Integrand size = 32, antiderivative size = 243 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}+\frac {(10 A b-13 a B) \left (a+b x^2\right )^{3/2}}{143 a^2 x^{11}}-\frac {\left (80 A b^2-104 a b B+143 a^2 C\right ) \left (a+b x^2\right )^{3/2}}{1287 a^3 x^9}+\frac {\left (160 A b^3-13 a \left (16 b^2 B-22 a b C+33 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{3003 a^4 x^7}-\frac {4 b \left (160 A b^3-13 a \left (16 b^2 B-22 a b C+33 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{15015 a^5 x^5}+\frac {8 b^2 \left (160 A b^3-13 a \left (16 b^2 B-22 a b C+33 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{45045 a^6 x^3} \] Output:
-1/13*A*(b*x^2+a)^(3/2)/a/x^13+1/143*(10*A*b-13*B*a)*(b*x^2+a)^(3/2)/a^2/x ^11-1/1287*(80*A*b^2-104*B*a*b+143*C*a^2)*(b*x^2+a)^(3/2)/a^3/x^9+1/3003*( 160*A*b^3-13*a*(16*B*b^2-22*C*a*b+33*D*a^2))*(b*x^2+a)^(3/2)/a^4/x^7-4/150 15*b*(160*A*b^3-13*a*(16*B*b^2-22*C*a*b+33*D*a^2))*(b*x^2+a)^(3/2)/a^5/x^5 +8/45045*b^2*(160*A*b^3-13*a*(16*B*b^2-22*C*a*b+33*D*a^2))*(b*x^2+a)^(3/2) /a^6/x^3
Time = 0.86 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=\frac {\left (a+b x^2\right )^{3/2} \left (1280 A b^5 x^{10}-128 a b^4 x^8 \left (15 A+13 B x^2\right )+16 a^2 b^3 x^6 \left (150 A+156 B x^2+143 C x^4\right )+2 a^4 b x^2 \left (1575 A+1820 B x^2+2145 C x^4+2574 D x^6\right )-8 a^3 b^2 x^4 \left (350 A+390 B x^2+429 x^4 \left (C+D x^2\right )\right )-5 a^5 \left (693 A+13 \left (63 B x^2+77 C x^4+99 D x^6\right )\right )\right )}{45045 a^6 x^{13}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/x^14,x]
Output:
((a + b*x^2)^(3/2)*(1280*A*b^5*x^10 - 128*a*b^4*x^8*(15*A + 13*B*x^2) + 16 *a^2*b^3*x^6*(150*A + 156*B*x^2 + 143*C*x^4) + 2*a^4*b*x^2*(1575*A + 1820* B*x^2 + 2145*C*x^4 + 2574*D*x^6) - 8*a^3*b^2*x^4*(350*A + 390*B*x^2 + 429* x^4*(C + D*x^2)) - 5*a^5*(693*A + 13*(63*B*x^2 + 77*C*x^4 + 99*D*x^6))))/( 45045*a^6*x^13)
Time = 0.52 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89, 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, 359, 245, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b x^2+a} \left (10 A b-13 a \left (D x^4+C x^2+B\right )\right )}{x^{12}}dx}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {\sqrt {b x^2+a} \left (-13 a D x^4-13 a C x^2+10 A b-13 a B\right )}{x^{12}}dx}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sqrt {b x^2+a} \left (143 D x^2 a^2+143 C a^2-104 b B a+80 A b^2\right )}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{3/2} (10 A b-13 a B)}{11 a x^{11}}}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (160 A b^3-13 a \left (33 a^2 D-22 a b C+16 b^2 B\right )\right ) \int \frac {\sqrt {b x^2+a}}{x^8}dx}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \left (143 a^2 C-104 a b B+80 A b^2\right )}{9 a x^9}}{11 a}-\frac {\left (a+b x^2\right )^{3/2} (10 A b-13 a B)}{11 a x^{11}}}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (160 A b^3-13 a \left (33 a^2 D-22 a b C+16 b^2 B\right )\right ) \left (-\frac {4 b \int \frac {\sqrt {b x^2+a}}{x^6}dx}{7 a}-\frac {\left (a+b x^2\right )^{3/2}}{7 a