Integrand size = 30, antiderivative size = 248 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=-\frac {(2 a C d-b (B c+A d)) \sqrt {a+b x^2}}{8 a x^4}-\frac {b (2 a C d-b (B c+A d)) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac {(B c+A d) \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac {(4 A b c-7 a (c C+B d)) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac {2 b (4 A b c-7 a (c C+B d)) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}+\frac {b^2 (2 a C d-b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}} \] Output:
-1/8*(2*a*C*d-b*(A*d+B*c))*(b*x^2+a)^(1/2)/a/x^4-1/16*b*(2*a*C*d-b*(A*d+B* c))*(b*x^2+a)^(1/2)/a^2/x^2-1/7*A*c*(b*x^2+a)^(3/2)/a/x^7-1/6*(A*d+B*c)*(b *x^2+a)^(3/2)/a/x^6+1/35*(4*A*b*c-7*a*(B*d+C*c))*(b*x^2+a)^(3/2)/a^2/x^5-2 /105*b*(4*A*b*c-7*a*(B*d+C*c))*(b*x^2+a)^(3/2)/a^3/x^3+1/16*b^2*(2*a*C*d-b *(A*d+B*c))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 2.31 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {\frac {\sqrt {a+b x^2} \left (-128 A b^3 c x^6+a b^2 x^4 (A (64 c+105 d x)+7 x (15 B c+32 c C x+32 B d x))-2 a^2 b x^2 \left (A (24 c+35 d x)+7 x \left (5 B c+8 c C x+8 B d x+15 C d x^2\right )\right )-4 a^3 (10 A (6 c+7 d x)+7 x (3 C x (4 c+5 d x)+2 B (5 c+6 d x)))\right )}{x^7}-420 a^{3/2} b^2 C d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-210 \sqrt {a} b^3 (B c+A d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1680 a^3} \] Input:
Integrate[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^8,x]
Output:
((Sqrt[a + b*x^2]*(-128*A*b^3*c*x^6 + a*b^2*x^4*(A*(64*c + 105*d*x) + 7*x* (15*B*c + 32*c*C*x + 32*B*d*x)) - 2*a^2*b*x^2*(A*(24*c + 35*d*x) + 7*x*(5* B*c + 8*c*C*x + 8*B*d*x + 15*C*d*x^2)) - 4*a^3*(10*A*(6*c + 7*d*x) + 7*x*( 3*C*x*(4*c + 5*d*x) + 2*B*(5*c + 6*d*x)))))/x^7 - 420*a^(3/2)*b^2*C*d*ArcT anh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] - 210*Sqrt[a]*b^3*(B*c + A*d)*A rcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(1680*a^3)
Time = 1.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2338, 25, 2338, 27, 539, 25, 539, 27, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x) \left (A+B x+C x^2\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {b x^2+a} \left (7 a C d x^2-(4 A b c-7 a (c C+B d)) x+7 a (B c+A d)\right )}{x^7}dx}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (7 a C d x^2-(4 A b c-7 a (c C+B d)) x+7 a (B c+A d)\right )}{x^7}dx}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int \frac {3 a (2 (4 A b c-7 a (c C+B d))+7 (b B c+A b d-2 a C d) x) \sqrt {b x^2+a}}{x^6}dx}{6 a}-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{2} \int \frac {(2 (4 A b c-7 a (c C+B d))+7 (b B c+A b d-2 a C d) x) \sqrt {b x^2+a}}{x^6}dx-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {\int -\frac {(35 a (b B c+A b d-2 a C d)-4 b (4 A b c-7 a (c C+B d)) x) \sqrt {b x^2+a}}{x^5}dx}{5 a}+\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {\int \frac {(35 a (b B c+A b d-2 a C d)-4 b (4 A b c-7 a (c C+B d)) x) \sqrt {b x^2+a}}{x^5}dx}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {\int \frac {a b (16 (4 A b c-7 a (c C+B d))+35 (b B c+A b d-2 a C d) x) \sqrt {b x^2+a}}{x^4}dx}{4 a}-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \int \frac {(16 (4 A b c-7 a (c C+B d))+35 (b B c+A b d-2 a C d) x) \sqrt {b x^2+a}}{x^4}dx-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \left (35 (-2 a C d+A b d+b B c) \int \frac {\sqrt {b x^2+a}}{x^3}dx-\frac {16 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{3 a x^3}\right )-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \left (\frac {35}{2} (-2 a C d+A b d+b B c) \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {16 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{3 a x^3}\right )-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \left (\frac {35}{2} (-2 a C d+A b d+b B c) \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{3 a x^3}\right )-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \left (\frac {35}{2} (-2 a C d+A b d+b B c) \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {16 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{3 a x^3}\right )-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{5 a x^5}-\frac {-\frac {1}{4} b \left (\frac {35}{2} \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right ) (-2 a C d+A b d+b B c)-\frac {16 \left (a+b x^2\right )^{3/2} (4 A b c-7 a (B d+c C))}{3 a x^3}\right )-\frac {35 \left (a+b x^2\right )^{3/2} (-2 a C d+A b d+b B c)}{4 x^4}}{5 a}\right )-\frac {7 \left (a+b x^2\right )^{3/2} (A d+B c)}{6 x^6}}{7 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
Input:
Int[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^8,x]
Output:
-1/7*(A*c*(a + b*x^2)^(3/2))/(a*x^7) + ((-7*(B*c + A*d)*(a + b*x^2)^(3/2)) /(6*x^6) + ((2*(4*A*b*c - 7*a*(c*C + B*d))*(a + b*x^2)^(3/2))/(5*a*x^5) - ((-35*(b*B*c + A*b*d - 2*a*C*d)*(a + b*x^2)^(3/2))/(4*x^4) - (b*((-16*(4*A *b*c - 7*a*(c*C + B*d))*(a + b*x^2)^(3/2))/(3*a*x^3) + (35*(b*B*c + A*b*d - 2*a*C*d)*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/ Sqrt[a]))/2))/4)/(5*a))/2)/(7*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 = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (128 A \,b^{3} c \,x^{6}-224 B a \,b^{2} d \,x^{6}-224 C a \,b^{2} c \,x^{6}-105 A a \,b^{2} d \,x^{5}-105 B a \,b^{2} c \,x^{5}+210 C \,a^{2} b d \,x^{5}-64 A a \,b^{2} c \,x^{4}+112 B \,a^{2} b d \,x^{4}+112 C \,a^{2} b c \,x^{4}+70 A \,a^{2} b d \,x^{3}+70 B \,a^{2} b c \,x^{3}+420 C \,a^{3} d \,x^{3}+48 A \,a^{2} b c \,x^{2}+336 B \,a^{3} d \,x^{2}+336 C \,a^{3} c \,x^{2}+280 A \,a^{3} d x +280 B \,a^{3} c x +240 A \,a^{3} c \right )}{1680 x^{7} a^{3}}-\frac {\left (A b d +B b c -2 a C d \right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {5}{2}}}\) | \(249\) |
| default | \(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )\) | \(316\) |
Input:
int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/1680*(b*x^2+a)^(1/2)*(128*A*b^3*c*x^6-224*B*a*b^2*d*x^6-224*C*a*b^2*c*x ^6-105*A*a*b^2*d*x^5-105*B*a*b^2*c*x^5+210*C*a^2*b*d*x^5-64*A*a*b^2*c*x^4+ 112*B*a^2*b*d*x^4+112*C*a^2*b*c*x^4+70*A*a^2*b*d*x^3+70*B*a^2*b*c*x^3+420* C*a^3*d*x^3+48*A*a^2*b*c*x^2+336*B*a^3*d*x^2+336*C*a^3*c*x^2+280*A*a^3*d*x +280*B*a^3*c*x+240*A*a^3*c)/x^7/a^3-1/16*(A*b*d+B*b*c-2*C*a*d)/a^(5/2)*b^2 *ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
Time = 0.19 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.01 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\left [\frac {105 \, {\left (B b^{3} c - {\left (2 \, C a b^{2} - A b^{3}\right )} d\right )} \sqrt {a} x^{7} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (32 \, {\left (7 \, B a b^{2} d + {\left (7 \, C a b^{2} - 4 \, A b^{3}\right )} c\right )} x^{6} + 105 \, {\left (B a b^{2} c - {\left (2 \, C a^{2} b - A a b^{2}\right )} d\right )} x^{5} - 240 \, A a^{3} c - 16 \, {\left (7 \, B a^{2} b d + {\left (7 \, C a^{2} b - 4 \, A a b^{2}\right )} c\right )} x^{4} - 70 \, {\left (B a^{2} b c + {\left (6 \, C a^{3} + A a^{2} b\right )} d\right )} x^{3} - 48 \, {\left (7 \, B a^{3} d + {\left (7 \, C a^{3} + A a^{2} b\right )} c\right )} x^{2} - 280 \, {\left (B a^{3} c + A a^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{3360 \, a^{3} x^{7}}, \frac {105 \, {\left (B b^{3} c - {\left (2 \, C a b^{2} - A b^{3}\right )} d\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (32 \, {\left (7 \, B a b^{2} d + {\left (7 \, C a b^{2} - 4 \, A b^{3}\right )} c\right )} x^{6} + 105 \, {\left (B a b^{2} c - {\left (2 \, C a^{2} b - A a b^{2}\right )} d\right )} x^{5} - 240 \, A a^{3} c - 16 \, {\left (7 \, B a^{2} b d + {\left (7 \, C a^{2} b - 4 \, A a b^{2}\right )} c\right )} x^{4} - 70 \, {\left (B a^{2} b c + {\left (6 \, C a^{3} + A a^{2} b\right )} d\right )} x^{3} - 48 \, {\left (7 \, B a^{3} d + {\left (7 \, C a^{3} + A a^{2} b\right )} c\right )} x^{2} - 280 \, {\left (B a^{3} c + A a^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, a^{3} x^{7}}\right ] \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x, algorithm="fricas")
Output:
[1/3360*(105*(B*b^3*c - (2*C*a*b^2 - A*b^3)*d)*sqrt(a)*x^7*log(-(b*x^2 - 2 *sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(32*(7*B*a*b^2*d + (7*C*a*b^2 - 4 *A*b^3)*c)*x^6 + 105*(B*a*b^2*c - (2*C*a^2*b - A*a*b^2)*d)*x^5 - 240*A*a^3 *c - 16*(7*B*a^2*b*d + (7*C*a^2*b - 4*A*a*b^2)*c)*x^4 - 70*(B*a^2*b*c + (6 *C*a^3 + A*a^2*b)*d)*x^3 - 48*(7*B*a^3*d + (7*C*a^3 + A*a^2*b)*c)*x^2 - 28 0*(B*a^3*c + A*a^3*d)*x)*sqrt(b*x^2 + a))/(a^3*x^7), 1/1680*(105*(B*b^3*c - (2*C*a*b^2 - A*b^3)*d)*sqrt(-a)*x^7*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (32*(7*B*a*b^2*d + (7*C*a*b^2 - 4*A*b^3)*c)*x^6 + 105*(B*a*b^2*c - (2*C*a ^2*b - A*a*b^2)*d)*x^5 - 240*A*a^3*c - 16*(7*B*a^2*b*d + (7*C*a^2*b - 4*A* a*b^2)*c)*x^4 - 70*(B*a^2*b*c + (6*C*a^3 + A*a^2*b)*d)*x^3 - 48*(7*B*a^3*d + (7*C*a^3 + A*a^2*b)*c)*x^2 - 280*(B*a^3*c + A*a^3*d)*x)*sqrt(b*x^2 + a) )/(a^3*x^7)]
Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (230) = 460\).
