Integrand size = 32, antiderivative size = 219 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {\left (A b^3-a \left (a^2 B+b^2 B-a b C\right )\right ) x}{7 b^4 \left (a+b x^2\right )^{7/2}}+\frac {\left (A b^3-a \left (22 a^2 B+8 b^2 B-15 a b C\right )\right ) x}{35 a b^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (4 A b^3+a \left (122 a^2 B+3 b^2 B-45 a b C\right )\right ) x}{105 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 A b^3-a \left (176 a^2 B-6 b^2 B-15 a b C\right )\right ) x}{105 a^3 b^4 \sqrt {a+b x^2}}+\frac {B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \] Output:
-1/7*(A*b^3-a*(B*a^2+B*b^2-C*a*b))*x/b^4/(b*x^2+a)^(7/2)+1/35*(A*b^3-a*(22 *B*a^2+8*B*b^2-15*C*a*b))*x/a/b^4/(b*x^2+a)^(5/2)+1/105*(4*A*b^3+a*(122*B* a^2+3*B*b^2-45*C*a*b))*x/a^2/b^4/(b*x^2+a)^(3/2)+1/105*(8*A*b^3-a*(176*B*a ^2-6*B*b^2-15*C*a*b))*x/a^3/b^4/(b*x^2+a)^(1/2)+B*arctanh(b^(1/2)*x/(b*x^2 +a)^(1/2))/b^(9/2)
Time = 0.76 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-105 a^6 B x-350 a^5 b B x^3-406 a^4 b^2 B x^5+8 A b^6 x^7-176 a^3 b^3 B x^7+2 a b^5 x^5 \left (14 A+3 B x^2\right )+a^2 b^4 x^3 \left (35 A+21 B x^2+15 C x^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{7/2}}-\frac {B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{9/2}} \] Input:
Integrate[(x^2*(A + B*x^2 + C*x^4 + B*x^6))/(a + b*x^2)^(9/2),x]
Output:
(-105*a^6*B*x - 350*a^5*b*B*x^3 - 406*a^4*b^2*B*x^5 + 8*A*b^6*x^7 - 176*a^ 3*b^3*B*x^7 + 2*a*b^5*x^5*(14*A + 3*B*x^2) + a^2*b^4*x^3*(35*A + 21*B*x^2 + 15*C*x^4))/(105*a^3*b^4*(a + b*x^2)^(7/2)) - (B*Log[-(Sqrt[b]*x) + Sqrt[ a + b*x^2]])/b^(9/2)
Time = 0.59 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2335, 9, 25, 1586, 9, 25, 27, 357, 252, 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.
| \(\displaystyle \int \frac {x^2 \left (A+B x^6+B x^2+C x^4\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x \left (7 a B x^5-7 a \left (\frac {a B}{b}-C\right ) x^3+\left (4 A b+\frac {3 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^2 \left (7 a B x^4-7 a \left (\frac {a B}{b}-C\right ) x^2+4 A b+\frac {3 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^2 \left (7 a B x^4-7 a \left (\frac {a B}{b}-C\right ) x^2+4 A b+\frac {3 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x \left (\frac {35 a^2 B x^3}{b}+\left (8 A b-\frac {3 a \left (12 B a^2-5 b C a-2 b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {x^2 \left (35 a^2 B x^2+b \left (8 A b-\frac {3 a \left (12 B a^2-5 b C a-2 b^2 B\right )}{b^2}\right )\right )}{b \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (8 A b^2+35 a^2 B x^2-3 a \left (\frac {12 B a^2}{b}-5 C a-2 b B\right )\right )}{b \left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (8 A b^2+35 a^2 B x^2-3 a \left (\frac {12 B a^2}{b}-5 C a-2 b B\right )\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 357 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 B \int \frac {x^2}{\left (b x^2+a\right )^{3/2}}dx}{b}+\frac {x^3 \left (8 A b^3-a \left (71 a^2 B-15 a b C-6 b^2 B\right )\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 B \left (\frac {\int \frac {1}{\sqrt {b x^2+a}}dx}{b}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}+\frac {x^3 \left (8 A b^3-a \left (71 a^2 B-15 a b C-6 b^2 B\right )\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {35 a^2 B \left (\frac {\int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}+\frac {x^3 \left (8 A b^3-a \left (71 a^2 B-15 a b C-6 b^2 B\right )\right )}{3 a b \left (a+b x^2\right )^{3/2}}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {x^3 \left (8 A b^3-a \left (71 a^2 B-15 a b C-6 b^2 B\right )\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {35 a^2 B \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x}{b \sqrt {a+b x^2}}\right )}{b}}{5 a b}+\frac {x^3 \left (\frac {a \left (17 a^2 B-10 a b C+3 b^2 B\right )}{b^2}+4 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^3 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(x^2*(A + B*x^2 + C*x^4 + B*x^6))/(a + b*x^2)^(9/2),x]
Output:
((A - (a*(a^2*B + b^2*B - a*b*C))/b^3)*x^3)/(7*a*(a + b*x^2)^(7/2)) + (((4 *A*b + (a*(17*a^2*B + 3*b^2*B - 10*a*b*C))/b^2)*x^3)/(5*a*(a + b*x^2)^(5/2 )) + (((8*A*b^3 - a*(71*a^2*B - 6*b^2*B - 15*a*b*C))*x^3)/(3*a*b*(a + b*x^ 2)^(3/2)) + (35*a^2*B*(-(x/(b*Sqrt[a + b*x^2])) + ArcTanh[(Sqrt[b]*x)/Sqrt [a + b*x^2]]/b^(3/2)))/b)/(5*a*b))/(7*a*b)
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_ Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*b*e*(m + 1))), x] + Simp[d/b Int[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
Int[((f_.)*(x_))^(m_.)*((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)*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(2*d *f*(q + 1))), x] + Simp[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]
