Integrand size = 32, antiderivative size = 313 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {(b c (B c+2 A d)-2 a d (2 c C+B d)) \sqrt {a+b x^2}}{8 a x^4}+\frac {b (b c (B c+2 A d)-2 a d (2 c C+B d)) \sqrt {a+b x^2}}{16 a^2 x^2}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac {\left (7 a c (c C+2 B d)-A \left (4 b c^2-7 a d^2\right )\right ) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac {\left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (c C+2 B d)\right )\right ) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}-\frac {b^2 (b c (B c+2 A d)-2 a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{5/2}} \] Output:
1/8*(b*c*(2*A*d+B*c)-2*a*d*(B*d+2*C*c))*(b*x^2+a)^(1/2)/a/x^4+1/16*b*(b*c* (2*A*d+B*c)-2*a*d*(B*d+2*C*c))*(b*x^2+a)^(1/2)/a^2/x^2-1/7*A*c^2*(b*x^2+a) ^(3/2)/a/x^7-1/6*c*(2*A*d+B*c)*(b*x^2+a)^(3/2)/a/x^6-1/35*(7*a*c*(2*B*d+C* c)-A*(-7*a*d^2+4*b*c^2))*(b*x^2+a)^(3/2)/a^2/x^5-1/105*(2*A*b*(-7*a*d^2+4* b*c^2)+7*a*(5*a*C*d^2-2*b*c*(2*B*d+C*c)))*(b*x^2+a)^(3/2)/a^3/x^3-1/16*b^2 *(b*c*(2*A*d+B*c)-2*a*d*(B*d+2*C*c))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5 /2)
Time = 2.96 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {\sqrt {a+b x^2} \left (-128 A b^3 c^2 x^6+a b^2 x^4 \left (7 c x (15 B c+32 c C x+64 B d x)+A \left (64 c^2+210 c d x+224 d^2 x^2\right )\right )-4 a^3 \left (4 A \left (15 c^2+35 c d x+21 d^2 x^2\right )+7 x \left (2 C x \left (6 c^2+15 c d x+10 d^2 x^2\right )+B \left (10 c^2+24 c d x+15 d^2 x^2\right )\right )\right )-2 a^2 b x^2 \left (A \left (24 c^2+70 c d x+56 d^2 x^2\right )+7 x \left (B \left (5 c^2+16 c d x+15 d^2 x^2\right )+2 C x \left (4 c^2+15 c d x+20 d^2 x^2\right )\right )\right )\right )}{1680 a^3 x^7}+\frac {b^2 (-b c (B c+2 A d)+2 a d (2 c C+B d)) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Input:
Integrate[((c + d*x)^2*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^2*x^6 + a*b^2*x^4*(7*c*x*(15*B*c + 32*c*C*x + 64*B*d*x) + A*(64*c^2 + 210*c*d*x + 224*d^2*x^2)) - 4*a^3*(4*A*(15*c^2 + 35*c*d*x + 21*d^2*x^2) + 7*x*(2*C*x*(6*c^2 + 15*c*d*x + 10*d^2*x^2) + B* (10*c^2 + 24*c*d*x + 15*d^2*x^2))) - 2*a^2*b*x^2*(A*(24*c^2 + 70*c*d*x + 5 6*d^2*x^2) + 7*x*(B*(5*c^2 + 16*c*d*x + 15*d^2*x^2) + 2*C*x*(4*c^2 + 15*c* d*x + 20*d^2*x^2)))))/(1680*a^3*x^7) + (b^2*(-(b*c*(B*c + 2*A*d)) + 2*a*d* (2*c*C + B*d))*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(8*a^(5/ 2))
Time = 1.76 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2338, 25, 2338, 27, 2338, 27, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x)^2 \left (A+B x+C x^2\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {b x^2+a} \left (7 a C d^2 x^3+7 a d (2 c C+B d) x^2+\left (7 a c (c C+2 B d)-A \left (4 b c^2-7 a d^2\right )\right ) x+7 a c (B c+2 A d)\right )}{x^7}dx}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (7 a C d^2 x^3+7 a d (2 c C+B d) x^2+\left (7 a c (c C+2 B d)-A \left (4 b c^2-7 a d^2\right )\right ) x+7 a c (B c+2 A d)\right )}{x^7}dx}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 \sqrt {b x^2+a} \left (14 a^2 C d^2 x^2-7 a (b c (B c+2 A d)-2 a d (2 c C+B d)) x+2 a \left (7 a c (c C+2 B d)-A \left (4 b c^2-7 a d^2\right )\right )\right )}{x^6}dx}{6 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (14 a^2 C d^2 x^2-7 a (b c (B c+2 A d)-2 a d (2 c C+B d)) x+2 a \left (7 a c (c C+2 B d)-A \left (4 b c^2-7 a d^2\right )\right )\right )}{x^6}dx}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (35 a (b c (B c+2 A d)-2 a d (2 c C+B d))-2 \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (c C+2 B d)\right )\right ) x\right ) \sqrt {b x^2+a}}{x^5}dx}{5 a}-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {1}{5} \int \frac {\left (35 a (b c (B c+2 A d)-2 a d (2 c C+B d))-2 \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (c C+2 B d)\right )\right ) x\right ) \sqrt {b x^2+a}}{x^5}dx-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {\int \frac {a \left (8 \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (c C+2 B d)\right )\right )+35 b (b c (B c+2 A d)-2 a d (2 c C+B d)) x\right ) \sqrt {b x^2+a}}{x^4}dx}{4 a}+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \int \frac {\left (8 \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (c C+2 B d)\right )\right )+35 b (b c (B c+2 A d)-2 a d (2 c C+B d)) x\right ) \sqrt {b x^2+a}}{x^4}dx+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \left (35 b (b c (2 A d+B c)-2 a d (B d+2 c C)) \int \frac {\sqrt {b x^2+a}}{x^3}dx-\frac {8 \left (a+b x^2\right )^{3/2} \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (2 B d+c C)\right )\right )}{3 a x^3}\right )+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {35}{2} b (b c (2 A d+B c)-2 a d (B d+2 c C)) \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {8 \left (a+b x^2\right )^{3/2} \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (2 B d+c C)\right )\right )}{3 a x^3}\right )+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {35}{2} b (b c (2 A d+B c)-2 a d (B d+2 c C)) \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {8 \left (a+b x^2\right )^{3/2} \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (2 B d+c C)\right )\right )}{3 a x^3}\right )+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {35}{2} b (b c (2 A d+B c)-2 a d (B d+2 c C)) \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {8 \left (a+b x^2\right )^{3/2} \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (2 B d+c C)\right )\right )}{3 a x^3}\right )+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {35}{2} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right ) (b c (2 A d+B c)-2 a d (B d+2 c C))-\frac {8 \left (a+b x^2\right )^{3/2} \left (2 A b \left (4 b c^2-7 a d^2\right )+7 a \left (5 a C d^2-2 b c (2 B d+c C)\right )\right )}{3 a x^3}\right )+\frac {35 \left (a+b x^2\right )^{3/2} (b c (2 A d+B c)-2 a d (B d+2 c C))}{4 x^4}\right )-\frac {2 \left (a+b x^2\right )^{3/2} \left (7 a c (2 B d+c C)-A \left (4 b c^2-7 a d^2\right )\right )}{5 x^5}}{2 a}-\frac {7 c \left (a+b x^2\right )^{3/2} (2 A d+B c)}{6 x^6}}{7 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{7 a x^7}\) |
Input:
Int[((c + d*x)^2*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^8,x]
Output:
