Integrand size = 30, antiderivative size = 288 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\frac {(5 A b c-8 a (c C+B d)) \sqrt {a+b x^2}}{48 a x^6}+\frac {b (5 A b c-8 a (c C+B d)) \sqrt {a+b x^2}}{192 a^2 x^4}-\frac {b^2 (5 A b c-8 a (c C+B d)) \sqrt {a+b x^2}}{128 a^3 x^2}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}-\frac {(B c+A d) \left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac {(7 a C d-4 b (B c+A d)) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}+\frac {2 b (7 a C d-4 b (B c+A d)) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}+\frac {b^3 (5 A b c-8 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{7/2}} \] Output:
1/48*(5*A*b*c-8*a*(B*d+C*c))*(b*x^2+a)^(1/2)/a/x^6+1/192*b*(5*A*b*c-8*a*(B *d+C*c))*(b*x^2+a)^(1/2)/a^2/x^4-1/128*b^2*(5*A*b*c-8*a*(B*d+C*c))*(b*x^2+ a)^(1/2)/a^3/x^2-1/8*A*c*(b*x^2+a)^(3/2)/a/x^8-1/7*(A*d+B*c)*(b*x^2+a)^(3/ 2)/a/x^7-1/35*(7*a*C*d-4*b*(A*d+B*c))*(b*x^2+a)^(3/2)/a^2/x^5+2/105*b*(7*a *C*d-4*b*(A*d+B*c))*(b*x^2+a)^(3/2)/a^3/x^3+1/128*b^3*(5*A*b*c-8*a*(B*d+C* c))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(7/2)
Time = 2.71 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^2} \left (b^3 x^6 (525 A c+1024 B c x+1024 A d x)+16 a^3 (15 A (7 c+8 d x)+4 x (7 C x (5 c+6 d x)+5 B (6 c+7 d x)))+8 a^2 b x^2 (A (35 c+48 d x)+2 x (7 C x (5 c+8 d x)+B (24 c+35 d x)))-2 a b^2 x^4 (A (175 c+256 d x)+4 x (7 C x (15 c+32 d x)+B (64 c+105 d x)))\right )}{x^8}-1050 A b^4 c \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-1680 a b^3 (c C+B d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{13440 a^{7/2}} \] Input:
Integrate[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^9,x]
Output:
(-((Sqrt[a]*Sqrt[a + b*x^2]*(b^3*x^6*(525*A*c + 1024*B*c*x + 1024*A*d*x) + 16*a^3*(15*A*(7*c + 8*d*x) + 4*x*(7*C*x*(5*c + 6*d*x) + 5*B*(6*c + 7*d*x) )) + 8*a^2*b*x^2*(A*(35*c + 48*d*x) + 2*x*(7*C*x*(5*c + 8*d*x) + B*(24*c + 35*d*x))) - 2*a*b^2*x^4*(A*(175*c + 256*d*x) + 4*x*(7*C*x*(15*c + 32*d*x) + B*(64*c + 105*d*x)))))/x^8) - 1050*A*b^4*c*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] - 1680*a*b^3*(c*C + B*d)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(13440*a^(7/2))
Time = 1.25 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2338, 25, 2338, 27, 539, 27, 539, 25, 27, 539, 534, 243, 51, 73, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x) \left (A+B x+C x^2\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {b x^2+a} \left (8 a C d x^2-(5 A b c-8 a (c C+B d)) x+8 a (B c+A d)\right )}{x^8}dx}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (8 a C d x^2-(5 A b c-8 a (c C+B d)) x+8 a (B c+A d)\right )}{x^8}dx}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int \frac {a (7 (5 A b c-8 a (c C+B d))-8 (7 a C d-4 b (B c+A d)) x) \sqrt {b x^2+a}}{x^7}dx}{7 a}-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{7} \int \frac {(7 (5 A b c-8 a (c C+B d))-8 (7 a C d-4 b (B c+A d)) x) \sqrt {b x^2+a}}{x^7}dx-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\int \frac {3 (16 a (7 a C d-4 b (B c+A d))+7 b (5 A b c-8 a (c C+B d)) x) \sqrt {b x^2+a}}{x^6}dx}{6 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\int \frac {(16 a (7 a C d-4 b (B c+A d))+7 b (5 A b c-8 a (c C+B d)) x) \sqrt {b x^2+a}}{x^6}dx}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {-\frac {\int -\frac {a b (35 (5 A b c-8 a (c C+B d))-32 (7 a C d-4 b (B c+A d)) x) \sqrt {b x^2+a}}{x^5}dx}{5 a}-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {\int \frac {a b (35 (5 A b c-8 a (c C+B d))-32 (7 a C d-4 b (B c+A d)) x) \sqrt {b x^2+a}}{x^5}dx}{5 a}-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \int \frac {(35 (5 A b c-8 a (c C+B d))-32 (7 a C d-4 b (B c+A d)) x) \sqrt {b x^2+a}}{x^5}dx-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {\int \frac {(128 a (7 a C d-4 b (B c+A d))+35 b (5 A b c-8 a (c C+B d)) x) \sqrt {b x^2+a}}{x^4}dx}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {35 b (5 A b c-8 a (B d+c C)) \int \frac {\sqrt {b x^2+a}}{x^3}dx-\frac {128 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{3 x^3}}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {\frac {35}{2} b (5 A b c-8 a (B d+c C)) \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {128 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{3 x^3}}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {\frac {35}{2} b (5 A b c-8 a (B d+c C)) \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {128 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{3 x^3}}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {\frac {35}{2} b (5 A b c-8 a (B d+c C)) \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {128 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{3 x^3}}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\frac {1}{5} b \left (-\frac {\frac {35}{2} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right ) (5 A b c-8 a (B d+c C))-\frac {128 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{3 x^3}}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{4 a x^4}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (7 a C d-4 b (A d+B c))}{5 x^5}}{2 a}+\frac {7 \left (a+b x^2\right )^{3/2} (5 A b c-8 a (B d+c C))}{6 a x^6}\right )-\frac {8 \left (a+b x^2\right )^{3/2} (A d+B c)}{7 x^7}}{8 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{8 a x^8}\) |
Input:
Int[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^9,x]
Output:
-1/8*(A*c*(a + b*x^2)^(3/2))/(a*x^8) + ((-8*(B*c + A*d)*(a + b*x^2)^(3/2)) /(7*x^7) + ((7*(5*A*b*c - 8*a*(c*C + B*d))*(a + b*x^2)^(3/2))/(6*a*x^6) + ((-16*(7*a*C*d - 4*b*(B*c + A*d))*(a + b*x^2)^(3/2))/(5*x^5) + (b*((-35*(5 *A*b*c - 8*a*(c*C + B*d))*(a + b*x^2)^(3/2))/(4*a*x^4) - ((-128*(7*a*C*d - 4*b*(B*c + A*d))*(a + b*x^2)^(3/2))/(3*x^3) + (35*b*(5*A*b*c - 8*a*(c*C + B*d))*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt [a]))/2)/(4*a)))/5)/(2*a))/7)/(8*a)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.26 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (1024 A \,b^{3} d \,x^{7}+1024 B \,b^{3} c \,x^{7}-1792 C a \,b^{2} d \,x^{7}+525 A \,b^{3} c \,x^{6}-840 B a \,b^{2} d \,x^{6}-840 