Integrand size = 30, antiderivative size = 262 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=-\frac {(3 A b+10 a C) \sqrt {a+b x^2}}{80 x^8}-\frac {b (A b+30 a C) \sqrt {a+b x^2}}{160 a x^6}+\frac {b^2 (A b-2 a C) \sqrt {a+b x^2}}{128 a^2 x^4}-\frac {3 b^3 (A b-2 a C) \sqrt {a+b x^2}}{256 a^3 x^2}-\frac {A \left (a+b x^2\right )^{3/2}}{10 x^{10}}-\frac {B \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 b B-9 a D) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 b B-9 a D) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}+\frac {3 b^4 (A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}} \] Output:
-1/80*(3*A*b+10*C*a)*(b*x^2+a)^(1/2)/x^8-1/160*b*(A*b+30*C*a)*(b*x^2+a)^(1 /2)/a/x^6+1/128*b^2*(A*b-2*C*a)*(b*x^2+a)^(1/2)/a^2/x^4-3/256*b^3*(A*b-2*C *a)*(b*x^2+a)^(1/2)/a^3/x^2-1/10*A*(b*x^2+a)^(3/2)/x^10-1/9*B*(b*x^2+a)^(5 /2)/a/x^9+1/63*(4*B*b-9*D*a)*(b*x^2+a)^(5/2)/a^2/x^7-2/315*b*(4*B*b-9*D*a) *(b*x^2+a)^(5/2)/a^3/x^5+3/256*b^4*(A*b-2*C*a)*arctanh((b*x^2+a)^(1/2)/a^( 1/2))/a^(7/2)
Time = 2.72 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=-\frac {\sqrt {a+b x^2} \left (b^4 x^8 (945 A+2048 B x)+32 a^4 (252 A+5 x (56 B+9 x (7 C+8 D x)))+12 a^2 b^2 x^4 (42 A+x (64 B+3 x (35 C+64 D x)))+16 a^3 b x^2 (693 A+x (800 B+9 x (105 C+128 D x)))-2 a b^3 x^6 (315 A+x (512 B+9 x (105 C+256 D x)))\right )}{80640 a^3 x^{10}}+\frac {3 b^4 (-A b+2 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{7/2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^11,x]
Output:
-1/80640*(Sqrt[a + b*x^2]*(b^4*x^8*(945*A + 2048*B*x) + 32*a^4*(252*A + 5* x*(56*B + 9*x*(7*C + 8*D*x))) + 12*a^2*b^2*x^4*(42*A + x*(64*B + 3*x*(35*C + 64*D*x))) + 16*a^3*b*x^2*(693*A + x*(800*B + 9*x*(105*C + 128*D*x))) - 2*a*b^3*x^6*(315*A + x*(512*B + 9*x*(105*C + 256*D*x)))))/(a^3*x^10) + (3* b^4*(-(A*b) + 2*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(128* a^(7/2))
Time = 0.64 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2338, 27, 2338, 27, 539, 25, 539, 27, 539, 27, 534, 243, 51, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {5 \left (b x^2+a\right )^{3/2} \left (2 a D x^2-(A b-2 a C) x+2 a B\right )}{x^{10}}dx}{10 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (2 a D x^2-(A b-2 a C) x+2 a B\right )}{x^{10}}dx}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int \frac {a (9 (A b-2 a C)+2 (4 b B-9 a D) x) \left (b x^2+a\right )^{3/2}}{x^9}dx}{9 a}-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{9} \int \frac {(9 (A b-2 a C)+2 (4 b B-9 a D) x) \left (b x^2+a\right )^{3/2}}{x^9}dx-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {\int -\frac {(16 a (4 b B-9 a D)-27 b (A b-2 a C) x) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}+\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {\int \frac {(16 a (4 b B-9 a D)-27 b (A b-2 a C) x) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {\int \frac {a b (189 (A b-2 a C)+32 (4 b B-9 a D) x) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \int \frac {(189 (A b-2 a C)+32 (4 b B-9 a D) x) \left (b x^2+a\right )^{3/2}}{x^7}dx-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (-\frac {\int -\frac {3 (64 a (4 b B-9 a D)-63 b (A b-2 a C) x) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {\int \frac {(64 a (4 b B-9 a D)-63 b (A b-2 a C) x) \left (b x^2+a\right )^{3/2}}{x^6}dx}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-63 b (A b-2 a C) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-\frac {63}{2} b (A b-2 a C) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-\frac {63}{2} b (A b-2 a C) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-\frac {63}{2} b (A b-2 a C) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-\frac {63}{2} b (A b-2 a C) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{8 a x^8}-\frac {-\frac {1}{7} b \left (\frac {-\frac {63}{2} b (A b-2 a C) \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{5 x^5}}{2 a}-\frac {63 \left (a+b x^2\right )^{5/2} (A b-2 a C)}{2 a x^6}\right )-\frac {16 \left (a+b x^2\right )^{5/2} (4 b B-9 a D)}{7 x^7}}{8 a}\right )-\frac {2 B \left (a+b x^2\right )^{5/2}}{9 x^9}}{2 a}-\frac {A \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
Input:
Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/x^11,x]
Output:
-1/10*(A*(a + b*x^2)^(5/2))/(a*x^10) + ((-2*B*(a + b*x^2)^(5/2))/(9*x^9) + ((9*(A*b - 2*a*C)*(a + b*x^2)^(5/2))/(8*a*x^8) - ((-16*(4*b*B - 9*a*D)*(a + b*x^2)^(5/2))/(7*x^7) - (b*((-63*(A*b - 2*a*C)*(a + b*x^2)^(5/2))/(2*a* x^6) + ((-64*(4*b*B - 9*a*D)*(a + b*x^2)^(5/2))/(5*x^5) - (63*b*(A*b - 2*a *C)*(-1/2*(a + b*x^2)^(3/2)/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTan h[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/4))/2)/(2*a)))/7)/(8*a))/9)/(2*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 = 0.76 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 a \,x^{10}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{2 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+D \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )\) | \(426\) |
Input:
int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^11,x,method=_RETURNVERBOSE)
Output:
A*(-1/10/a/x^10*(b*x^2+a)^(5/2)-1/2*b/a*(-1/8/a/x^8*(b*x^2+a)^(5/2)-3/8*b/ a*(-1/6/a/x^6*(b*x^2+a)^(5/2)-1/6*b/a*(-1/4/a/x^4*(b*x^2+a)^(5/2)+1/4*b/a* (-1/2/a/x^2*(b*x^2+a)^(5/2)+3/2*b/a*(1/3*(b*x^2+a)^(3/2)+a*((b*x^2+a)^(1/2 )-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))))))))+B*(-1/9/a/x^9*(b*x^ 2+a)^(5/2)-4/9*b/a*(-1/7/a/x^7*(b*x^2+a)^(5/2)+2/35*b/a^2/x^5*(b*x^2+a)^(5 /2)))+C*(-1/8/a/x^8*(b*x^2+a)^(5/2)-3/8*b/a*(-1/6/a/x^6*(b*x^2+a)^(5/2)-1/ 6*b/a*(-1/4/a/x^4*(b*x^2+a)^(5/2)+1/4*b/a*(-1/2/a/x^2*(b*x^2+a)^(5/2)+3/2* b/a*(1/3*(b*x^2+a)^(3/2)+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*(-1/7/a/x^7*(b*x^2+a)^(5/2)+2/35*b/a^2/x^5*(b*x^2+ a)^(5/2))
Time = 0.20 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=\left [-\frac {945 \, {\left (2 \, C a b^{4} - A b^{5}\right )} \sqrt {a} x^{10} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (512 \, {\left (9 \, D a^{2} b^{3} - 4 \, B a b^{4}\right )} x^{9} + 945 \, {\left (2 \, C a^{2} b^{3} - A a b^{4}\right )} x^{8} - 256 \, {\left (9 \, D a^{3} b^{2} - 4 \, B a^{2} b^{3}\right )} x^{7} - 8960 \, B a^{5} x - 630 \, {\left (2 \, C a^{3} b^{2} - A a^{2} b^{3}\right )} x^{6} - 8064 \, A a^{5} - 768 \, {\left (24 \, D a^{4} b + B a^{3} b^{2}\right )} x^{5} - 504 \, {\left (30 \, C a^{4} b + A a^{3} b^{2}\right )} x^{4} - 1280 \, {\left (9 \, D a^{5} + 10 \, B a^{4} b\right )} x^{3} - 1008 \, {\left (10 \, C a^{5} + 11 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{161280 \, a^{4} x^{10}}, \frac {945 \, {\left (2 \, C a b^{4} - A b^{5}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (512 \, {\left (9 \, D a^{2} b^{3} - 4 \, B a b^{4}\right )} x^{9} + 945 \, {\left (2 \, C a^{2} b^{3} - A a b^{4}\right )} x^{8} - 256 \, {\left (9 \, D a^{3} b^{2} - 4 \, B a^{2} b^{3}\right )} x^{7} - 8960 \, B a^{5} x - 630 \, {\left (2 \, C a^{3} b^{2} - A a^{2} b^{3}\right )} x^{6} - 8064 \, A a^{5} - 768 \, {\left (24 \, D a^{4} b + B a^{3} b^{2}\right )} x^{5} - 504 \, {\left (30 \, C a^{4} b + A a^{3} b^{2}\right )} x^{4} - 1280 \, {\left (9 \, D a^{5} + 10 \, B a^{4} b\right )} x^{3} - 1008 \, {\left (10 \, C a^{5} + 11 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{80640 \, a^{4} x^{10}}\right ] \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^11,x, algorithm="fricas")
