Integrand size = 32, antiderivative size = 249 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {a \left (A b^3-a \left (a^2 B+b^2 B-a b C\right )\right ) x}{7 b^5 \left (a+b x^2\right )^{7/2}}-\frac {\left (8 A b^3-a \left (29 a^2 B+15 b^2 B-22 a b C\right )\right ) x}{35 b^5 \left (a+b x^2\right )^{5/2}}+\frac {\left (3 A b^3-a \left (234 a^2 B+45 b^2 B-122 a b C\right )\right ) x}{105 a b^5 \left (a+b x^2\right )^{3/2}}+\frac {\left (6 A b^3+a \left (582 a^2 B+15 b^2 B-176 a b C\right )\right ) x}{105 a^2 b^5 \sqrt {a+b x^2}}+\frac {B x \sqrt {a+b x^2}}{2 b^5}-\frac {(9 a B-2 b C) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \] Output:
1/7*a*(A*b^3-a*(B*a^2+B*b^2-C*a*b))*x/b^5/(b*x^2+a)^(7/2)-1/35*(8*A*b^3-a* (29*B*a^2+15*B*b^2-22*C*a*b))*x/b^5/(b*x^2+a)^(5/2)+1/105*(3*A*b^3-a*(234* B*a^2+45*B*b^2-122*C*a*b))*x/a/b^5/(b*x^2+a)^(3/2)+1/105*(6*A*b^3+a*(582*B *a^2+15*B*b^2-176*C*a*b))*x/a^2/b^5/(b*x^2+a)^(1/2)+1/2*B*x*(b*x^2+a)^(1/2 )/b^5-1/2*(9*B*a-2*C*b)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(11/2)
Time = 1.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.71 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (945 a^6 B+12 A b^6 x^6-210 a^5 b \left (C-15 B x^2\right )+6 a b^5 x^4 \left (7 A+5 B x^2\right )+a^2 b^4 x^6 \left (-352 C+105 B x^2\right )+14 a^4 b^2 x^2 \left (-50 C+261 B x^2\right )+4 a^3 b^3 x^4 \left (-203 C+396 B x^2\right )\right )}{210 a^2 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(-9 a B+2 b C) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{11/2}} \] Input:
Integrate[(x^4*(A + B*x^2 + C*x^4 + B*x^6))/(a + b*x^2)^(9/2),x]
Output:
(x*(945*a^6*B + 12*A*b^6*x^6 - 210*a^5*b*(C - 15*B*x^2) + 6*a*b^5*x^4*(7*A + 5*B*x^2) + a^2*b^4*x^6*(-352*C + 105*B*x^2) + 14*a^4*b^2*x^2*(-50*C + 2 61*B*x^2) + 4*a^3*b^3*x^4*(-203*C + 396*B*x^2)))/(210*a^2*b^5*(a + b*x^2)^ (7/2)) + ((-9*a*B + 2*b*C)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2] )])/b^(11/2)
Time = 0.63 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2335, 9, 25, 1586, 9, 27, 360, 1471, 27, 299, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^6+B x^2+C x^4\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^3 \left (7 a B x^5-7 a \left (\frac {a B}{b}-C\right ) x^3+\left (2 A b+\frac {5 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^4 \left (7 a B x^4-7 a \left (\frac {a B}{b}-C\right ) x^2+2 A b+\frac {5 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^4 \left (7 a B x^4-7 a \left (\frac {a B}{b}-C\right ) x^2+2 A b+\frac {5 a \left (B a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {35 x^3 \left (\frac {a^2 (2 a B-b C) x}{b^2}-\frac {a^2 B x^3}{b}\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {35 a^2 x^4 \left (-b B x^2+2 a B-b C\right )}{b^2 \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \int \frac {x^4 \left (-b B x^2+2 a B-b C\right )}{\left (b x^2+a\right )^{5/2}}dx}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {3 b^3 B x^4-3 b^2 (3 a B-b C) x^2+a b (3 a B-b C)}{\left (b x^2+a\right )^{3/2}}dx}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (15 a B-4 b C)}{\sqrt {a+b x^2}}-\frac {\int \frac {3 a b \left (-b B x^2+4 a B-b C\right )}{\sqrt {b x^2+a}}dx}{a}}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (15 a B-4 b C)}{\sqrt {a+b x^2}}-3 b \int \frac {-b B x^2+4 a B-b C}{\sqrt {b x^2+a}}dx}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (15 a B-4 b C)}{\sqrt {a+b x^2}}-3 b \left (\frac {1}{2} (9 a B-2 b C) \int \frac {1}{\sqrt {b x^2+a}}dx-\frac {1}{2} B x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (15 a B-4 b C)}{\sqrt {a+b x^2}}-3 b \left (\frac {1}{2} (9 a B-2 b C) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-\frac {1}{2} B x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 B-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {7 a \left (\frac {a x (3 a B-b C)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (15 a B-4 b C)}{\sqrt {a+b x^2}}-3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (9 a B-2 b C)}{2 \sqrt {b}}-\frac {1}{2} B x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 