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] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[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]
Time = 0.58 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.67
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} B \,a^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \left (x^{2} a^{2} \left (\frac {3}{7} C \,x^{4}+\frac {3}{5} x^{2} B +A \right ) b^{\frac {9}{2}}+\frac {4 \left (\frac {3 x^{2} B}{14}+A \right ) a \,x^{4} b^{\frac {11}{2}}}{5}+\frac {8 A \,b^{\frac {13}{2}} x^{6}}{35}-10 a^{3} \left (\frac {88 b^{\frac {7}{2}} x^{6}}{175}+\frac {29 a \,b^{\frac {5}{2}} x^{4}}{25}+a^{2} b^{\frac {3}{2}} x^{2}+\frac {3 a^{3} \sqrt {b}}{10}\right ) B \right )}{3 b^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3}}\) | \(146\) |
| default | \(A \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )+B \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+B \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+C \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )\) | \(469\) |
Input:
int(x^2*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/3/b^(9/2)/(b*x^2+a)^(7/2)*(3*(b*x^2+a)^(7/2)*B*a^3*arctanh((b*x^2+a)^(1/ 2)/x/b^(1/2))+x*(x^2*a^2*(3/7*C*x^4+3/5*x^2*B+A)*b^(9/2)+4/5*(3/14*x^2*B+A )*a*x^4*b^(11/2)+8/35*A*b^(13/2)*x^6-10*a^3*(88/175*b^(7/2)*x^6+29/25*a*b^ (5/2)*x^4+a^2*b^(3/2)*x^2+3/10*a^3*b^(1/2))*B))/a^3
Time = 0.14 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.24 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (B a^{3} b^{4} x^{8} + 4 \, B a^{4} b^{3} x^{6} + 6 \, B a^{5} b^{2} x^{4} + 4 \, B a^{6} b x^{2} + B a^{7}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (105 \, B a^{6} b x + {\left (176 \, B a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, B a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, B a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, -\frac {105 \, {\left (B a^{3} b^{4} x^{8} + 4 \, B a^{4} b^{3} x^{6} + 6 \, B a^{5} b^{2} x^{4} + 4 \, B a^{6} b x^{2} + B a^{7}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a^{6} b x + {\left (176 \, B a^{3} b^{4} - 15 \, C a^{2} b^{5} - 6 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 7 \, {\left (58 \, B a^{4} b^{3} - 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )} x^{5} + 35 \, {\left (10 \, B a^{5} b^{2} - A a^{2} b^{5}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \] Input:
integrate(x^2*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/210*(105*(B*a^3*b^4*x^8 + 4*B*a^4*b^3*x^6 + 6*B*a^5*b^2*x^4 + 4*B*a^6*b *x^2 + B*a^7)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2* (105*B*a^6*b*x + (176*B*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^7 + 7*(58*B*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*(10*B*a^5*b^2 - A*a^ 2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 + a^7*b^5), -1/105*(105*(B*a^3*b^4*x^8 + 4*B*a^4*b^3*x^6 + 6*B*a^5*b^2*x^4 + 4*B*a^6*b*x^2 + B*a^7)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt( b*x^2 + a)) + (105*B*a^6*b*x + (176*B*a^3*b^4 - 15*C*a^2*b^5 - 6*B*a*b^6 - 8*A*b^7)*x^7 + 7*(58*B*a^4*b^3 - 3*B*a^2*b^5 - 4*A*a*b^6)*x^5 + 35*(10*B* a^5*b^2 - A*a^2*b^5)*x^3)*sqrt(b*x^2 + a))/(a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 + a^7*b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 3803 vs. \(2 (214) = 428\).
Time = 74.11 (sec) , antiderivative size = 3803, normalized size of antiderivative = 17.37 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(x**2*(B*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
Output:
A*(35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*s qrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(1 3/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x** 2/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b *x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 42 0*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a **(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x* *2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x* *8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a ) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt (1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2 )*b**4*x**8*sqrt(1 + b*x**2/a))) + B*(7*a*x**5/(35*a**(11/2)*sqrt(1 + b*x* *2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sq rt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + 2*b*x**7/(3 5*a**(11/2)*sqrt(1 + b*x**2/a) + 105*a**(9/2)*b*x**2*sqrt(1 + b*x**2/a) + 105*a**(7/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3*x**6*sqrt(1 + b*x**2/a))) + B*(105*a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/ sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b** (101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(...