x^7}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \left (143 a^2 C-104 a b B+80 A b^2\right )}{9 a x^9}}{11 a}-\frac {\left (a+b x^2\right )^{3/2} (10 A b-13 a B)}{11 a x^{11}}}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (160 A b^3-13 a \left (33 a^2 D-22 a b C+16 b^2 B\right )\right ) \left (-\frac {4 b \left (-\frac {2 b \int \frac {\sqrt {b x^2+a}}{x^4}dx}{5 a}-\frac {\left (a+b x^2\right )^{3/2}}{5 a x^5}\right )}{7 a}-\frac {\left (a+b x^2\right )^{3/2}}{7 a x^7}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \left (143 a^2 C-104 a b B+80 A b^2\right )}{9 a x^9}}{11 a}-\frac {\left (a+b x^2\right )^{3/2} (10 A b-13 a B)}{11 a x^{11}}}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {-\frac {-\frac {\left (a+b x^2\right )^{3/2} \left (143 a^2 C-104 a b B+80 A b^2\right )}{9 a x^9}-\frac {\left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{3/2}}{15 a^2 x^3}-\frac {\left (a+b x^2\right )^{3/2}}{5 a x^5}\right )}{7 a}-\frac {\left (a+b x^2\right )^{3/2}}{7 a x^7}\right ) \left (160 A b^3-13 a \left (33 a^2 D-22 a b C+16 b^2 B\right )\right )}{3 a}}{11 a}-\frac {\left (a+b x^2\right )^{3/2} (10 A b-13 a B)}{11 a x^{11}}}{13 a}-\frac {A \left (a+b x^2\right )^{3/2}}{13 a x^{13}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4 + D*x^6))/x^14,x]
Output:
-1/13*(A*(a + b*x^2)^(3/2))/(a*x^13) - (-1/11*((10*A*b - 13*a*B)*(a + b*x^ 2)^(3/2))/(a*x^11) - (-1/9*((80*A*b^2 - 104*a*b*B + 143*a^2*C)*(a + b*x^2) ^(3/2))/(a*x^9) - ((160*A*b^3 - 13*a*(16*b^2*B - 22*a*b*C + 33*a^2*D))*(-1 /7*(a + b*x^2)^(3/2)/(a*x^7) - (4*b*(-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))/(13*a)
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 = 0.57 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.63
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {13}{7} D x^{6}+\frac {13}{9} C \,x^{4}+\frac {13}{11} x^{2} B +A \right ) a^{5}-\frac {10 \left (\frac {286}{175} D x^{6}+\frac {143}{105} C \,x^{4}+\frac {52}{45} x^{2} B +A \right ) x^{2} b \,a^{4}}{11}+\frac {80 \left (\frac {429}{350} D x^{6}+\frac {429}{350} C \,x^{4}+\frac {39}{35} x^{2} B +A \right ) x^{4} b^{2} a^{3}}{99}-\frac {160 \left (\frac {143}{150} C \,x^{4}+\frac {26}{25} x^{2} B +A \right ) x^{6} b^{3} a^{2}}{231}+\frac {128 \left (\frac {13 x^{2} B}{15}+A \right ) x^{8} b^{4} a}{231}-\frac {256 A \,b^{5} x^{10}}{693}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{13 x^{13} a^{6}}\) | \(154\) |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{10}+1664 B a \,b^{4} x^{10}-2288 C \,a^{2} b^{3} x^{10}+3432 D a^{3} b^{2} x^{10}+1920 a A \,b^{4} x^{8}-2496 B \,a^{2} b^{3} x^{8}+3432 C \,a^{3} b^{2} x^{8}-5148 D a^{4} b \,x^{8}-2400 a^{2} A \,b^{3} x^{6}+3120 B \,a^{3} b^{2} x^{6}-4290 C \,a^{4} b \,x^{6}+6435 D a^{5} x^{6}+2800 a^{3} A \,b^{2} x^{4}-3640 B \,a^{4} b \,x^{4}+5005 C \,a^{5} x^{4}-3150 a^{4} A b \,x^{2}+4095 B \,a^{5} x^{2}+3465 a^{5} A \right )}{45045 x^{13} a^{6}}\) | \(205\) |
| orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{10}+1664 B a \,b^{4} x^{10}-2288 C \,a^{2} b^{3} x^{10}+3432 D a^{3} b^{2} x^{10}+1920 a A \,b^{4} x^{8}-2496 B \,a^{2} b^{3} x^{8}+3432 C \,a^{3} b^{2} x^{8}-5148 D a^{4} b \,x^{8}-2400 a^{2} A \,b^{3} x^{6}+3120 B \,a^{3} b^{2} x^{6}-4290 C \,a^{4} b \,x^{6}+6435 D a^{5} x^{6}+2800 a^{3} A \,b^{2} x^{4}-3640 B \,a^{4} b \,x^{4}+5005 C \,a^{5} x^{4}-3150 a^{4} A b \,x^{2}+4095 B \,a^{5} x^{2}+3465 a^{5} A \right )}{45045 x^{13} a^{6}}\) | \(205\) |
| trager | \(-\frac {\left (-1280 A \,b^{6} x^{12}+1664 B a \,b^{5} x^{12}-2288 C \,a^{2} b^{4} x^{12}+3432 D a^{3} b^{3} x^{12}+640 A a \,b^{5} x^{10}-832 B \,a^{2} b^{4} x^{10}+1144 C \,a^{3} b^{3} x^{10}-1716 D a^{4} b^{2} x^{10}-480 A \,a^{2} b^{4} x^{8}+624 B \,a^{3} b^{3} x^{8}-858 C \,a^{4} b^{2} x^{8}+1287 D a^{5} b \,x^{8}+400 A \,a^{3} b^{3} x^{6}-520 B \,a^{4} b^{2} x^{6}+715 C \,a^{5} b \,x^{6}+6435 D a^{6} x^{6}-350 A \,a^{4} b^{2} x^{4}+455 B \,a^{5} b \,x^{4}+5005 C \,a^{6} x^{4}+315 A \,a^{5} b \,x^{2}+4095 B \,a^{6} x^{2}+3465 a^{6} A \right ) \sqrt {b \,x^{2}+a}}{45045 x^{13} a^{6}}\) | \(253\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{13 a \,x^{13}}-\frac {10 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{11 a \,x^{11}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{11 a \,x^{11}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 a \,x^{9}}-\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )}{3 a}\right )+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )\) | \(394\) |
Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^14,x,method=_RETURNVERBOSE)
Output:
-1/13*((13/7*D*x^6+13/9*C*x^4+13/11*x^2*B+A)*a^5-10/11*(286/175*D*x^6+143/ 105*C*x^4+52/45*x^2*B+A)*x^2*b*a^4+80/99*(429/350*D*x^6+429/350*C*x^4+39/3 5*x^2*B+A)*x^4*b^2*a^3-160/231*(143/150*C*x^4+26/25*x^2*B+A)*x^6*b^3*a^2+1 28/231*(13/15*x^2*B+A)*x^8*b^4*a-256/693*A*b^5*x^10)*(b*x^2+a)^(3/2)/x^13/ a^6
Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=-\frac {{\left (8 \, {\left (429 \, D a^{3} b^{3} - 286 \, C a^{2} b^{4} + 208 \, B a b^{5} - 160 \, A b^{6}\right )} x^{12} - 4 \, {\left (429 \, D a^{4} b^{2} - 286 \, C a^{3} b^{3} + 208 \, B a^{2} b^{4} - 160 \, A a b^{5}\right )} x^{10} + 3 \, {\left (429 \, D a^{5} b - 286 \, C a^{4} b^{2} + 208 \, B a^{3} b^{3} - 160 \, A a^{2} b^{4}\right )} x^{8} + 3465 \, A a^{6} + 5 \, {\left (1287 \, D a^{6} + 143 \, C a^{5} b - 104 \, B a^{4} b^{2} + 80 \, A a^{3} b^{3}\right )} x^{6} + 35 \, {\left (143 \, C a^{6} + 13 \, B a^{5} b - 10 \, A a^{4} b^{2}\right )} x^{4} + 315 \, {\left (13 \, B a^{6} + A a^{5} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{45045 \, a^{6} x^{13}} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^14,x, algorithm="fricas" )
Output:
-1/45045*(8*(429*D*a^3*b^3 - 286*C*a^2*b^4 + 208*B*a*b^5 - 160*A*b^6)*x^12 - 4*(429*D*a^4*b^2 - 286*C*a^3*b^3 + 208*B*a^2*b^4 - 160*A*a*b^5)*x^10 + 3*(429*D*a^5*b - 286*C*a^4*b^2 + 208*B*a^3*b^3 - 160*A*a^2*b^4)*x^8 + 3465 *A*a^6 + 5*(1287*D*a^6 + 143*C*a^5*b - 104*B*a^4*b^2 + 80*A*a^3*b^3)*x^6 + 35*(143*C*a^6 + 13*B*a^5*b - 10*A*a^4*b^2)*x^4 + 315*(13*B*a^6 + A*a^5*b) *x^2)*sqrt(b*x^2 + a)/(a^6*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 2990 vs. \(2 (243) = 486\).
