*a^8)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2* 
((176*F*a^4*b^4 - 15*D*a^3*b^5 - 6*C*a^2*b^6 - 8*B*a*b^7 - 48*A*b^8)*x^7 + 
 7*(58*F*a^5*b^3 - 3*C*a^3*b^5 - 4*B*a^2*b^6 - 24*A*a*b^7)*x^5 + 35*(10*F* 
a^6*b^2 - B*a^3*b^5 - 6*A*a^2*b^6)*x^3 + 105*(F*a^7*b - A*a^3*b^5)*x)*sqrt 
(b*x^2 + a))/(a^4*b^9*x^8 + 4*a^5*b^8*x^6 + 6*a^6*b^7*x^4 + 4*a^7*b^6*x^2 
+ a^8*b^5), -1/105*(105*(F*a^4*b^4*x^8 + 4*F*a^5*b^3*x^6 + 6*F*a^6*b^2*x^4 
 + 4*F*a^7*b*x^2 + F*a^8)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (( 
176*F*a^4*b^4 - 15*D*a^3*b^5 - 6*C*a^2*b^6 - 8*B*a*b^7 - 48*A*b^8)*x^7 + 7 
*(58*F*a^5*b^3 - 3*C*a^3*b^5 - 4*B*a^2*b^6 - 24*A*a*b^7)*x^5 + 35*(10*F*a^ 
6*b^2 - B*a^3*b^5 - 6*A*a^2*b^6)*x^3 + 105*(F*a^7*b - A*a^3*b^5)*x)*sqrt(b 
*x^2 + a))/(a^4*b^9*x^8 + 4*a^5*b^8*x^6 + 6*a^6*b^7*x^4 + 4*a^7*b^6*x^2 + 
a^8*b^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5071 vs. \(2 (246) = 492\).

Time = 73.19 (sec) , antiderivative size = 5071, normalized size of antiderivative = 20.20 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

A*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt 
(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2 
)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a 
) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12* 
sqrt(1 + b*x**2/a)) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 
210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + 
 b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b* 
*4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 
 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*a** 
(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525* 
a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + 
b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b** 
5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 
 429*a**11*b**3*x**7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x* 
*2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a 
**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b 
*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6 
*x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)*sqrt(1 + b* 
x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x** 
4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (231) = 462\).

Time = 0.06 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.38 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((F*x^8+D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima 
")
 

Output:

-1/35*(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))*F*x - 
1/15*F*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*D*x^5/((b*x^2 + a)^(7/2)*b) - 1/3 
*F*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 - F*a 
*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*D*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1/4*C 
*x^3/((b*x^2 + a)^(7/2)*b) + 16/35*A*x/(sqrt(b*x^2 + a)*a^4) + 8/35*A*x/(( 
b*x^2 + a)^(3/2)*a^3) + 6/35*A*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*A*x/((b*x^2 
 + a)^(7/2)*a) + 139/105*F*x/(sqrt(b*x^2 + a)*b^4) + 17/105*F*a*x/((b*x^2 
+ a)^(3/2)*b^4) - 29/35*F*a^2*x/((b*x^2 + a)^(5/2)*b^4) + 1/14*D*x/((b*x^2 
 + a)^(3/2)*b^3) + 1/7*D*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*D*a*x/((b*x^2 + 
a)^(5/2)*b^3) - 15/56*D*a^2*x/((b*x^2 + a)^(7/2)*b^3) + 3/140*C*x/((b*x^2 
+ a)^(5/2)*b^2) + 2/35*C*x/(sqrt(b*x^2 + a)*a^2*b^2) + 1/35*C*x/((b*x^2 + 
a)^(3/2)*a*b^2) - 3/28*C*a*x/((b*x^2 + a)^(7/2)*b^2) - 1/7*B*x/((b*x^2 + a 
)^(7/2)*b) + 8/105*B*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*B*x/((b*x^2 + a)^(3 
/2)*a^2*b) + 1/35*B*x/((b*x^2 + a)^(5/2)*a*b) + F*arcsinh(b*x/sqrt(a*b))/b 
^(9/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {{\left (176 \, F a^{4} b^{6} - 15 \, D a^{3} b^{7} - 6 \, C a^{2} b^{8} - 8 \, B a b^{9} - 48 \, A b^{10}\right )} x^{2}}{a^{4} b^{7}} + \frac {7 \, {\left (58 \, F a^{5} b^{5} - 3 \, C a^{3} b^{7} - 4 \, B a^{2} b^{8} - 24 \, A a b^{9}\right )}}{a^{4} b^{7}}\right )} + \frac {35 \, {\left (10 \, F a^{6} b^{4} - B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )}}{a^{4} b^{7}}\right )} x^{2} + \frac {105 \, {\left (F a^{7} b^{3} - A a^{3} b^{7}\right )}}{a^{4} b^{7}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {F \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \] Input:

integrate((F*x^8+D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
 

Output:

-1/105*((x^2*((176*F*a^4*b^6 - 15*D*a^3*b^7 - 6*C*a^2*b^8 - 8*B*a*b^9 - 48 
*A*b^10)*x^2/(a^4*b^7) + 7*(58*F*a^5*b^5 - 3*C*a^3*b^7 - 4*B*a^2*b^8 - 24* 
A*a*b^9)/(a^4*b^7)) + 35*(10*F*a^6*b^4 - B*a^3*b^7 - 6*A*a^2*b^8)/(a^4*b^7 
))*x^2 + 105*(F*a^7*b^3 - A*a^3*b^7)/(a^4*b^7))*x/(b*x^2 + a)^(7/2) - F*lo 
g(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+F\,x^8+x^6\,D}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.63 \[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {15 \sqrt {b}\, a^{2} b^{5} d \,x^{8}-6 \sqrt {b}\, a \,b^{6} c \,x^{8}+56 \sqrt {b}\, a^{3} b^{4} f \,x^{8}-24 \sqrt {b}\, a^{2} b^{5} c \,x^{6}-105 \sqrt {b \,x^{2}+a}\, a^{6} b f x -350 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} f \,x^{3}-406 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} f \,x^{5}-176 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} f \,x^{7}+15 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} d \,x^{7}+6 \sqrt {b \,x^{2}+a}\, a \,b^{6} c \,x^{7}+224 \sqrt {b}\, a^{6} b f \,x^{2}+60 \sqrt {b}\, a^{5} b^{2} d \,x^{2}+336 \sqrt {b}\, a^{5} b^{2} f \,x^{4}-24 \sqrt {b}\, a^{4} b^{3} c \,x^{2}+90 \sqrt {b}\, a^{4} b^{3} d \,x^{4}+224 \sqrt {b}\, a^{4} b^{3} f \,x^{6}-36 \sqrt {b}\, a^{3} b^{4} c \,x^{4}+60 \sqrt {b}\, a^{3} b^{4} d \,x^{6}+56 \sqrt {b \,x^{2}+a}\, b^{8} x^{7}+56 \sqrt {b}\, a^{7} f -56 \sqrt {b}\, a^{4} b^{4}-56 \sqrt {b}\, b^{8} x^{8}+105 \sqrt {b \,x^{2}+a}\, a^{3} b^{5} x +196 \sqrt {b \,x^{2}+a}\, a \,b^{7} x^{5}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{7} f +15 \sqrt {b}\, a^{6} b d -6 \sqrt {b}\, a^{5} b^{2} c -224 \sqrt {b}\, a^{3} b^{5} x^{2}-336 \sqrt {b}\, a^{2} b^{6} x^{4}-224 \sqrt {b}\, a \,b^{7} x^{6}+245 \sqrt {b \,x^{2}+a}\, a^{2} b^{6} x^{3}+21 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} c \,x^{5}+630 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} b^{2} f \,x^{4}+420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b^{3} f \,x^{6}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{4} f \,x^{8}+420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{6} b f \,x^{2}}{105 a^{3} b^{5} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

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

Output:

( - 105*sqrt(a + b*x**2)*a**6*b*f*x - 350*sqrt(a + b*x**2)*a**5*b**2*f*x** 
3 - 406*sqrt(a + b*x**2)*a**4*b**3*f*x**5 + 105*sqrt(a + b*x**2)*a**3*b**5 
*x - 176*sqrt(a + b*x**2)*a**3*b**4*f*x**7 + 245*sqrt(a + b*x**2)*a**2*b** 
6*x**3 + 21*sqrt(a + b*x**2)*a**2*b**5*c*x**5 + 15*sqrt(a + b*x**2)*a**2*b 
**5*d*x**7 + 196*sqrt(a + b*x**2)*a*b**7*x**5 + 6*sqrt(a + b*x**2)*a*b**6* 
c*x**7 + 56*sqrt(a + b*x**2)*b**8*x**7 + 105*sqrt(b)*log((sqrt(a + b*x**2) 
 + sqrt(b)*x)/sqrt(a))*a**7*f + 420*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b 
)*x)/sqrt(a))*a**6*b*f*x**2 + 630*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)* 
x)/sqrt(a))*a**5*b**2*f*x**4 + 420*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b) 
*x)/sqrt(a))*a**4*b**3*f*x**6 + 105*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b 
)*x)/sqrt(a))*a**3*b**4*f*x**8 + 56*sqrt(b)*a**7*f + 15*sqrt(b)*a**6*b*d + 
 224*sqrt(b)*a**6*b*f*x**2 - 6*sqrt(b)*a**5*b**2*c + 60*sqrt(b)*a**5*b**2* 
d*x**2 + 336*sqrt(b)*a**5*b**2*f*x**4 - 56*sqrt(b)*a**4*b**4 - 24*sqrt(b)* 
a**4*b**3*c*x**2 + 90*sqrt(b)*a**4*b**3*d*x**4 + 224*sqrt(b)*a**4*b**3*f*x 
**6 - 224*sqrt(b)*a**3*b**5*x**2 - 36*sqrt(b)*a**3*b**4*c*x**4 + 60*sqrt(b 
)*a**3*b**4*d*x**6 + 56*sqrt(b)*a**3*b**4*f*x**8 - 336*sqrt(b)*a**2*b**6*x 
**4 - 24*sqrt(b)*a**2*b**5*c*x**6 + 15*sqrt(b)*a**2*b**5*d*x**8 - 224*sqrt 
(b)*a*b**7*x**6 - 6*sqrt(b)*a*b**6*c*x**8 - 56*sqrt(b)*b**8*x**8)/(105*a** 
3*b**5*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x** 
8))