Integrand size = 32, antiderivative size = 191 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}+\frac {(8 A b-11 a B) \left (a+b x^2\right )^{3/2}}{99 a^2 x^9}-\frac {\left (16 A b^2-22 a b B+33 a^2 C\right ) \left (a+b x^2\right )^{3/2}}{231 a^3 x^7}+\frac {\left (64 A b^3-11 a \left (8 b^2 B-12 a b C+21 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{1155 a^4 x^5}-\frac {2 b \left (64 A b^3-11 a \left (8 b^2 B-12 a b C+21 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{3465 a^5 x^3} \] Output:
-1/11*A*(b*x^2+a)^(3/2)/a/x^11+1/99*(8*A*b-11*B*a)*(b*x^2+a)^(3/2)/a^2/x^9 -1/231*(16*A*b^2-22*B*a*b+33*C*a^2)*(b*x^2+a)^(3/2)/a^3/x^7+1/1155*(64*A*b ^3-11*a*(8*B*b^2-12*C*a*b+21*D*a^2))*(b*x^2+a)^(3/2)/a^4/x^5-2/3465*b*(64* A*b^3-11*a*(8*B*b^2-12*C*a*b+21*D*a^2))*(b*x^2+a)^(3/2)/a^5/x^3
Time = 0.69 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=-\frac {\left (a+b x^2\right )^{3/2} \left (128 A b^4 x^8-16 a b^3 x^6 \left (12 A+11 B x^2\right )+a^4 \left (315 A+385 B x^2+495 C x^4+693 D x^6\right )+24 a^2 b^2 x^4 \left (10 A+11 x^2 \left (B+C x^2\right )\right )-2 a^3 b x^2 \left (140 A+33 \left (5 B x^2+6 C x^4+7 D x^6\right )\right )\right )}{3465 a^5 x^{11}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/x^12,x]
Output:
-1/3465*((a + b*x^2)^(3/2)*(128*A*b^4*x^8 - 16*a*b^3*x^6*(12*A + 11*B*x^2) + a^4*(315*A + 385*B*x^2 + 495*C*x^4 + 693*D*x^6) + 24*a^2*b^2*x^4*(10*A + 11*x^2*(B + C*x^2)) - 2*a^3*b*x^2*(140*A + 33*(5*B*x^2 + 6*C*x^4 + 7*D*x ^6))))/(a^5*x^11)
Time = 0.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2334, 2089, 1588, 27, 359, 245, 242}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b x^2+a} \left (8 A b-11 a \left (D x^4+C x^2+B\right )\right )}{x^{10}}dx}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b x^2+a} \left (-11 a D x^4-11 a C x^2+8 A b-11 a B\right )}{x^{10}}dx}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \sqrt {b x^2+a} \left (16 A b^2+33 a^2 D x^2-11 a (2 b B-3 a C)\right )}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{3/2} (8 A b-11 a B)}{9 a x^9}}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sqrt {b x^2+a} \left (33 D x^2 a^2+33 C a^2-22 b B a+16 A b^2\right )}{x^8}dx}{3 a}-\frac {\left (a+b x^2\right )^{3/2} (8 A b-11 a B)}{9 a x^9}}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (64 A b^3-11 a \left (21 a^2 D-12 a b C+8 b^2 B\right )\right ) \int \frac {\sqrt {b x^2+a}}{x^6}dx}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \left (33 a^2 C-22 a b B+16 A b^2\right )}{7 a x^7}}{3 a}-\frac {\left (a+b x^2\right )^{3/2} (8 A b-11 a B)}{9 a x^9}}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (64 A b^3-11 a \left (21 a^2 D-12 a b C+8 b^2 B\right )\right ) \left (-\frac {2 b \int \frac {\sqrt {b x^2+a}}{x^4}dx}{5 a}-\frac {\left (a+b x^2\right )^{3/2}}{5 a x^5}\right )}{7 a}-\frac {\left (a+b x^2\right )^{3/2} \left (33 a^2 C-22 a b B+16 A b^2\right )}{7 a x^7}}{3 a}-\frac {\left (a+b x^2\right )^{3/2} (8 A b-11 a B)}{9 a x^9}}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (a+b x^2\right )^{3/2} \left (33 a^2 C-22 a b B+16 A b^2\right )}{7 a x^7}-\frac {\left (\frac {2 b \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {\left (a+b x^2\right )^{3/2}}{5 a x^5}\right ) \left (64 A b^3-11 a \left (21 a^2 D-12 a b C+8 b^2 B\right )\right )}{7 a}}{3 a}-\frac {\left (a+b x^2\right )^{3/2} (8 A b-11 a B)}{9 a x^9}}{11 a}-\frac {A \left (a+b x^2\right )^{3/2}}{11 a x^{11}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/x^12,x]
Output:
-1/11*(A*(a + b*x^2)^(3/2))/(a*x^11) - (-1/9*((8*A*b - 11*a*B)*(a + b*x^2) ^(3/2))/(a*x^9) - (-1/7*((16*A*b^2 - 22*a*b*B + 33*a^2*C)*(a + b*x^2)^(3/2 ))/(a*x^7) - ((64*A*b^3 - 11*a*(8*b^2*B - 12*a*b*C + 21*a^2*D))*(-1/5*(a + b*x^2)^(3/2)/(a*x^5) + (2*b*(a + b*x^2)^(3/2))/(15*a^2*x^3)))/(7*a))/(3*a ))/(11*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi alQ[z, x] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ [u, x])
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[ x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Time = 0.56 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.64
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {11}{5} D x^{6}+\frac {11}{7} C \,x^{4}+\frac {11}{9} x^{2} B +A \right ) a^{4}-\frac {8 x^{2} b \left (\frac {33}{20} D x^{6}+\frac {99}{70} C \,x^{4}+\frac {33}{28} x^{2} B +A \right ) a^{3}}{9}+\frac {16 \left (\frac {11}{10} C \,x^{4}+\frac {11}{10} x^{2} B +A \right ) x^{4} b^{2} a^{2}}{21}-\frac {64 \left (\frac {11 x^{2} B}{12}+A \right ) x^{6} b^{3} a}{105}+\frac {128 A \,x^{8} b^{4}}{315}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{11 x^{11} a^{5}}\) | \(123\) |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (128 A \,x^{8} b^{4}-176 B \,x^{8} a \,b^{3}+264 C \,a^{2} b^{2} x^{8}-462 D a^{3} b \,x^{8}-192 A \,x^{6} a \,b^{3}+264 B \,x^{6} a^{2} b^{2}-396 C \,a^{3} b \,x^{6}+693 D a^{4} x^{6}+240 A \,x^{4} a^{2} b^{2}-330 B \,x^{4} a^{3} b +495 C \,a^{4} x^{4}-280 A \,x^{2} a^{3} b +385 B \,x^{2} a^{4}+315 A \,a^{4}\right )}{3465 x^{11} a^{5}}\) | \(157\) |
| orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (128 A \,x^{8} b^{4}-176 B \,x^{8} a \,b^{3}+264 C \,a^{2} b^{2} x^{8}-462 D a^{3} b \,x^{8}-192 A \,x^{6} a \,b^{3}+264 B \,x^{6} a^{2} b^{2}-396 C \,a^{3} b \,x^{6}+693 D a^{4} x^{6}+240 A \,x^{4} a^{2} b^{2}-330 B \,x^{4} a^{3} b +495 C \,a^{4} x^{4}-280 A \,x^{2} a^{3} b +385 B \,x^{2} a^{4}+315 A \,a^{4}\right )}{3465 x^{11} a^{5}}\) | \(157\) |
| trager | \(-\frac {\left (128 A \,b^{5} x^{10}-176 B a \,b^{4} x^{10}+264 C \,a^{2} b^{3} x^{10}-462 D a^{3} b^{2} x^{10}-64 a A \,b^{4} x^{8}+88 B \,a^{2} b^{3} x^{8}-132 C \,a^{3} b^{2} x^{8}+231 D a^{4} b \,x^{8}+48 a^{2} A \,b^{3} x^{6}-66 B \,a^{3} b^{2} x^{6}+99 C \,a^{4} b \,x^{6}+693 D a^{5} x^{6}-40 a^{3} A \,b^{2} x^{4}+55 B \,a^{4} b \,x^{4}+495 C \,a^{5} x^{4}+35 a^{4} A b \,x^{2}+385 B \,a^{5} x^{2}+315 a^{5} A \right ) \sqrt {b \,x^{2}+a}}{3465 x^{11} a^{5}}\) | \(205\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{11 a \,x^{11}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )\) | \(298\) |
Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^12,x,method=_RETURNVERBOSE)
Output:
-1/11*((11/5*D*x^6+11/7*C*x^4+11/9*x^2*B+A)*a^4-8/9*x^2*b*(33/20*D*x^6+99/ 70*C*x^4+33/28*x^2*B+A)*a^3+16/21*(11/10*C*x^4+11/10*x^2*B+A)*x^4*b^2*a^2- 64/105*(11/12*x^2*B+A)*x^6*b^3*a+128/315*A*x^8*b^4)*(b*x^2+a)^(3/2)/x^11/a ^5
Time = 0.20 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=\frac {{\left (2 \, {\left (231 \, D a^{3} b^{2} - 132 \, C a^{2} b^{3} + 88 \, B a b^{4} - 64 \, A b^{5}\right )} x^{10} - {\left (231 \, D a^{4} b - 132 \, C a^{3} b^{2} + 88 \, B a^{2} b^{3} - 64 \, A a b^{4}\right )} x^{8} - 3 \, {\left (231 \, D a^{5} + 33 \, C a^{4} b - 22 \, B a^{3} b^{2} + 16 \, A a^{2} b^{3}\right )} x^{6} - 315 \, A a^{5} - 5 \, {\left (99 \, C a^{5} + 11 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x^{4} - 35 \, {\left (11 \, B a^{5} + A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, a^{5} x^{11}} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^12,x, algorithm="fricas" )