-1/7*(A*c^2*(a + b*x^2)^(3/2))/(a*x^7) + ((-7*c*(B*c + 2*A*d)*(a + b*x^2)^ (3/2))/(6*x^6) + ((-2*(7*a*c*(c*C + 2*B*d) - A*(4*b*c^2 - 7*a*d^2))*(a + b *x^2)^(3/2))/(5*x^5) + ((35*(b*c*(B*c + 2*A*d) - 2*a*d*(2*c*C + B*d))*(a + b*x^2)^(3/2))/(4*x^4) + ((-8*(2*A*b*(4*b*c^2 - 7*a*d^2) + 7*a*(5*a*C*d^2 - 2*b*c*(c*C + 2*B*d)))*(a + b*x^2)^(3/2))/(3*a*x^3) + (35*b*(b*c*(B*c + 2 *A*d) - 2*a*d*(2*c*C + B*d))*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2)/4)/5)/(2*a))/(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 = 0.59 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )-\frac {C \,d^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}}+c \left (2 A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )}{2 a}\right )+d \left (B d +2 C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )\) | \(357\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-224 A a \,b^{2} d^{2} x^{6}+128 A \,b^{3} c^{2} x^{6}-448 B a \,b^{2} c d \,x^{6}+560 C \,a^{2} b \,d^{2} x^{6}-224 C a \,b^{2} c^{2} x^{6}-210 A a \,b^{2} c d \,x^{5}+210 B \,a^{2} b \,d^{2} x^{5}-105 B a \,b^{2} c^{2} x^{5}+420 C \,a^{2} b c d \,x^{5}+112 A \,a^{2} b \,d^{2} x^{4}-64 A a \,b^{2} c^{2} x^{4}+224 B \,a^{2} b c d \,x^{4}+560 C \,a^{3} d^{2} x^{4}+112 C \,a^{2} b \,c^{2} x^{4}+140 A \,a^{2} b c d \,x^{3}+420 B \,a^{3} d^{2} x^{3}+70 B \,a^{2} b \,c^{2} x^{3}+840 C \,a^{3} c d \,x^{3}+336 A \,a^{3} d^{2} x^{2}+48 A \,a^{2} b \,c^{2} x^{2}+672 B \,a^{3} c d \,x^{2}+336 C \,a^{3} c^{2} x^{2}+560 A \,a^{3} c d x +280 B \,a^{3} c^{2} x +240 A \,a^{3} c^{2}\right )}{1680 x^{7} a^{3}}-\frac {\left (2 A b c d -2 a B \,d^{2}+b B \,c^{2}-4 C a c d \right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {5}{2}}}\) | \(377\) |
Input:
int((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x,method=_RETURNVERBOSE)
Output:
(A*d^2+2*B*c*d+C*c^2)*(-1/5/a/x^5*(b*x^2+a)^(3/2)+2/15*b/a^2/x^3*(b*x^2+a) ^(3/2))+A*c^2*(-1/7/a/x^7*(b*x^2+a)^(3/2)-4/7*b/a*(-1/5/a/x^5*(b*x^2+a)^(3 /2)+2/15*b/a^2/x^3*(b*x^2+a)^(3/2)))-1/3*C*d^2/a/x^3*(b*x^2+a)^(3/2)+c*(2* A*d+B*c)*(-1/6/a/x^6*(b*x^2+a)^(3/2)-1/2*b/a*(-1/4/a/x^4*(b*x^2+a)^(3/2)-1 /4*b/a*(-1/2/a/x^2*(b*x^2+a)^(3/2)+1/2*b/a*((b*x^2+a)^(1/2)-a^(1/2)*ln((2* a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))))+d*(B*d+2*C*c)*(-1/4/a/x^4*(b*x^2+a)^(3 /2)-1/4*b/a*(-1/2/a/x^2*(b*x^2+a)^(3/2)+1/2*b/a*((b*x^2+a)^(1/2)-a^(1/2)*l n((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))))
Time = 0.33 (sec) , antiderivative size = 702, normalized size of antiderivative = 2.24 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2*(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 - 2*B*a*b^2*d^2 - 2*(2*C*a*b^2 - A*b^3)*c*d)*sqrt (a)*x^7*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(16*(28*B* a*b^2*c*d + 2*(7*C*a*b^2 - 4*A*b^3)*c^2 - 7*(5*C*a^2*b - 2*A*a*b^2)*d^2)*x ^6 - 240*A*a^3*c^2 + 105*(B*a*b^2*c^2 - 2*B*a^2*b*d^2 - 2*(2*C*a^2*b - A*a *b^2)*c*d)*x^5 - 16*(14*B*a^2*b*c*d + (7*C*a^2*b - 4*A*a*b^2)*c^2 + 7*(5*C *a^3 + A*a^2*b)*d^2)*x^4 - 70*(B*a^2*b*c^2 + 6*B*a^3*d^2 + 2*(6*C*a^3 + A* a^2*b)*c*d)*x^3 - 48*(14*B*a^3*c*d + 7*A*a^3*d^2 + (7*C*a^3 + A*a^2*b)*c^2 )*x^2 - 280*(B*a^3*c^2 + 2*A*a^3*c*d)*x)*sqrt(b*x^2 + a))/(a^3*x^7), 1/168 0*(105*(B*b^3*c^2 - 2*B*a*b^2*d^2 - 2*(2*C*a*b^2 - A*b^3)*c*d)*sqrt(-a)*x^ 7*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (16*(28*B*a*b^2*c*d + 2*(7*C*a*b^2 - 4*A*b^3)*c^2 - 7*(5*C*a^2*b - 2*A*a*b^2)*d^2)*x^6 - 240*A*a^3*c^2 + 105* (B*a*b^2*c^2 - 2*B*a^2*b*d^2 - 2*(2*C*a^2*b - A*a*b^2)*c*d)*x^5 - 16*(14*B *a^2*b*c*d + (7*C*a^2*b - 4*A*a*b^2)*c^2 + 7*(5*C*a^3 + A*a^2*b)*d^2)*x^4 - 70*(B*a^2*b*c^2 + 6*B*a^3*d^2 + 2*(6*C*a^3 + A*a^2*b)*c*d)*x^3 - 48*(14* B*a^3*c*d + 7*A*a^3*d^2 + (7*C*a^3 + A*a^2*b)*c^2)*x^2 - 280*(B*a^3*c^2 + 2*A*a^3*c*d)*x)*sqrt(b*x^2 + a))/(a^3*x^7)]
Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (299) = 598\).