B-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(x^4*(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^5)/(7*a*(a + b*x^2)^(7/2)) + (((2 *A*b + (a*(19*a^2*B + 5*b^2*B - 12*a*b*C))/b^2)*x^5)/(5*a*(a + b*x^2)^(5/2 )) - (7*a*((a*(3*a*B - b*C)*x)/(3*b^2*(a + b*x^2)^(3/2)) - ((b*(15*a*B - 4 *b*C)*x)/Sqrt[a + b*x^2] - 3*b*(-1/2*(B*x*Sqrt[a + b*x^2]) + ((9*a*B - 2*b *C)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(3*b^3)))/b^2)/(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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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[(a + b*x^2)^(p + 1)*Expan dToSum[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])
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]
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 = 0.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.69
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {9 a^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (B a -\frac {2 C b}{9}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}+\frac {2 \left (\frac {315 B \sqrt {b}\, a^{6}}{4}+\left (\frac {35 \left (15 x^{2} B -C \right ) a^{5}}{2}+\frac {609 \left (x^{2} B -\frac {50 C}{261}\right ) x^{2} b \,a^{4}}{2}+132 \left (x^{2} B -\frac {203 C}{396}\right ) x^{4} b^{2} a^{3}+\frac {35 \left (x^{2} B -\frac {352 C}{105}\right ) x^{6} b^{3} a^{2}}{4}+\frac {7 \left (\frac {5 x^{2} B}{7}+A \right ) x^{4} b^{4} a}{2}+b^{5} A \,x^{6}\right ) b^{\frac {3}{2}}\right ) x}{35}}{b^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2}}\) | \(173\) |
| default | \(A \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+B \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )+C \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )\) | \(500\) |
| risch | \(\text {Expression too large to display}\) | \(2150\) |
Input:
int(x^4*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
2/35/(b*x^2+a)^(7/2)/b^(11/2)*(-315/4*a^2*(b*x^2+a)^(7/2)*(B*a-2/9*C*b)*ar ctanh((b*x^2+a)^(1/2)/x/b^(1/2))+(315/4*B*b^(1/2)*a^6+(35/2*(15*B*x^2-C)*a ^5+609/2*(x^2*B-50/261*C)*x^2*b*a^4+132*(x^2*B-203/396*C)*x^4*b^2*a^3+35/4 *(x^2*B-352/105*C)*x^6*b^3*a^2+7/2*(5/7*x^2*B+A)*x^4*b^4*a+b^5*A*x^6)*b^(3 /2))*x)/a^2
Time = 0.16 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.62 \[ \int \frac {x^4 \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 ({\left (9 \, B a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, B a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, B a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, B a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, B a^{6} b - 2 \, C a^{5} 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 \, B a^{2} b^{5} x^{9} + 2 \, {\left (792 \, B a^{3} b^{4} - 176 \, C a^{2} b^{5} + 15 \, B a b^{6} + 6 \, A b^{7}\right )} x^{7} + 14 \, {\left (261 \, B a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, B a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, B a^{6} b - 2 \, C a^{5} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{420 \, {\left (a^{2} b^{10} x^{8} + 4 \, a^{3} b^{9} x^{6} + 6 \, a^{4} b^{8} x^{4} + 4 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}, \frac {105 \, {\left ({\left (9 \, B a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, B a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, B a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, B a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, B a^{6} b - 2 \, C a^{5} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a^{2} b^{5} x^{9} + 2 \, {\left (792 \, B a^{3} b^{4} - 176 \, C a^{2} b^{5} + 15 \, B a b^{6} + 6 \, A b^{7}\right )} x^{7} + 14 \, {\left (261 \, B a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, B a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, B a^{6} b - 2 \, C a^{5} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{2} b^{10} x^{8} + 4 \, a^{3} b^{9} x^{6} + 6 \, a^{4} b^{8} x^{4} + 4 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}\right ] \] Input:
integrate(x^4*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[-1/420*(105*((9*B*a^3*b^4 - 2*C*a^2*b^5)*x^8 + 9*B*a^7 - 2*C*a^6*b + 4*(9 *B*a^4*b^3 - 2*C*a^3*b^4)*x^6 + 6*(9*B*a^5*b^2 - 2*C*a^4*b^3)*x^4 + 4*(9*B *a^6*b - 2*C*a^5*b^2)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b )*x - a) - 2*(105*B*a^2*b^5*x^9 + 2*(792*B*a^3*b^4 - 176*C*a^2*b^5 + 15*B* a*b^6 + 6*A*b^7)*x^7 + 14*(261*B*a^4*b^3 - 58*C*a^3*b^4 + 3*A*a*b^6)*x^5 + 350*(9*B*a^5*b^2 - 2*C*a^4*b^3)*x^3 + 105*(9*B*a^6*b - 2*C*a^5*b^2)*x)*sq rt(b*x^2 + a))/(a^2*b^10*x^8 + 4*a^3*b^9*x^6 + 6*a^4*b^8*x^4 + 4*a^5*b^7*x ^2 + a^6*b^6), 1/210*(105*((9*B*a^3*b^4 - 2*C*a^2*b^5)*x^8 + 9*B*a^7 - 2*C *a^6*b + 4*(9*B*a^4*b^3 - 2*C*a^3*b^4)*x^6 + 6*(9*B*a^5*b^2 - 2*C*a^4*b^3) *x^4 + 4*(9*B*a^6*b - 2*C*a^5*b^2)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b* x^2 + a)) + (105*B*a^2*b^5*x^9 + 2*(792*B*a^3*b^4 - 176*C*a^2*b^5 + 15*B*a *b^6 + 6*A*b^7)*x^7 + 14*(261*B*a^4*b^3 - 58*C*a^3*b^4 + 3*A*a*b^6)*x^5 + 350*(9*B*a^5*b^2 - 2*C*a^4*b^3)*x^3 + 105*(9*B*a^6*b - 2*C*a^5*b^2)*x)*sqr t(b*x^2 + a))/(a^2*b^10*x^8 + 4*a^3*b^9*x^6 + 6*a^4*b^8*x^4 + 4*a^5*b^7*x^ 2 + a^6*b^6)]
Leaf count of result is larger than twice the leaf count of optimal. 6467 vs. \(2 (238) = 476\).
Time = 86.91 (sec) , antiderivative size = 6467, normalized size of antiderivative = 25.97 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(x**4*(B*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
Output:
A*(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*sqrt(1 + b*x**2/a) + 35*a**(5/2)*b**3 *x**6*sqrt(1 + b*x**2/a)) + 2*b*x**7/(35*a**(11/2)*sqrt(1 + b*x**2/a) + 10 5*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*x**7/(7*a**(9/2)*s qrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2*sqrt(1 + b*x**2/a) + 21*a**(5/2)*b* *2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt(1 + b*x**2/a)) + B* (-315*a**(311/2)*b**66*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a** (309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqr t(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 140 0*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/ 2)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x* *2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 1890*a**(309/ 2)*b**67*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b **(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x **2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303 /2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*s qrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 7 0*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 4725*a**(307/2)*b**68* x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143...
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (221) = 442\).
Time = 0.06 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.02 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
1/2*B*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))*C*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))*B*a*x/b + 3/10*B*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*C*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*B*x^5/((b*x^2 + a)^(7/2)*b) + 3/2*B*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*C*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*B*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - C*a*x^3/((b*x^2 + a)^(5/2)*b^3) - 5/8*B*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 1/4*A*x^3/((b*x^2 + a)^(7/2)*b) - 417/70*B*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*B*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*B*a^3*x/((b*x^2 + a)^(5/2)*b^5) + 139/105*C*x/(sqrt(b*x^2 + a)*b ^4) + 17/105*C*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*C*a^2*x/((b*x^2 + a)^(5 /2)*b^4) + 1/14*B*x/((b*x^2 + a)^(3/2)*b^3) + 1/7*B*x/(sqrt(b*x^2 + a)*a*b ^3) + 3/56*B*a*x/((b*x^2 + a)^(5/2)*b^3) - 15/56*B*a^2*x/((b*x^2 + a)^(7/2 )*b^3) + 3/140*A*x/((b*x^2 + a)^(5/2)*b^2) + 2/35*A*x/(sqrt(b*x^2 + a)*a^2 *b^2) + 1/35*A*x/((b*x^2 + a)^(3/2)*a*b^2) - 3/28*A*a*x/((b*x^2 + a)^(7/2) *b^2) - 9/2*B*a*arcsinh(b*x/sqrt(a*b))/b^(11/2) + C*arcsinh(b*x/sqrt(a*...