C a \,b^{2} c \,x^{6}-512 A a \,b^{2} d \,x^{5}-512 B a \,b^{2} c \,x^{5}+896 C \,a^{2} b d \,x^{5}-350 A a \,b^{2} c \,x^{4}+560 B \,a^{2} b d \,x^{4}+560 C \,a^{2} b c \,x^{4}+384 A \,a^{2} b d \,x^{3}+384 B \,a^{2} b c \,x^{3}+2688 C \,a^{3} d \,x^{3}+280 A \,a^{2} b c \,x^{2}+2240 B \,a^{3} d \,x^{2}+2240 C \,a^{3} c \,x^{2}+1920 A \,a^{3} d x +1920 B \,a^{3} c x +1680 A \,a^{3} c \right )}{13440 x^{8} a^{3}}+\frac {\left (5 A b c -8 B a d -8 C a c \right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {7}{2}}}\) | \(282\) |
| default | \(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}-\frac {5 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )}{8 a}\right )+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )\) | \(364\) |
Input:
int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^9,x,method=_RETURNVERBOSE)
Output:
-1/13440*(b*x^2+a)^(1/2)*(1024*A*b^3*d*x^7+1024*B*b^3*c*x^7-1792*C*a*b^2*d *x^7+525*A*b^3*c*x^6-840*B*a*b^2*d*x^6-840*C*a*b^2*c*x^6-512*A*a*b^2*d*x^5 -512*B*a*b^2*c*x^5+896*C*a^2*b*d*x^5-350*A*a*b^2*c*x^4+560*B*a^2*b*d*x^4+5 60*C*a^2*b*c*x^4+384*A*a^2*b*d*x^3+384*B*a^2*b*c*x^3+2688*C*a^3*d*x^3+280* A*a^2*b*c*x^2+2240*B*a^3*d*x^2+2240*C*a^3*c*x^2+1920*A*a^3*d*x+1920*B*a^3* c*x+1680*A*a^3*c)/x^8/a^3+1/128*(5*A*b*c-8*B*a*d-8*C*a*c)*b^3/a^(7/2)*ln(( 2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
Time = 0.29 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.06 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\left [\frac {105 \, {\left (8 \, B a b^{3} d + {\left (8 \, C a b^{3} - 5 \, A b^{4}\right )} c\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (256 \, {\left (4 \, B a b^{3} c - {\left (7 \, C a^{2} b^{2} - 4 \, A a b^{3}\right )} d\right )} x^{7} - 105 \, {\left (8 \, B a^{2} b^{2} d + {\left (8 \, C a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} x^{6} + 1680 \, A a^{4} c - 128 \, {\left (4 \, B a^{2} b^{2} c - {\left (7 \, C a^{3} b - 4 \, A a^{2} b^{2}\right )} d\right )} x^{5} + 70 \, {\left (8 \, B a^{3} b d + {\left (8 \, C a^{3} b - 5 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 384 \, {\left (B a^{3} b c + {\left (7 \, C a^{4} + A a^{3} b\right )} d\right )} x^{3} + 280 \, {\left (8 \, B a^{4} d + {\left (8 \, C a^{4} + A a^{3} b\right )} c\right )} x^{2} + 1920 \, {\left (B a^{4} c + A a^{4} d\right )} x\right )} \sqrt {b x^{2} + a}}{26880 \, a^{4} x^{8}}, \frac {105 \, {\left (8 \, B a b^{3} d + {\left (8 \, C a b^{3} - 5 \, A b^{4}\right )} c\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (256 \, {\left (4 \, B a b^{3} c - {\left (7 \, C a^{2} b^{2} - 4 \, A a b^{3}\right )} d\right )} x^{7} - 105 \, {\left (8 \, B a^{2} b^{2} d + {\left (8 \, C a^{2} b^{2} - 5 \, A a b^{3}\right )} c\right )} x^{6} + 1680 \, A a^{4} c - 128 \, {\left (4 \, B a^{2} b^{2} c - {\left (7 \, C a^{3} b - 4 \, A a^{2} b^{2}\right )} d\right )} x^{5} + 70 \, {\left (8 \, B a^{3} b d + {\left (8 \, C a^{3} b - 5 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 384 \, {\left (B a^{3} b c + {\left (7 \, C a^{4} + A a^{3} b\right )} d\right )} x^{3} + 280 \, {\left (8 \, B a^{4} d + {\left (8 \, C a^{4} + A a^{3} b\right )} c\right )} x^{2} + 1920 \, {\left (B a^{4} c + A a^{4} d\right )} x\right )} \sqrt {b x^{2} + a}}{13440 \, a^{4} x^{8}}\right ] \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^9,x, algorithm="fricas")