Time = 4.32 (sec) , antiderivative size = 2990, normalized size of antiderivative = 12.30 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**(1/2)*(D*x**6+C*x**4+B*x**2+A)/x**14,x)
Output:
-693*A*a**11*b**(51/2)*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 4504 5*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x**18 + 45 045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 3528*A*a**10*b**(53/2)*x** 2*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 7175*A*a**9*b**(55/2)*x**4*sqrt(a/(b*x**2) + 1) /(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**1 6 + 90090*a**8*b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**2 2) - 7290*A*a**8*b**(57/2)*x**6*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x** 12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x **18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 3699*A*a**7*b**(5 9/2)*x**8*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26 *x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x**18 + 45045*a**7*b**2 9*x**20 + 9009*a**6*b**30*x**22) - 756*A*a**6*b**(61/2)*x**10*sqrt(a/(b*x* *2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b* *27*x**16 + 90090*a**8*b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b* *30*x**22) + 63*A*a**5*b**(63/2)*x**12*sqrt(a/(b*x**2) + 1)/(9009*a**11*b* *25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8* b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) + 630*A*a**4 *b**(65/2)*x**14*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a...
Time = 0.04 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=-\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D b^{2}}{105 \, a^{3} x^{3}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{3}}{315 \, a^{4} x^{3}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{4}}{3465 \, a^{5} x^{3}} + \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5}}{9009 \, a^{6} x^{3}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{2}}{105 \, a^{3} x^{5}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{1155 \, a^{4} x^{5}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{3003 \, a^{5} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b}{21 \, a^{2} x^{7}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{231 \, a^{3} x^{7}} + \frac {160 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{3003 \, a^{4} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C}{9 \, a x^{9}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{99 \, a^{2} x^{9}} - \frac {80 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{1287 \, a^{3} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{11 \, a x^{11}} + \frac {10 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{13 \, a x^{13}} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^14,x, algorithm="maxima" )
Output:
-8/105*(b*x^2 + a)^(3/2)*D*b^2/(a^3*x^3) + 16/315*(b*x^2 + a)^(3/2)*C*b^3/ (a^4*x^3) - 128/3465*(b*x^2 + a)^(3/2)*B*b^4/(a^5*x^3) + 256/9009*(b*x^2 + a)^(3/2)*A*b^5/(a^6*x^3) + 4/35*(b*x^2 + a)^(3/2)*D*b/(a^2*x^5) - 8/105*( b*x^2 + a)^(3/2)*C*b^2/(a^3*x^5) + 64/1155*(b*x^2 + a)^(3/2)*B*b^3/(a^4*x^ 5) - 128/3003*(b*x^2 + a)^(3/2)*A*b^4/(a^5*x^5) - 1/7*(b*x^2 + a)^(3/2)*D/ (a*x^7) + 2/21*(b*x^2 + a)^(3/2)*C*b/(a^2*x^7) - 16/231*(b*x^2 + a)^(3/2)* B*b^2/(a^3*x^7) + 160/3003*(b*x^2 + a)^(3/2)*A*b^3/(a^4*x^7) - 1/9*(b*x^2 + a)^(3/2)*C/(a*x^9) + 8/99*(b*x^2 + a)^(3/2)*B*b/(a^2*x^9) - 80/1287*(b*x ^2 + a)^(3/2)*A*b^2/(a^3*x^9) - 1/11*(b*x^2 + a)^(3/2)*B/(a*x^11) + 10/143 *(b*x^2 + a)^(3/2)*A*b/(a^2*x^11) - 1/13*(b*x^2 + a)^(3/2)*A/(a*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 996 vs. \(2 (219) = 438\).