Output:
1/3465*(2*(231*D*a^3*b^2 - 132*C*a^2*b^3 + 88*B*a*b^4 - 64*A*b^5)*x^10 - ( 231*D*a^4*b - 132*C*a^3*b^2 + 88*B*a^2*b^3 - 64*A*a*b^4)*x^8 - 3*(231*D*a^ 5 + 33*C*a^4*b - 22*B*a^3*b^2 + 16*A*a^2*b^3)*x^6 - 315*A*a^5 - 5*(99*C*a^ 5 + 11*B*a^4*b - 8*A*a^3*b^2)*x^4 - 35*(11*B*a^5 + A*a^4*b)*x^2)*sqrt(b*x^ 2 + a)/(a^5*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 1890 vs. \(2 (189) = 378\).
Time = 3.25 (sec) , antiderivative size = 1890, normalized size of antiderivative = 9.90 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**(1/2)*(D*x**6+C*x**4+B*x**2+A)/x**12,x)
Output:
-315*A*a**9*b**(33/2)*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860* a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465* a**5*b**20*x**18) - 1295*A*a**8*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(3465* a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860 *a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 1990*A*a**7*b**(37/2)*x**4*sq rt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790 *a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 1358 *A*a**6*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860 *a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465 *a**5*b**20*x**18) - 343*A*a**5*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(3465* a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860 *a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 35*A*a**4*b**(43/2)*x**10*sqr t(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790* a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 280*A *a**3*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860* a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465* a**5*b**20*x**18) - 560*A*a**2*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(3465* a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860 *a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 448*A*a*b**(49/2)*x**16*sqrt( a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790...
Time = 0.04 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D b}{15 \, a^{2} x^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{2}}{105 \, a^{3} x^{3}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{315 \, a^{4} x^{3}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{3465 \, a^{5} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D}{5 \, a x^{5}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{105 \, a^{3} x^{5}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{1155 \, a^{4} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{21 \, a^{2} x^{7}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{231 \, a^{3} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{9 \, a x^{9}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{99 \, a^{2} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{11 \, a x^{11}} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^12,x, algorithm="maxima" )
Output:
2/15*(b*x^2 + a)^(3/2)*D*b/(a^2*x^3) - 8/105*(b*x^2 + a)^(3/2)*C*b^2/(a^3* x^3) + 16/315*(b*x^2 + a)^(3/2)*B*b^3/(a^4*x^3) - 128/3465*(b*x^2 + a)^(3/ 2)*A*b^4/(a^5*x^3) - 1/5*(b*x^2 + a)^(3/2)*D/(a*x^5) + 4/35*(b*x^2 + a)^(3 /2)*C*b/(a^2*x^5) - 8/105*(b*x^2 + a)^(3/2)*B*b^2/(a^3*x^5) + 64/1155*(b*x ^2 + a)^(3/2)*A*b^3/(a^4*x^5) - 1/7*(b*x^2 + a)^(3/2)*C/(a*x^7) + 2/21*(b* x^2 + a)^(3/2)*B*b/(a^2*x^7) - 16/231*(b*x^2 + a)^(3/2)*A*b^2/(a^3*x^7) - 1/9*(b*x^2 + a)^(3/2)*B/(a*x^9) + 8/99*(b*x^2 + a)^(3/2)*A*b/(a^2*x^9) - 1 /11*(b*x^2 + a)^(3/2)*A/(a*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (171) = 342\).