Output:
[1/26880*(105*(8*B*a*b^3*d + (8*C*a*b^3 - 5*A*b^4)*c)*sqrt(a)*x^8*log(-(b* x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(256*(4*B*a*b^3*c - (7*C*a ^2*b^2 - 4*A*a*b^3)*d)*x^7 - 105*(8*B*a^2*b^2*d + (8*C*a^2*b^2 - 5*A*a*b^3 )*c)*x^6 + 1680*A*a^4*c - 128*(4*B*a^2*b^2*c - (7*C*a^3*b - 4*A*a^2*b^2)*d )*x^5 + 70*(8*B*a^3*b*d + (8*C*a^3*b - 5*A*a^2*b^2)*c)*x^4 + 384*(B*a^3*b* c + (7*C*a^4 + A*a^3*b)*d)*x^3 + 280*(8*B*a^4*d + (8*C*a^4 + A*a^3*b)*c)*x ^2 + 1920*(B*a^4*c + A*a^4*d)*x)*sqrt(b*x^2 + a))/(a^4*x^8), 1/13440*(105* (8*B*a*b^3*d + (8*C*a*b^3 - 5*A*b^4)*c)*sqrt(-a)*x^8*arctan(sqrt(b*x^2 + a )*sqrt(-a)/a) - (256*(4*B*a*b^3*c - (7*C*a^2*b^2 - 4*A*a*b^3)*d)*x^7 - 105 *(8*B*a^2*b^2*d + (8*C*a^2*b^2 - 5*A*a*b^3)*c)*x^6 + 1680*A*a^4*c - 128*(4 *B*a^2*b^2*c - (7*C*a^3*b - 4*A*a^2*b^2)*d)*x^5 + 70*(8*B*a^3*b*d + (8*C*a ^3*b - 5*A*a^2*b^2)*c)*x^4 + 384*(B*a^3*b*c + (7*C*a^4 + A*a^3*b)*d)*x^3 + 280*(8*B*a^4*d + (8*C*a^4 + A*a^3*b)*c)*x^2 + 1920*(B*a^4*c + A*a^4*d)*x) *sqrt(b*x^2 + a))/(a^4*x^8)]
Leaf count of result is larger than twice the leaf count of optimal. 1278 vs. \(2 (274) = 548\).
Time = 29.04 (sec) , antiderivative size = 1278, normalized size of antiderivative = 4.44 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**9,x)
Output:
-15*A*a**5*b**(9/2)*d*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4* b**5*x**8 + 105*a**3*b**6*x**10) - 33*A*a**4*b**(11/2)*d*x**2*sqrt(a/(b*x* *2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 17*A*a**3*b**(13/2)*d*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210 *a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 3*A*a**2*b**(15/2)*d*x**6*sqrt(a/ (b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x** 10) - 12*A*a*b**(17/2)*d*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 2 10*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - A*a*c/(8*sqrt(b)*x**9*sqrt(a/(b *x**2) + 1)) - 8*A*b**(19/2)*d*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x **6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 7*A*sqrt(b)*c/(48*x**7*s qrt(a/(b*x**2) + 1)) + A*b**(3/2)*c/(192*a*x**5*sqrt(a/(b*x**2) + 1)) - 5* A*b**(5/2)*c/(384*a**2*x**3*sqrt(a/(b*x**2) + 1)) - 5*A*b**(7/2)*c/(128*a* *3*x*sqrt(a/(b*x**2) + 1)) + 5*A*b**4*c*asinh(sqrt(a)/(sqrt(b)*x))/(128*a* *(7/2)) - 15*B*a**5*b**(9/2)*c*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 33*B*a**4*b**(11/2)*c*x**2*sqr t(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6 *x**10) - 17*B*a**3*b**(13/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x **6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 3*B*a**2*b**(15/2)*c*x** 6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3 *b**6*x**10) - 12*B*a*b**(17/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b...