Time = 0.15 (sec) , antiderivative size = 996, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^14,x, algorithm="giac")
Output:
16/45045*(30030*(sqrt(b)*x - sqrt(b*x^2 + a))^20*D*b^(7/2) - 165165*(sqrt( b)*x - sqrt(b*x^2 + a))^18*D*a*b^(7/2) + 90090*(sqrt(b)*x - sqrt(b*x^2 + a ))^18*C*b^(9/2) + 369369*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^2*b^(7/2) - 306306*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a*b^(9/2) + 288288*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*b^(11/2) - 432432*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D *a^3*b^(7/2) + 348348*(sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^2*b^(9/2) - 384 384*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a*b^(11/2) + 960960*(sqrt(b)*x - sq rt(b*x^2 + a))^14*A*b^(13/2) + 303732*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a ^4*b^(7/2) - 142428*(sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^3*b^(9/2) - 27456 *(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^2*b^(11/2) + 686400*(sqrt(b)*x - sqr t(b*x^2 + a))^12*A*a*b^(13/2) - 182754*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D* a^5*b^(7/2) + 61776*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^4*b^(9/2) + 20592 *(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^3*b^(11/2) + 205920*(sqrt(b)*x - sqr t(b*x^2 + a))^10*A*a^2*b^(13/2) + 141570*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D *a^6*b^(7/2) - 114400*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^5*b^(9/2) + 1487 20*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^4*b^(11/2) - 114400*(sqrt(b)*x - sq rt(b*x^2 + a))^8*A*a^3*b^(13/2) - 92664*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D* a^7*b^(7/2) + 81796*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^6*b^(9/2) - 59488* (sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^5*b^(11/2) + 45760*(sqrt(b)*x - sqrt(b *x^2 + a))^6*A*a^4*b^(13/2) + 33462*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{x^{14}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/x^14,x)
Output:
int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4 + x^6*D))/x^14, x)
Time = 0.18 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4+D x^6\right )}{x^{14}} \, dx=\frac {-3465 \sqrt {b \,x^{2}+a}\, a^{6}-4410 \sqrt {b \,x^{2}+a}\, a^{5} b \,x^{2}-5005 \sqrt {b \,x^{2}+a}\, a^{5} c \,x^{4}-6435 \sqrt {b \,x^{2}+a}\, a^{5} d \,x^{6}-105 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} x^{4}-715 \sqrt {b \,x^{2}+a}\, a^{4} b c \,x^{6}-1287 \sqrt {b \,x^{2}+a}\, a^{4} b d \,x^{8}+120 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{6}+858 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,x^{8}+1716 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d \,x^{10}-144 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{8}-1144 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{10}-3432 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{12}+192 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{10}+2288 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{12}-384 \sqrt {b \,x^{2}+a}\, b^{6} x^{12}+3432 \sqrt {b}\, a^{2} b^{3} d \,x^{13}-2288 \sqrt {b}\, a \,b^{4} c \,x^{13}+384 \sqrt {b}\, b^{6} x^{13}}{45045 a^{5} x^{13}} \] Input:
int((b*x^2+a)^(1/2)*(D*x^6+C*x^4+B*x^2+A)/x^14,x)
Output:
( - 3465*sqrt(a + b*x**2)*a**6 - 4410*sqrt(a + b*x**2)*a**5*b*x**2 - 5005* sqrt(a + b*x**2)*a**5*c*x**4 - 6435*sqrt(a + b*x**2)*a**5*d*x**6 - 105*sqr t(a + b*x**2)*a**4*b**2*x**4 - 715*sqrt(a + b*x**2)*a**4*b*c*x**6 - 1287*s qrt(a + b*x**2)*a**4*b*d*x**8 + 120*sqrt(a + b*x**2)*a**3*b**3*x**6 + 858* sqrt(a + b*x**2)*a**3*b**2*c*x**8 + 1716*sqrt(a + b*x**2)*a**3*b**2*d*x**1 0 - 144*sqrt(a + b*x**2)*a**2*b**4*x**8 - 1144*sqrt(a + b*x**2)*a**2*b**3* c*x**10 - 3432*sqrt(a + b*x**2)*a**2*b**3*d*x**12 + 192*sqrt(a + b*x**2)*a *b**5*x**10 + 2288*sqrt(a + b*x**2)*a*b**4*c*x**12 - 384*sqrt(a + b*x**2)* b**6*x**12 + 3432*sqrt(b)*a**2*b**3*d*x**13 - 2288*sqrt(b)*a*b**4*c*x**13 + 384*sqrt(b)*b**6*x**13)/(45045*a**5*x**13)