Time = 0.15 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.82 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (\frac {105 \, B x^{2}}{b} + \frac {2 \, {\left (792 \, B a^{4} b^{7} - 176 \, C a^{3} b^{8} + 15 \, B a^{2} b^{9} + 6 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {14 \, {\left (261 \, B a^{5} b^{6} - 58 \, C a^{4} b^{7} + 3 \, A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {350 \, {\left (9 \, B a^{6} b^{5} - 2 \, C a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {105 \, {\left (9 \, B a^{7} b^{4} - 2 \, C a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (9 \, B a - 2 \, C b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \] Input:
integrate(x^4*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
1/210*((((105*B*x^2/b + 2*(792*B*a^4*b^7 - 176*C*a^3*b^8 + 15*B*a^2*b^9 + 6*A*a*b^10)/(a^3*b^9))*x^2 + 14*(261*B*a^5*b^6 - 58*C*a^4*b^7 + 3*A*a^2*b^ 9)/(a^3*b^9))*x^2 + 350*(9*B*a^6*b^5 - 2*C*a^5*b^6)/(a^3*b^9))*x^2 + 105*( 9*B*a^7*b^4 - 2*C*a^6*b^5)/(a^3*b^9))*x/(b*x^2 + a)^(7/2) + 1/2*(9*B*a - 2 *C*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^4\,\left (B\,x^6+C\,x^4+B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:
int((x^4*(A + B*x^2 + B*x^6 + C*x^4))/(a + b*x^2)^(9/2),x)
Output:
int((x^4*(A + B*x^2 + B*x^6 + C*x^4))/(a + b*x^2)^(9/2), x)
Time = 8.85 (sec) , antiderivative size = 728, normalized size of antiderivative = 2.92 \[ \int \frac {x^4 \left (A+B x^2+C x^4+B x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:
int(x^4*(B*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)
Output:
(945*sqrt(a + b*x**2)*a**5*b*x + 3150*sqrt(a + b*x**2)*a**4*b**2*x**3 - 21 0*sqrt(a + b*x**2)*a**4*b*c*x + 3654*sqrt(a + b*x**2)*a**3*b**3*x**5 - 700 *sqrt(a + b*x**2)*a**3*b**2*c*x**3 + 1584*sqrt(a + b*x**2)*a**2*b**4*x**7 - 812*sqrt(a + b*x**2)*a**2*b**3*c*x**5 + 105*sqrt(a + b*x**2)*a*b**5*x**9 + 42*sqrt(a + b*x**2)*a*b**5*x**5 - 352*sqrt(a + b*x**2)*a*b**4*c*x**7 + 42*sqrt(a + b*x**2)*b**6*x**7 - 945*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b )*x)/sqrt(a))*a**6 - 3780*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt( a))*a**5*b*x**2 + 210*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))* a**5*c - 5670*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b** 2*x**4 + 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*c* x**2 - 3780*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**3* x**6 + 1260*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2* c*x**4 - 945*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**4 *x**8 + 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**3* c*x**6 + 210*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**4*c* x**8 - 639*sqrt(b)*a**6 - 2556*sqrt(b)*a**5*b*x**2 + 112*sqrt(b)*a**5*c - 3834*sqrt(b)*a**4*b**2*x**4 + 18*sqrt(b)*a**4*b**2 + 448*sqrt(b)*a**4*b*c* x**2 - 2556*sqrt(b)*a**3*b**3*x**6 + 72*sqrt(b)*a**3*b**3*x**2 + 672*sqrt( b)*a**3*b**2*c*x**4 - 639*sqrt(b)*a**2*b**4*x**8 + 108*sqrt(b)*a**2*b**4*x **4 + 448*sqrt(b)*a**2*b**3*c*x**6 + 72*sqrt(b)*a*b**5*x**6 + 112*sqrt(...