Time = 0.04 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\frac {5 \, A b^{4} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {7}{2}}} - \frac {5 \, \sqrt {b x^{2} + a} A b^{4} c}{128 \, a^{4}} - \frac {{\left (C c + B d\right )} b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )} b^{3}}{16 \, a^{3}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3} c}{128 \, a^{4} x^{2}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C b d}{15 \, a^{2} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} b^{2}}{16 \, a^{3} x^{2}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} c}{64 \, a^{3} x^{4}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b^{2}}{105 \, a^{3} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C d}{5 \, a x^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} b}{8 \, a^{2} x^{4}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b c}{48 \, a^{2} x^{6}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )}}{6 \, a x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c}{8 \, a x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )}}{7 \, a x^{7}} \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^9,x, algorithm="maxima")
Output:
5/128*A*b^4*c*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) - 5/128*sqrt(b*x^2 + a )*A*b^4*c/a^4 - 1/16*(C*c + B*d)*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/16*sqrt(b*x^2 + a)*(C*c + B*d)*b^3/a^3 + 5/128*(b*x^2 + a)^(3/2)*A*b^ 3*c/(a^4*x^2) + 2/15*(b*x^2 + a)^(3/2)*C*b*d/(a^2*x^3) - 1/16*(b*x^2 + a)^ (3/2)*(C*c + B*d)*b^2/(a^3*x^2) - 5/64*(b*x^2 + a)^(3/2)*A*b^2*c/(a^3*x^4) - 8/105*(b*x^2 + a)^(3/2)*(B*c + A*d)*b^2/(a^3*x^3) - 1/5*(b*x^2 + a)^(3/ 2)*C*d/(a*x^5) + 1/8*(b*x^2 + a)^(3/2)*(C*c + B*d)*b/(a^2*x^4) + 5/48*(b*x ^2 + a)^(3/2)*A*b*c/(a^2*x^6) + 4/35*(b*x^2 + a)^(3/2)*(B*c + A*d)*b/(a^2* x^5) - 1/6*(b*x^2 + a)^(3/2)*(C*c + B*d)/(a*x^6) - 1/8*(b*x^2 + a)^(3/2)*A *c/(a*x^8) - 1/7*(b*x^2 + a)^(3/2)*(B*c + A*d)/(a*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 1266 vs. \(2 (252) = 504\).
Time = 0.21 (sec) , antiderivative size = 1266, normalized size of antiderivative = 4.40 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^9,x, algorithm="giac")
Output:
1/64*(8*C*a*b^3*c - 5*A*b^4*c + 8*B*a*b^3*d)*arctan(-(sqrt(b)*x - sqrt(b*x ^2 + a))/sqrt(-a))/(sqrt(-a)*a^3) - 1/6720*(840*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a*b^3*c - 525*(sqrt(b)*x - sqrt(b*x^2 + a))^15*A*b^4*c + 840*(sqr t(b)*x - sqrt(b*x^2 + a))^15*B*a*b^3*d - 6440*(sqrt(b)*x - sqrt(b*x^2 + a) )^13*C*a^2*b^3*c + 4025*(sqrt(b)*x - sqrt(b*x^2 + a))^13*A*a*b^4*c - 6440* (sqrt(b)*x - sqrt(b*x^2 + a))^13*B*a^2*b^3*d - 26880*(sqrt(b)*x - sqrt(b*x ^2 + a))^12*C*a^3*b^(5/2)*d - 21560*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a^3 *b^3*c - 13405*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*a^2*b^4*c - 21560*(sqrt( b)*x - sqrt(b*x^2 + a))^11*B*a^3*b^3*d - 71680*(sqrt(b)*x - sqrt(b*x^2 + a ))^10*B*a^3*b^(7/2)*c + 71680*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^4*b^(5/ 2)*d - 71680*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(7/2)*d + 27160*(sqr t(b)*x - sqrt(b*x^2 + a))^9*C*a^4*b^3*c - 97615*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a^3*b^4*c + 27160*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a^4*b^3*d + 35 840*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^4*b^(7/2)*c - 62720*(sqrt(b)*x - s qrt(b*x^2 + a))^8*C*a^5*b^(5/2)*d + 35840*(sqrt(b)*x - sqrt(b*x^2 + a))^8* A*a^4*b^(7/2)*d + 27160*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^5*b^3*c - 9761 5*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^4*b^4*c + 27160*(sqrt(b)*x - sqrt(b* x^2 + a))^7*B*a^5*b^3*d + 14336*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^5*b^(7 /2)*c + 28672*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^6*b^(5/2)*d + 14336*(sqr t(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(7/2)*d - 21560*(sqrt(b)*x - sqrt(b...