Time = 15.03 (sec) , antiderivative size = 1216, normalized size of antiderivative = 3.88 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)**2*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**8,x)
Output:
-15*A*a**5*b**(9/2)*c**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) - 33*A*a**4*b**(11/2)*c**2*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**2*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**2* 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**2*x**8*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) - A*a*c*d/(3*sqrt( b)*x**7*sqrt(a/(b*x**2) + 1)) - 8*A*b**(19/2)*c**2*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)*c*d/(12*x**5*sqrt(a/(b*x**2) + 1)) - A*sqrt(b)*d**2*sqrt(a/(b*x**2 ) + 1)/(5*x**4) + A*b**(3/2)*c*d/(24*a*x**3*sqrt(a/(b*x**2) + 1)) - A*b**( 3/2)*d**2*sqrt(a/(b*x**2) + 1)/(15*a*x**2) + A*b**(5/2)*c*d/(8*a**2*x*sqrt (a/(b*x**2) + 1)) + 2*A*b**(5/2)*d**2*sqrt(a/(b*x**2) + 1)/(15*a**2) - A*b **3*c*d*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) - B*a*c**2/(6*sqrt(b)*x**7 *sqrt(a/(b*x**2) + 1)) - B*a*d**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 5*B*sqrt(b)*c**2/(24*x**5*sqrt(a/(b*x**2) + 1)) - 2*B*sqrt(b)*c*d*sqrt(a/( b*x**2) + 1)/(5*x**4) - 3*B*sqrt(b)*d**2/(8*x**3*sqrt(a/(b*x**2) + 1)) + B *b**(3/2)*c**2/(48*a*x**3*sqrt(a/(b*x**2) + 1)) - 2*B*b**(3/2)*c*d*sqrt(a/ (b*x**2) + 1)/(15*a*x**2) - B*b**(3/2)*d**2/(8*a*x*sqrt(a/(b*x**2) + 1)...
Time = 0.04 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\frac {{\left (2 \, C c d + B d^{2}\right )} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {5}{2}}} - \frac {{\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} b^{2}}{8 \, a^{2}} + \frac {{\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a} b^{3}}{16 \, a^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} c^{2}}{105 \, a^{3} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C d^{2}}{3 \, a x^{3}} + \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{8 \, a^{2} x^{2}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}{16 \, a^{3} x^{2}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b c^{2}}{35 \, a^{2} x^{5}} + \frac {2 \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{15 \, a^{2} x^{3}} - \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{4 \, a x^{4}} + \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{8 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{2}}{7 \, a x^{7}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{5 \, a x^{5}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{6 \, a x^{6}} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^8,x, algorithm="maxima ")
Output:
1/8*(2*C*c*d + B*d^2)*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/16*(B* c^2 + 2*A*c*d)*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 1/8*(2*C*c*d + B*d^2)*sqrt(b*x^2 + a)*b^2/a^2 + 1/16*(B*c^2 + 2*A*c*d)*sqrt(b*x^2 + a)*b^ 3/a^3 - 8/105*(b*x^2 + a)^(3/2)*A*b^2*c^2/(a^3*x^3) - 1/3*(b*x^2 + a)^(3/2 )*C*d^2/(a*x^3) + 1/8*(2*C*c*d + B*d^2)*(b*x^2 + a)^(3/2)*b/(a^2*x^2) - 1/ 16*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(3/2)*b^2/(a^3*x^2) + 4/35*(b*x^2 + a)^(3 /2)*A*b*c^2/(a^2*x^5) + 2/15*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a)^(3/2)*b /(a^2*x^3) - 1/4*(2*C*c*d + B*d^2)*(b*x^2 + a)^(3/2)/(a*x^4) + 1/8*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(3/2)*b/(a^2*x^4) - 1/7*(b*x^2 + a)^(3/2)*A*c^2/(a* x^7) - 1/5*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a)^(3/2)/(a*x^5) - 1/6*(B*c^ 2 + 2*A*c*d)*(b*x^2 + a)^(3/2)/(a*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (281) = 562\).
