Integrand size = 32, antiderivative size = 210 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b C-9 a D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \]
1/7*(A-a*(B*b^2-C*a*b+D*a^2)/b^3)*x^5/a/(b*x^2+a)^(7/2)+1/35*(2*A*b^3+a*(5 *B*b^2-12*C*a*b+19*D*a^2))*x^5/a^2/b^3/(b*x^2+a)^(5/2)+1/3*a*(C*b-3*D*a)*x /b^5/(b*x^2+a)^(3/2)+1/2*(2*C*b-9*D*a)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/ b^(11/2)-1/3*(4*C*b-15*D*a)*x/b^5/(b*x^2+a)^(1/2)+1/2*D*x*(b*x^2+a)^(1/2)/ b^5
Time = 1.01 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x \left (945 a^6 D+12 A b^6 x^6+6 a b^5 x^4 \left (7 A+5 B x^2\right )-210 a^5 b \left (C-15 D x^2\right )+a^2 b^4 x^6 \left (-352 C+105 D x^2\right )+14 a^4 b^2 x^2 \left (-50 C+261 D x^2\right )+4 a^3 b^3 x^4 \left (-203 C+396 D x^2\right )\right )}{210 a^2 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(2 b C-9 a D) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{11/2}} \]
(x*(945*a^6*D + 12*A*b^6*x^6 + 6*a*b^5*x^4*(7*A + 5*B*x^2) - 210*a^5*b*(C - 15*D*x^2) + a^2*b^4*x^6*(-352*C + 105*D*x^2) + 14*a^4*b^2*x^2*(-50*C + 2 61*D*x^2) + 4*a^3*b^3*x^4*(-203*C + 396*D*x^2)))/(210*a^2*b^5*(a + b*x^2)^ (7/2)) + ((2*b*C - 9*a*D)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2]) ])/b^(11/2)
Time = 0.61 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^3 \left (7 a D x^5+7 a \left (C-\frac {a D}{b}\right ) x^3+\left (2 A b+\frac {5 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right ) x\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {x^4 \left (7 a D x^4+7 a \left (C-\frac {a D}{b}\right ) x^2+2 A b+\frac {5 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^4 \left (7 a D x^4+7 a \left (C-\frac {a D}{b}\right ) x^2+2 A b+\frac {5 a \left (D a^2-b C a+b^2 B\right )}{b^2}\right )}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1586 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {35 x^3 \left (\frac {a^2 D x^3}{b}+\frac {a^2 (b C-2 a D) x}{b^2}\right )}{\left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {35 a^2 x^4 \left (b D x^2+b C-2 a D\right )}{b^2 \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 a \int \frac {x^4 \left (b D x^2+b C-2 a D\right )}{\left (b x^2+a\right )^{5/2}}dx}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle \frac {\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-3 b^3 D x^4-3 b^2 (b C-3 a D) x^2+a b (b C-3 a D)}{\left (b x^2+a\right )^{3/2}}dx}{3 b^3}\right )}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (4 b C-15 a D)}{\sqrt {a+b x^2}}-\frac {\int \frac {3 a b \left (b D x^2+b C-4 a D\right )}{\sqrt {b x^2+a}}dx}{a}}{3 b^3}\right )}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (4 b C-15 a D)}{\sqrt {a+b x^2}}-3 b \int \frac {b D x^2+b C-4 a D}{\sqrt {b x^2+a}}dx}{3 b^3}\right )}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (4 b C-15 a D)}{\sqrt {a+b x^2}}-3 b \left (\frac {1}{2} (2 b C-9 a D) \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} D x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (4 b C-15 a D)}{\sqrt {a+b x^2}}-3 b \left (\frac {1}{2} (2 b C-9 a D) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} D x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}+\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {x^5 \left (\frac {a \left (19 a^2 D-12 a b C+5 b^2 B\right )}{b^2}+2 A b\right )}{5 a \left (a+b x^2\right )^{5/2}}+\frac {7 a \left (\frac {a x (b C-3 a D)}{3 b^2 \left (a+b x^2\right )^{3/2}}-\frac {\frac {b x (4 b C-15 a D)}{\sqrt {a+b x^2}}-3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b C-9 a D)}{2 \sqrt {b}}+\frac {1}{2} D x \sqrt {a+b x^2}\right )}{3 b^3}\right )}{b^2}}{7 a b}+\frac {x^5 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}\) |
((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^5)/(7*a*(a + b*x^2)^(7/2)) + (((2 *A*b + (a*(5*b^2*B - 12*a*b*C + 19*a^2*D))/b^2)*x^5)/(5*a*(a + b*x^2)^(5/2 )) + (7*a*((a*(b*C - 3*a*D)*x)/(3*b^2*(a + b*x^2)^(3/2)) - ((b*(4*b*C - 15 *a*D)*x)/Sqrt[a + b*x^2] - 3*b*((D*x*Sqrt[a + b*x^2])/2 + ((2*b*C - 9*a*D) *ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(3*b^3)))/b^2)/(7*a*b )
3.2.61.3.1 Defintions of rubi rules used
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 = 3.65 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.78
| method | result | size |
| pseudoelliptic | \(\frac {\frac {a \left (\frac {5 x^{2} B}{7}+A \right ) x^{5} b^{\frac {11}{2}}}{5}+\frac {2 A \,b^{\frac {13}{2}} x^{7}}{35}+a^{2} \left (-x \,a^{3} \left (-15 D x^{2}+C \right ) b^{\frac {3}{2}}-\frac {10 a^{2} \left (-\frac {261 D x^{2}}{50}+C \right ) x^{3} b^{\frac {5}{2}}}{3}-\frac {58 a \,x^{5} \left (-\frac {396 D x^{2}}{203}+C \right ) b^{\frac {7}{2}}}{15}+\left (\frac {1}{2} D x^{9}-\frac {176}{105} C \,x^{7}\right ) b^{\frac {9}{2}}+\frac {9 D \sqrt {b}\, a^{4} x}{2}+\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (C b -\frac {9 D a}{2}\right )\right )}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {11}{2}} a^{2}}\) | \(164\) |