Timed out. \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^9} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^9,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^9, x)
Time = 0.27 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.19 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^9} \, dx=\frac {1024 \sqrt {b}\, b^{4} c \,x^{8}+840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{8}+525 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{8}-840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} d \,x^{8}-840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{8}+1024 \sqrt {b}\, a \,b^{3} d \,x^{8}-280 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{2}-1920 \sqrt {b \,x^{2}+a}\, a^{3} b c x -384 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{3}-1920 \sqrt {b \,x^{2}+a}\, a^{4} d x -2240 \sqrt {b \,x^{2}+a}\, a^{3} c^{2} x^{2}-2240 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{2}-384 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{3}+512 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{5}+1792 \sqrt {b \,x^{2}+a}\, a \,b^{2} c d \,x^{7}-1792 \sqrt {b}\, a \,b^{2} c d \,x^{8}-1024 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{7}-525 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{8}+840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} d \,x^{8}-2688 \sqrt {b \,x^{2}+a}\, a^{3} c d \,x^{3}+350 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{4}+512 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{5}-560 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{4}-560 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} x^{4}-525 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{6}+840 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{6}+840 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{6}-1024 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{7}-1680 \sqrt {b \,x^{2}+a}\, a^{4} c -896 \sqrt {b \,x^{2}+a}\, a^{2} b c d \,x^{5}}{13440 a^{3} x^{8}} \] Input:
int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^9,x)
Output:
( - 1680*sqrt(a + b*x**2)*a**4*c - 1920*sqrt(a + b*x**2)*a**4*d*x - 280*sq rt(a + b*x**2)*a**3*b*c*x**2 - 1920*sqrt(a + b*x**2)*a**3*b*c*x - 384*sqrt (a + b*x**2)*a**3*b*d*x**3 - 2240*sqrt(a + b*x**2)*a**3*b*d*x**2 - 2240*sq rt(a + b*x**2)*a**3*c**2*x**2 - 2688*sqrt(a + b*x**2)*a**3*c*d*x**3 + 350* sqrt(a + b*x**2)*a**2*b**2*c*x**4 - 384*sqrt(a + b*x**2)*a**2*b**2*c*x**3 + 512*sqrt(a + b*x**2)*a**2*b**2*d*x**5 - 560*sqrt(a + b*x**2)*a**2*b**2*d *x**4 - 560*sqrt(a + b*x**2)*a**2*b*c**2*x**4 - 896*sqrt(a + b*x**2)*a**2* b*c*d*x**5 - 525*sqrt(a + b*x**2)*a*b**3*c*x**6 + 512*sqrt(a + b*x**2)*a*b **3*c*x**5 - 1024*sqrt(a + b*x**2)*a*b**3*d*x**7 + 840*sqrt(a + b*x**2)*a* b**3*d*x**6 + 840*sqrt(a + b*x**2)*a*b**2*c**2*x**6 + 1792*sqrt(a + b*x**2 )*a*b**2*c*d*x**7 - 1024*sqrt(a + b*x**2)*b**4*c*x**7 - 525*sqrt(a)*log((s qrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*c*x**8 + 840*sqrt(a)* log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*d*x**8 + 840*sq rt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c**2*x**8 + 525*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4* c*x**8 - 840*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a)) *b**4*d*x**8 - 840*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sq rt(a))*b**3*c**2*x**8 + 1024*sqrt(b)*a*b**3*d*x**8 - 1792*sqrt(b)*a*b**2*c *d*x**8 + 1024*sqrt(b)*b**4*c*x**8)/(13440*a**3*x**8)