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (197) = 394\).
Time = 0.06 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.43 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:
integrate(x^2*(B*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))*B*x - 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*C*x^5/((b*x^2 + a)^(7/2)*b) - 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 - B*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) + 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 *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) + B*arcsinh(b*x/ sqrt(a*b))/b^(9/2)
Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {{\left (176 \, B a^{3} b^{6} - 15 \, C a^{2} b^{7} - 6 \, B a b^{8} - 8 \, A b^{9}\right )} x^{2}}{a^{3} b^{7}} + \frac {7 \, {\left (58 \, B a^{4} b^{5} - 3 \, B a^{2} b^{7} - 4 \, A a b^{8}\right )}}{a^{3} b^{7}}\right )} + \frac {35 \, {\left (10 \, B a^{5} b^{4} - A a^{2} b^{7}\right )}}{a^{3} b^{7}}\right )} x^{2} + \frac {105 \, B a^{3}}{b^{4}}\right )} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {B \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \] Input:
integrate(x^2*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-1/105*((x^2*((176*B*a^3*b^6 - 15*C*a^2*b^7 - 6*B*a*b^8 - 8*A*b^9)*x^2/(a^ 3*b^7) + 7*(58*B*a^4*b^5 - 3*B*a^2*b^7 - 4*A*a*b^8)/(a^3*b^7)) + 35*(10*B* a^5*b^4 - A*a^2*b^7)/(a^3*b^7))*x^2 + 105*B*a^3/b^4)*x/(b*x^2 + a)^(7/2) - B*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^2\,\left (B\,x^6+C\,x^4+B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:
int((x^2*(A + B*x^2 + B*x^6 + C*x^4))/(a + b*x^2)^(9/2),x)
Output:
int((x^2*(A + B*x^2 + B*x^6 + C*x^4))/(a + b*x^2)^(9/2), x)
Time = 0.32 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.37 \[ \int \frac {x^2 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-105 \sqrt {b \,x^{2}+a}\, a^{5} b x -350 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} x^{3}-406 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{5}-176 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{7}+224 \sqrt {b}\, a^{5} b \,x^{2}+336 \sqrt {b}\, a^{4} b^{2} x^{4}+224 \sqrt {b}\, a^{3} b^{3} x^{6}+56 \sqrt {b}\, a^{2} b^{4} x^{8}+14 \sqrt {b \,x^{2}+a}\, b^{6} x^{7}+15 \sqrt {b}\, a^{5} c -14 \sqrt {b}\, a^{4} b^{2}-14 \sqrt {b}\, b^{6} x^{8}+56 \sqrt {b}\, a^{6}+15 \sqrt {b}\, a \,b^{4} c \,x^{8}+420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} b \,x^{2}+630 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b^{2} x^{4}+420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{3} x^{6}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{4} x^{8}+15 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{7}+60 \sqrt {b}\, a^{4} b c \,x^{2}+90 \sqrt {b}\, a^{3} b^{2} c \,x^{4}+60 \sqrt {b}\, a^{2} b^{3} c \,x^{6}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{6}+35 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{3}+49 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{5}-56 \sqrt {b}\, a^{3} b^{3} x^{2}-84 \sqrt {b}\, a^{2} b^{4} x^{4}-56 \sqrt {b}\, a \,b^{5} x^{6}}{105 a^{2} b^{4} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(x^2*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)
Output:
( - 105*sqrt(a + b*x**2)*a**5*b*x - 350*sqrt(a + b*x**2)*a**4*b**2*x**3 - 406*sqrt(a + b*x**2)*a**3*b**3*x**5 - 176*sqrt(a + b*x**2)*a**2*b**4*x**7 + 35*sqrt(a + b*x**2)*a**2*b**4*x**3 + 49*sqrt(a + b*x**2)*a*b**5*x**5 + 1 5*sqrt(a + b*x**2)*a*b**4*c*x**7 + 14*sqrt(a + b*x**2)*b**6*x**7 + 105*sqr t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**6 + 420*sqrt(b)*log((s qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*b*x**2 + 630*sqrt(b)*log((sqrt( a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*x**4 + 420*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**3*x**6 + 105*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**4*x**8 + 56*sqrt(b)*a**6 + 224*sqr t(b)*a**5*b*x**2 + 15*sqrt(b)*a**5*c + 336*sqrt(b)*a**4*b**2*x**4 - 14*sqr t(b)*a**4*b**2 + 60*sqrt(b)*a**4*b*c*x**2 + 224*sqrt(b)*a**3*b**3*x**6 - 5 6*sqrt(b)*a**3*b**3*x**2 + 90*sqrt(b)*a**3*b**2*c*x**4 + 56*sqrt(b)*a**2*b **4*x**8 - 84*sqrt(b)*a**2*b**4*x**4 + 60*sqrt(b)*a**2*b**3*c*x**6 - 56*sq rt(b)*a*b**5*x**6 + 15*sqrt(b)*a*b**4*c*x**8 - 14*sqrt(b)*b**6*x**8)/(105* a**2*b**4*(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))