| default | \(C \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+B \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+A \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )+D \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )\) | \(500\) |
1/(b*x^2+a)^(7/2)*(1/5*a*(5/7*x^2*B+A)*x^5*b^(11/2)+2/35*A*b^(13/2)*x^7+a^ 2*(-x*a^3*(-15*D*x^2+C)*b^(3/2)-10/3*a^2*(-261/50*D*x^2+C)*x^3*b^(5/2)-58/ 15*a*x^5*(-396/203*D*x^2+C)*b^(7/2)+(1/2*D*x^9-176/105*C*x^7)*b^(9/2)+9/2* D*b^(1/2)*a^4*x+arctanh((b*x^2+a)^(1/2)/x/b^(1/2))*(b*x^2+a)^(7/2)*(C*b-9/ 2*D*a)))/b^(11/2)/a^2
Time = 0.40 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.11 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left ({\left (9 \, D a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, D a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, D a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, D 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 \, D a^{2} b^{5} x^{9} + 2 \, {\left (792 \, D 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 \, D a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, D 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 \, D a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, D a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, D a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, D 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 \, D a^{2} b^{5} x^{9} + 2 \, {\left (792 \, D 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 \, D a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, D 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 ] \]
[1/420*(105*((9*D*a^3*b^4 - 2*C*a^2*b^5)*x^8 + 9*D*a^7 - 2*C*a^6*b + 4*(9* D*a^4*b^3 - 2*C*a^3*b^4)*x^6 + 6*(9*D*a^5*b^2 - 2*C*a^4*b^3)*x^4 + 4*(9*D* 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*D*a^2*b^5*x^9 + 2*(792*D*a^3*b^4 - 176*C*a^2*b^5 + 15*B*a *b^6 + 6*A*b^7)*x^7 + 14*(261*D*a^4*b^3 - 58*C*a^3*b^4 + 3*A*a*b^6)*x^5 + 350*(9*D*a^5*b^2 - 2*C*a^4*b^3)*x^3 + 105*(9*D*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), 1/210*(105*((9*D*a^3*b^4 - 2*C*a^2*b^5)*x^8 + 9*D*a^7 - 2*C* a^6*b + 4*(9*D*a^4*b^3 - 2*C*a^3*b^4)*x^6 + 6*(9*D*a^5*b^2 - 2*C*a^4*b^3)* x^4 + 4*(9*D*a^6*b - 2*C*a^5*b^2)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x ^2 + a)) + (105*D*a^2*b^5*x^9 + 2*(792*D*a^3*b^4 - 176*C*a^2*b^5 + 15*B*a* b^6 + 6*A*b^7)*x^7 + 14*(261*D*a^4*b^3 - 58*C*a^3*b^4 + 3*A*a*b^6)*x^5 + 3 50*(9*D*a^5*b^2 - 2*C*a^4*b^3)*x^3 + 105*(9*D*a^6*b - 2*C*a^5*b^2)*x)*sqrt (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. \(2 (199) = 398\).
Time = 94.78 (sec) , antiderivative size = 6467, normalized size of antiderivative = 30.80 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
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)) + C* (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(1 + b*x**2/a) + 2100 *a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2 )*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x** 2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 630*a**(203/2 )*b**46*x**2*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(1 + b*x**2/a) + 2100*a**(199/ 2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sq rt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 10 5*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(201/2)*b**47* x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99...
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (183) = 366\).
Time = 0.23 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.59 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
1/2*D*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))*D*a*x/b + 3/10*D*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*D*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*D*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*D*a*x/(sqrt(b*x^2 + a)*b^5) - 51/70*D*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*D*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*D*a*arcsinh(b*x/sqrt(a*b))/b^(11/2) + C*arcsinh(b*x/sqrt(a*...
Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (\frac {105 \, D x^{2}}{b} + \frac {2 \, {\left (792 \, D a^{4} b^{7} - 176 \, C a^{3} b^{8} + 15 \, B a^{2} b^{9} + 6 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {14 \, {\left (261 \, D a^{5} b^{6} - 58 \, C a^{4} b^{7} + 3 \, A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {350 \, {\left (9 \, D a^{6} b^{5} - 2 \, C a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {105 \, {\left (9 \, D a^{7} b^{4} - 2 \, C a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (9 \, D a - 2 \, C b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \]
1/210*((((105*D*x^2/b + 2*(792*D*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*D*a^5*b^6 - 58*C*a^4*b^7 + 3*A*a^2*b^ 9)/(a^3*b^9))*x^2 + 350*(9*D*a^6*b^5 - 2*C*a^5*b^6)/(a^3*b^9))*x^2 + 105*( 9*D*a^7*b^4 - 2*C*a^6*b^5)/(a^3*b^9))*x/(b*x^2 + a)^(7/2) + 1/2*(9*D*a - 2 *C*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^4\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]