Output:
[-1/161280*(945*(2*C*a*b^4 - A*b^5)*sqrt(a)*x^10*log(-(b*x^2 + 2*sqrt(b*x^ 2 + a)*sqrt(a) + 2*a)/x^2) - 2*(512*(9*D*a^2*b^3 - 4*B*a*b^4)*x^9 + 945*(2 *C*a^2*b^3 - A*a*b^4)*x^8 - 256*(9*D*a^3*b^2 - 4*B*a^2*b^3)*x^7 - 8960*B*a ^5*x - 630*(2*C*a^3*b^2 - A*a^2*b^3)*x^6 - 8064*A*a^5 - 768*(24*D*a^4*b + B*a^3*b^2)*x^5 - 504*(30*C*a^4*b + A*a^3*b^2)*x^4 - 1280*(9*D*a^5 + 10*B*a ^4*b)*x^3 - 1008*(10*C*a^5 + 11*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^4*x^10), 1/80640*(945*(2*C*a*b^4 - A*b^5)*sqrt(-a)*x^10*arctan(sqrt(b*x^2 + a)*sqr t(-a)/a) + (512*(9*D*a^2*b^3 - 4*B*a*b^4)*x^9 + 945*(2*C*a^2*b^3 - A*a*b^4 )*x^8 - 256*(9*D*a^3*b^2 - 4*B*a^2*b^3)*x^7 - 8960*B*a^5*x - 630*(2*C*a^3* b^2 - A*a^2*b^3)*x^6 - 8064*A*a^5 - 768*(24*D*a^4*b + B*a^3*b^2)*x^5 - 504 *(30*C*a^4*b + A*a^3*b^2)*x^4 - 1280*(9*D*a^5 + 10*B*a^4*b)*x^3 - 1008*(10 *C*a^5 + 11*A*a^4*b)*x^2)*sqrt(b*x^2 + a))/(a^4*x^10)]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/x**11,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=-\frac {3 \, C b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} + \frac {3 \, A b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, a^{\frac {7}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} C b^{4}}{128 \, a^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5}}{256 \, a^{5}} - \frac {3 \, \sqrt {b x^{2} + a} A b^{5}}{256 \, a^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4}}{256 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b^{2}}{64 \, a^{3} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{128 \, a^{4} x^{4}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{315 \, a^{3} x^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{32 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D}{7 \, a x^{7}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C}{8 \, a x^{8}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{16 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{9 \, a x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{10 \, a x^{10}} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^11,x, algorithm="maxima")
Output:
-3/128*C*b^4*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 3/256*A*b^5*arcsinh(a /(sqrt(a*b)*abs(x)))/a^(7/2) + 1/128*(b*x^2 + a)^(3/2)*C*b^4/a^4 + 3/128*s qrt(b*x^2 + a)*C*b^4/a^3 - 1/256*(b*x^2 + a)^(3/2)*A*b^5/a^5 - 3/256*sqrt( b*x^2 + a)*A*b^5/a^4 - 1/128*(b*x^2 + a)^(5/2)*C*b^3/(a^4*x^2) + 1/256*(b* x^2 + a)^(5/2)*A*b^4/(a^5*x^2) - 1/64*(b*x^2 + a)^(5/2)*C*b^2/(a^3*x^4) + 1/128*(b*x^2 + a)^(5/2)*A*b^3/(a^4*x^4) + 2/35*(b*x^2 + a)^(5/2)*D*b/(a^2* x^5) - 8/315*(b*x^2 + a)^(5/2)*B*b^2/(a^3*x^5) + 1/16*(b*x^2 + a)^(5/2)*C* b/(a^2*x^6) - 1/32*(b*x^2 + a)^(5/2)*A*b^2/(a^3*x^6) - 1/7*(b*x^2 + a)^(5/ 2)*D/(a*x^7) + 4/63*(b*x^2 + a)^(5/2)*B*b/(a^2*x^7) - 1/8*(b*x^2 + a)^(5/2 )*C/(a*x^8) + 1/16*(b*x^2 + a)^(5/2)*A*b/(a^2*x^8) - 1/9*(b*x^2 + a)^(5/2) *B/(a*x^9) - 1/10*(b*x^2 + a)^(5/2)*A/(a*x^10)
Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (225) = 450\).