Time = 0.15 (sec) , antiderivative size = 884, normalized size of antiderivative = 4.63 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^12,x, algorithm="giac")
Output:
4/3465*(3465*(sqrt(b)*x - sqrt(b*x^2 + a))^18*D*b^(5/2) - 19635*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a*b^(5/2) + 9240*(sqrt(b)*x - sqrt(b*x^2 + a))^16 *C*b^(7/2) + 46200*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^2*b^(5/2) - 32340* (sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a*b^(7/2) + 27720*(sqrt(b)*x - sqrt(b*x ^2 + a))^14*B*b^(9/2) - 59136*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^3*b^(5/ 2) + 39732*(sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^2*b^(7/2) - 38808*(sqrt(b) *x - sqrt(b*x^2 + a))^12*B*a*b^(9/2) + 88704*(sqrt(b)*x - sqrt(b*x^2 + a)) ^12*A*b^(11/2) + 47586*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^4*b^(5/2) - 21 252*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^3*b^(7/2) + 1848*(sqrt(b)*x - sqr t(b*x^2 + a))^10*B*a^2*b^(9/2) + 59136*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A* a*b^(11/2) - 30030*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^5*b^(5/2) + 11220*( sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^4*b^(7/2) - 1320*(sqrt(b)*x - sqrt(b*x^ 2 + a))^8*B*a^3*b^(9/2) + 21120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(1 1/2) + 18480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^6*b^(5/2) - 12540*(sqrt(b )*x - sqrt(b*x^2 + a))^6*C*a^5*b^(7/2) + 14520*(sqrt(b)*x - sqrt(b*x^2 + a ))^6*B*a^4*b^(9/2) - 10560*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^3*b^(11/2) - 9240*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^7*b^(5/2) + 7260*(sqrt(b)*x - s qrt(b*x^2 + a))^4*C*a^6*b^(7/2) - 4840*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a ^5*b^(9/2) + 3520*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^4*b^(11/2) + 2541*(s qrt(b)*x - sqrt(b*x^2 + a))^2*D*a^8*b^(5/2) - 1452*(sqrt(b)*x - sqrt(b*...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{x^{12}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/x^12,x)
Output:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/x^12, x)
Time = 0.17 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{12}} \, dx=\frac {-105 \sqrt {b \,x^{2}+a}\, a^{5}-140 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-165 \sqrt {b \,x^{2}+a}\, a^{4} c \,x^{4}-231 \sqrt {b \,x^{2}+a}\, a^{4} d \,x^{6}-5 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-33 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{6}-77 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{8}+6 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}+44 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{8}+154 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{10}-8 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-88 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{10}+16 \sqrt {b \,x^{2}+a}\, b^{5} x^{10}-154 \sqrt {b}\, a^{2} b^{2} d \,x^{11}+88 \sqrt {b}\, a \,b^{3} c \,x^{11}-16 \sqrt {b}\, b^{5} x^{11}}{1155 a^{4} x^{11}} \] Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^12,x)
Output:
( - 105*sqrt(a + b*x**2)*a**5 - 140*sqrt(a + b*x**2)*a**4*b*x**2 - 165*sqr t(a + b*x**2)*a**4*c*x**4 - 231*sqrt(a + b*x**2)*a**4*d*x**6 - 5*sqrt(a + b*x**2)*a**3*b**2*x**4 - 33*sqrt(a + b*x**2)*a**3*b*c*x**6 - 77*sqrt(a + b *x**2)*a**3*b*d*x**8 + 6*sqrt(a + b*x**2)*a**2*b**3*x**6 + 44*sqrt(a + b*x **2)*a**2*b**2*c*x**8 + 154*sqrt(a + b*x**2)*a**2*b**2*d*x**10 - 8*sqrt(a + b*x**2)*a*b**4*x**8 - 88*sqrt(a + b*x**2)*a*b**3*c*x**10 + 16*sqrt(a + b *x**2)*b**5*x**10 - 154*sqrt(b)*a**2*b**2*d*x**11 + 88*sqrt(b)*a*b**3*c*x* *11 - 16*sqrt(b)*b**5*x**11)/(1155*a**4*x**11)