Time = 12.84 (sec) , antiderivative size = 916, normalized size of antiderivative = 3.69 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**8,x)
Output:
-15*A*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*A*a**4*b**(11/2)*c*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)*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*A*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*A*a*b**(17/2)*c*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*d/(6*sqrt(b)*x**7*sqrt(a/(b *x**2) + 1)) - 8*A*b**(19/2)*c*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) - 5*A*sqrt(b)*d/(24*x**5*s qrt(a/(b*x**2) + 1)) + A*b**(3/2)*d/(48*a*x**3*sqrt(a/(b*x**2) + 1)) + A*b **(5/2)*d/(16*a**2*x*sqrt(a/(b*x**2) + 1)) - A*b**3*d*asinh(sqrt(a)/(sqrt( b)*x))/(16*a**(5/2)) - B*a*c/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 5*B*s qrt(b)*c/(24*x**5*sqrt(a/(b*x**2) + 1)) - B*sqrt(b)*d*sqrt(a/(b*x**2) + 1) /(5*x**4) + B*b**(3/2)*c/(48*a*x**3*sqrt(a/(b*x**2) + 1)) - B*b**(3/2)*d*s qrt(a/(b*x**2) + 1)/(15*a*x**2) + B*b**(5/2)*c/(16*a**2*x*sqrt(a/(b*x**2) + 1)) + 2*B*b**(5/2)*d*sqrt(a/(b*x**2) + 1)/(15*a**2) - B*b**3*c*asinh(sqr t(a)/(sqrt(b)*x))/(16*a**(5/2)) - C*a*d/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - C*sqrt(b)*c*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*C*sqrt(b)*d/(8*x**3*sq rt(a/(b*x**2) + 1)) - C*b**(3/2)*c*sqrt(a/(b*x**2) + 1)/(15*a*x**2) - C...
Time = 0.05 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {C b^{2} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} C b^{2} d}{8 \, a^{2}} - \frac {{\left (B c + A d\right )} b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )} b^{3}}{16 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b d}{8 \, a^{2} x^{2}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} c}{105 \, a^{3} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C d}{4 \, a x^{4}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} b}{15 \, a^{2} x^{3}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b c}{35 \, a^{2} x^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b}{8 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )}}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c}{7 \, a x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )}}{6 \, a x^{6}} \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x, algorithm="maxima")
Output:
1/8*C*b^2*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/8*sqrt(b*x^2 + a)*C* b^2*d/a^2 - 1/16*(B*c + A*d)*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1 /16*sqrt(b*x^2 + a)*(B*c + A*d)*b^3/a^3 + 1/8*(b*x^2 + a)^(3/2)*C*b*d/(a^2 *x^2) - 8/105*(b*x^2 + a)^(3/2)*A*b^2*c/(a^3*x^3) - 1/16*(b*x^2 + a)^(3/2) *(B*c + A*d)*b^2/(a^3*x^2) - 1/4*(b*x^2 + a)^(3/2)*C*d/(a*x^4) + 2/15*(b*x ^2 + a)^(3/2)*(C*c + B*d)*b/(a^2*x^3) + 4/35*(b*x^2 + a)^(3/2)*A*b*c/(a^2* x^5) + 1/8*(b*x^2 + a)^(3/2)*(B*c + A*d)*b/(a^2*x^4) - 1/5*(b*x^2 + a)^(3/ 2)*(C*c + B*d)/(a*x^5) - 1/7*(b*x^2 + a)^(3/2)*A*c/(a*x^7) - 1/6*(b*x^2 + a)^(3/2)*(B*c + A*d)/(a*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (216) = 432\).