Time = 0.14 (sec) , antiderivative size = 1068, normalized size of antiderivative = 4.08 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^11,x, algorithm="giac")
Output:
3/128*(2*C*a*b^4 - A*b^5)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/ (sqrt(-a)*a^3) - 1/40320*(1890*(sqrt(b)*x - sqrt(b*x^2 + a))^19*C*a*b^4 - 945*(sqrt(b)*x - sqrt(b*x^2 + a))^19*A*b^5 - 18270*(sqrt(b)*x - sqrt(b*x^2 + a))^17*C*a^2*b^4 + 9135*(sqrt(b)*x - sqrt(b*x^2 + a))^17*A*a*b^5 - 1612 80*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^3*b^(7/2) - 178920*(sqrt(b)*x - sq rt(b*x^2 + a))^15*C*a^3*b^4 - 39564*(sqrt(b)*x - sqrt(b*x^2 + a))^15*A*a^2 *b^5 + 322560*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(7/2) - 430080*(sqr t(b)*x - sqrt(b*x^2 + a))^14*B*a^3*b^(9/2) - 17640*(sqrt(b)*x - sqrt(b*x^2 + a))^13*C*a^4*b^4 - 636300*(sqrt(b)*x - sqrt(b*x^2 + a))^13*A*a^3*b^5 - 322560*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^5*b^(7/2) - 215040*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^4*b^(9/2) + 212940*(sqrt(b)*x - sqrt(b*x^2 + a)) ^11*C*a^5*b^4 - 1396710*(sqrt(b)*x - sqrt(b*x^2 + a))^11*A*a^4*b^5 + 58060 8*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^6*b^(7/2) - 258048*(sqrt(b)*x - sqr t(b*x^2 + a))^10*B*a^5*b^(9/2) + 212940*(sqrt(b)*x - sqrt(b*x^2 + a))^9*C* a^6*b^4 - 1396710*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a^5*b^5 - 645120*(sqrt (b)*x - sqrt(b*x^2 + a))^8*D*a^7*b^(7/2) + 645120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^6*b^(9/2) - 17640*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^7*b^4 - 636300*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a^6*b^5 + 230400*(sqrt(b)*x - sqr t(b*x^2 + a))^6*D*a^8*b^(7/2) + 184320*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a ^7*b^(9/2) - 178920*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^8*b^4 - 39564*(...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{x^{11}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^11,x)
Output:
int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/x^11, x)
Time = 0.42 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{x^{11}} \, dx=\frac {-8064 \sqrt {b \,x^{2}+a}\, a^{5}-11088 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-8960 \sqrt {b \,x^{2}+a}\, a^{4} b x -10080 \sqrt {b \,x^{2}+a}\, a^{4} c \,x^{2}-11520 \sqrt {b \,x^{2}+a}\, a^{4} d \,x^{3}-504 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-12800 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{3}-15120 \sqrt {b \,x^{2}+a}\, a^{3} b c \,x^{4}-18432 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{5}+630 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-768 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{5}-1260 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{6}-2304 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{7}-945 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}+1024 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{7}+1890 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{8}+4608 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{9}-2048 \sqrt {b \,x^{2}+a}\, b^{5} x^{9}-945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}+1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{10}+945 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}-1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{10}-4608 \sqrt {b}\, a \,b^{3} d \,x^{10}+2048 \sqrt {b}\, b^{5} x^{10}}{80640 a^{3} x^{10}} \] Input:
int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/x^11,x)
Output:
( - 8064*sqrt(a + b*x**2)*a**5 - 11088*sqrt(a + b*x**2)*a**4*b*x**2 - 8960 *sqrt(a + b*x**2)*a**4*b*x - 10080*sqrt(a + b*x**2)*a**4*c*x**2 - 11520*sq rt(a + b*x**2)*a**4*d*x**3 - 504*sqrt(a + b*x**2)*a**3*b**2*x**4 - 12800*s qrt(a + b*x**2)*a**3*b**2*x**3 - 15120*sqrt(a + b*x**2)*a**3*b*c*x**4 - 18 432*sqrt(a + b*x**2)*a**3*b*d*x**5 + 630*sqrt(a + b*x**2)*a**2*b**3*x**6 - 768*sqrt(a + b*x**2)*a**2*b**3*x**5 - 1260*sqrt(a + b*x**2)*a**2*b**2*c*x **6 - 2304*sqrt(a + b*x**2)*a**2*b**2*d*x**7 - 945*sqrt(a + b*x**2)*a*b**4 *x**8 + 1024*sqrt(a + b*x**2)*a*b**4*x**7 + 1890*sqrt(a + b*x**2)*a*b**3*c *x**8 + 4608*sqrt(a + b*x**2)*a*b**3*d*x**9 - 2048*sqrt(a + b*x**2)*b**5*x **9 - 945*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b* *5*x**10 + 1890*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt( a))*b**4*c*x**10 + 945*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x )/sqrt(a))*b**5*x**10 - 1890*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqr t(b)*x)/sqrt(a))*b**4*c*x**10 - 4608*sqrt(b)*a*b**3*d*x**10 + 2048*sqrt(b) *b**5*x**10)/(80640*a**3*x**10)