Time = 0.21 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.25 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(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 - 4*C*a*b^2*c*d + 2*A*b^3*c*d - 2*B*a*b^2*d^2)*arctan(-(sqr t(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^2 - 420*(sqrt(b)*x - sqrt(b*x^2 + a))^13*C* a*b^2*c*d + 210*(sqrt(b)*x - sqrt(b*x^2 + a))^13*A*b^3*c*d - 210*(sqrt(b)* x - sqrt(b*x^2 + a))^13*B*a*b^2*d^2 - 1680*(sqrt(b)*x - sqrt(b*x^2 + a))^1 2*C*a^2*b^(3/2)*d^2 - 700*(sqrt(b)*x - sqrt(b*x^2 + a))^11*B*a*b^3*c^2 - 1 680*(sqrt(b)*x - sqrt(b*x^2 + a))^11*C*a^2*b^2*c*d - 1400*(sqrt(b)*x - sqr t(b*x^2 + a))^11*A*a*b^3*c*d - 840*(sqrt(b)*x - sqrt(b*x^2 + a))^11*B*a^2* b^2*d^2 - 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^2*b^(5/2)*c^2 - 6720*( sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2)*c*d + 6720*(sqrt(b)*x - sqrt (b*x^2 + a))^10*C*a^3*b^(3/2)*d^2 - 3360*(sqrt(b)*x - sqrt(b*x^2 + a))^10* A*a^2*b^(5/2)*d^2 - 3395*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a^2*b^3*c^2 + 4 620*(sqrt(b)*x - sqrt(b*x^2 + a))^9*C*a^3*b^2*c*d - 6790*(sqrt(b)*x - sqrt (b*x^2 + a))^9*A*a^2*b^3*c*d + 2310*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a^3* b^2*d^2 + 5600*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^3*b^(5/2)*c^2 - 8960*(s qrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2)*c^2 + 11200*(sqrt(b)*x - sqrt( b*x^2 + a))^8*B*a^3*b^(5/2)*c*d - 10640*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C* a^4*b^(3/2)*d^2 + 5600*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^3*b^(5/2)*d^2 - 2240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^4*b^(5/2)*c^2 - 4480*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^3*b^(7/2)*c^2 - 4480*(sqrt(b)*x - sqrt(b*x^2 +...
Timed out. \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^2\,\left (C\,x^2+B\,x+A\right )}{x^8} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^8,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^8, x)
Time = 1.73 (sec) , antiderivative size = 864, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^8} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(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**2 - 560*sqrt(a + b*x**2)*a**4*c*d*x - 336 *sqrt(a + b*x**2)*a**4*d**2*x**2 - 48*sqrt(a + b*x**2)*a**3*b*c**2*x**2 - 280*sqrt(a + b*x**2)*a**3*b*c**2*x - 140*sqrt(a + b*x**2)*a**3*b*c*d*x**3 - 672*sqrt(a + b*x**2)*a**3*b*c*d*x**2 - 112*sqrt(a + b*x**2)*a**3*b*d**2* x**4 - 420*sqrt(a + b*x**2)*a**3*b*d**2*x**3 - 336*sqrt(a + b*x**2)*a**3*c **3*x**2 - 840*sqrt(a + b*x**2)*a**3*c**2*d*x**3 - 560*sqrt(a + b*x**2)*a* *3*c*d**2*x**4 + 64*sqrt(a + b*x**2)*a**2*b**2*c**2*x**4 - 70*sqrt(a + b*x **2)*a**2*b**2*c**2*x**3 + 210*sqrt(a + b*x**2)*a**2*b**2*c*d*x**5 - 224*s qrt(a + b*x**2)*a**2*b**2*c*d*x**4 + 224*sqrt(a + b*x**2)*a**2*b**2*d**2*x **6 - 210*sqrt(a + b*x**2)*a**2*b**2*d**2*x**5 - 112*sqrt(a + b*x**2)*a**2 *b*c**3*x**4 - 420*sqrt(a + b*x**2)*a**2*b*c**2*d*x**5 - 560*sqrt(a + b*x* *2)*a**2*b*c*d**2*x**6 - 128*sqrt(a + b*x**2)*a*b**3*c**2*x**6 + 105*sqrt( a + b*x**2)*a*b**3*c**2*x**5 + 448*sqrt(a + b*x**2)*a*b**3*c*d*x**6 + 224* sqrt(a + b*x**2)*a*b**2*c**3*x**6 + 210*sqrt(a)*log((sqrt(a + b*x**2) - sq rt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*c*d*x**7 - 210*sqrt(a)*log((sqrt(a + b* x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*d**2*x**7 - 420*sqrt(a)*log(( sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*c**2*d*x**7 + 105* sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*c**2*x* *7 - 210*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b **3*c*d*x**7 + 210*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)...