Time = 0.23 (sec) , antiderivative size = 1028, normalized size of antiderivative = 4.15 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x, algorithm="giac")
Output:
1/8*(B*b^3*c - 2*C*a*b^2*d + A*b^3*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a) )/sqrt(-a))/(sqrt(-a)*a^2) - 1/840*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^13*B *b^3*c - 210*(sqrt(b)*x - sqrt(b*x^2 + a))^13*C*a*b^2*d + 105*(sqrt(b)*x - sqrt(b*x^2 + a))^13*A*b^3*d - 700*(sqrt(b)*x - sqrt(b*x^2 + a))^11*B*a*b^ 3*c - 840*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a^2*b^2*d - 700*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*a*b^3*d - 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^ 2*b^(5/2)*c - 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2)*d - 3395 *(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a^2*b^3*c + 2310*(sqrt(b)*x - sqrt(b*x^ 2 + a))^9*C*a^3*b^2*d - 3395*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a^2*b^3*d + 5600*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^3*b^(5/2)*c - 8960*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2)*c + 5600*(sqrt(b)*x - sqrt(b*x^2 + a))^8* B*a^3*b^(5/2)*d - 2240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^4*b^(5/2)*c - 4 480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^3*b^(7/2)*c - 2240*(sqrt(b)*x - sq rt(b*x^2 + a))^6*B*a^4*b^(5/2)*d + 3395*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B* a^4*b^3*c - 2310*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^5*b^2*d + 3395*(sqrt( b)*x - sqrt(b*x^2 + a))^5*A*a^4*b^3*d + 1344*(sqrt(b)*x - sqrt(b*x^2 + a)) ^4*C*a^5*b^(5/2)*c - 2688*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^4*b^(7/2)*c + 1344*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2)*d + 700*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a^5*b^3*c + 840*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*a^6 *b^2*d + 700*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^5*b^3*d - 1568*(sqrt(b...
Timed out. \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^8} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^8,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^8, x)
Time = 0.23 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.34 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {-48 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{2}-280 \sqrt {b \,x^{2}+a}\, a^{3} b c x -70 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{3}-280 \sqrt {b \,x^{2}+a}\, a^{4} d x -336 \sqrt {b \,x^{2}+a}\, a^{3} c^{2} x^{2}-336 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{2}-70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{3}+105 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{5}-420 \sqrt {b \,x^{2}+a}\, a^{3} c d \,x^{3}+64 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{4}+105 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{5}-112 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{4}-112 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} x^{4}-128 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{6}+224 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{6}+224 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{6}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{7}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{7}+128 \sqrt {b}\, a \,b^{3} c \,x^{7}-224 \sqrt {b}\, a \,b^{3} d \,x^{7}-224 \sqrt {b}\, a \,b^{2} c^{2} x^{7}-240 \sqrt {b \,x^{2}+a}\, a^{4} c -210 \sqrt {b \,x^{2}+a}\, a^{2} b c d \,x^{5}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,x^{7}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,x^{7}-210 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c d \,x^{7}+210 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c d \,x^{7}}{1680 a^{3} x^{7}} \] Input:
int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x)
Output:
( - 240*sqrt(a + b*x**2)*a**4*c - 280*sqrt(a + b*x**2)*a**4*d*x - 48*sqrt( a + b*x**2)*a**3*b*c*x**2 - 280*sqrt(a + b*x**2)*a**3*b*c*x - 70*sqrt(a + b*x**2)*a**3*b*d*x**3 - 336*sqrt(a + b*x**2)*a**3*b*d*x**2 - 336*sqrt(a + b*x**2)*a**3*c**2*x**2 - 420*sqrt(a + b*x**2)*a**3*c*d*x**3 + 64*sqrt(a + b*x**2)*a**2*b**2*c*x**4 - 70*sqrt(a + b*x**2)*a**2*b**2*c*x**3 + 105*sqrt (a + b*x**2)*a**2*b**2*d*x**5 - 112*sqrt(a + b*x**2)*a**2*b**2*d*x**4 - 11 2*sqrt(a + b*x**2)*a**2*b*c**2*x**4 - 210*sqrt(a + b*x**2)*a**2*b*c*d*x**5 - 128*sqrt(a + b*x**2)*a*b**3*c*x**6 + 105*sqrt(a + b*x**2)*a*b**3*c*x**5 + 224*sqrt(a + b*x**2)*a*b**3*d*x**6 + 224*sqrt(a + b*x**2)*a*b**2*c**2*x **6 + 105*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a* b**3*d*x**7 - 210*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqr t(a))*a*b**2*c*d*x**7 + 105*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt (b)*x)/sqrt(a))*b**4*c*x**7 - 105*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*d*x**7 + 210*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*c*d*x**7 - 105*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*c*x**7 + 128*sqrt(b)*a*b**3*c *x**7 - 224*sqrt(b)*a*b**3*d*x**7 - 224*sqrt(b)*a*b**2*c**2*x**7)/(1680*a* *3*x**7)