Integrand size = 32, antiderivative size = 216 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {a^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {a+b x^2}}{b^6}-\frac {a \left (2 A b^3-a \left (3 b^2 B-4 a b C+5 a^2 D\right )\right ) \left (a+b x^2\right )^{3/2}}{3 b^6}+\frac {\left (A b^3-a \left (3 b^2 B-6 a b C+10 a^2 D\right )\right ) \left (a+b x^2\right )^{5/2}}{5 b^6}+\frac {\left (b^2 B-4 a b C+10 a^2 D\right ) \left (a+b x^2\right )^{7/2}}{7 b^6}+\frac {(b C-5 a D) \left (a+b x^2\right )^{9/2}}{9 b^6}+\frac {D \left (a+b x^2\right )^{11/2}}{11 b^6} \] Output:
a^2*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x^2+a)^(1/2)/b^6-1/3*a*(2*A*b^3-a*(3* B*b^2-4*C*a*b+5*D*a^2))*(b*x^2+a)^(3/2)/b^6+1/5*(A*b^3-a*(3*B*b^2-6*C*a*b+ 10*D*a^2))*(b*x^2+a)^(5/2)/b^6+1/7*(B*b^2-4*C*a*b+10*D*a^2)*(b*x^2+a)^(7/2 )/b^6+1/9*(C*b-5*D*a)*(b*x^2+a)^(9/2)/b^6+1/11*D*(b*x^2+a)^(11/2)/b^6
Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.73 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-1280 a^5 D+128 a^4 b \left (11 C+5 D x^2\right )-16 a^3 b^2 \left (99 B+44 C x^2+30 D x^4\right )+8 a^2 b^3 \left (231 A+99 B x^2+66 C x^4+50 D x^6\right )-2 a b^4 x^2 \left (462 A+297 B x^2+220 C x^4+175 D x^6\right )+b^5 x^4 \left (693 A+5 \left (99 B x^2+77 C x^4+63 D x^6\right )\right )\right )}{3465 b^6} \] Input:
Integrate[(x^5*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[a + b*x^2],x]
Output:
(Sqrt[a + b*x^2]*(-1280*a^5*D + 128*a^4*b*(11*C + 5*D*x^2) - 16*a^3*b^2*(9 9*B + 44*C*x^2 + 30*D*x^4) + 8*a^2*b^3*(231*A + 99*B*x^2 + 66*C*x^4 + 50*D *x^6) - 2*a*b^4*x^2*(462*A + 297*B*x^2 + 220*C*x^4 + 175*D*x^6) + b^5*x^4* (693*A + 5*(99*B*x^2 + 77*C*x^4 + 63*D*x^6))))/(3465*b^6)
Time = 0.56 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2331, 2123, 2009}
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.
| \(\displaystyle \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \left (D x^6+C x^4+B x^2+A\right )}{\sqrt {b x^2+a}}dx^2\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {D \left (b x^2+a\right )^{9/2}}{b^5}+\frac {(b C-5 a D) \left (b x^2+a\right )^{7/2}}{b^5}+\frac {\left (10 D a^2-4 b C a+b^2 B\right ) \left (b x^2+a\right )^{5/2}}{b^5}+\frac {\left (A b^3-a \left (10 D a^2-6 b C a+3 b^2 B\right )\right ) \left (b x^2+a\right )^{3/2}}{b^5}+\frac {a \left (a \left (5 D a^2-4 b C a+3 b^2 B\right )-2 A b^3\right ) \sqrt {b x^2+a}}{b^5}+\frac {a^2 \left (A b^3-a \left (D a^2-b C a+b^2 B\right )\right )}{b^5 \sqrt {b x^2+a}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{5/2} \left (A b^3-a \left (10 a^2 D-6 a b C+3 b^2 B\right )\right )}{5 b^6}-\frac {2 a \left (a+b x^2\right )^{3/2} \left (2 A b^3-a \left (5 a^2 D-4 a b C+3 b^2 B\right )\right )}{3 b^6}+\frac {2 a^2 \sqrt {a+b x^2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^6}+\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a^2 D-4 a b C+b^2 B\right )}{7 b^6}+\frac {2 \left (a+b x^2\right )^{9/2} (b C-5 a D)}{9 b^6}+\frac {2 D \left (a+b x^2\right )^{11/2}}{11 b^6}\right )\) |
Input:
Int[(x^5*(A + B*x^2 + C*x^4 + D*x^6))/Sqrt[a + b*x^2],x]
Output:
((2*a^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[a + b*x^2])/b^6 - (2*a*(2 *A*b^3 - a*(3*b^2*B - 4*a*b*C + 5*a^2*D))*(a + b*x^2)^(3/2))/(3*b^6) + (2* (A*b^3 - a*(3*b^2*B - 6*a*b*C + 10*a^2*D))*(a + b*x^2)^(5/2))/(5*b^6) + (2 *(b^2*B - 4*a*b*C + 10*a^2*D)*(a + b*x^2)^(7/2))/(7*b^6) + (2*(b*C - 5*a*D )*(a + b*x^2)^(9/2))/(9*b^6) + (2*D*(a + b*x^2)^(11/2))/(11*b^6))/2
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Time = 0.69 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {8 \left (\frac {3 \left (\frac {5}{11} D x^{6}+\frac {5}{9} C \,x^{4}+\frac {5}{7} x^{2} B +A \right ) x^{4} b^{5}}{8}-\frac {\left (\frac {25}{66} D x^{6}+\frac {10}{21} C \,x^{4}+\frac {9}{14} x^{2} B +A \right ) x^{2} a \,b^{4}}{2}+a^{2} \left (\frac {50}{231} D x^{6}+\frac {2}{7} C \,x^{4}+\frac {3}{7} x^{2} B +A \right ) b^{3}-\frac {6 \left (\frac {10}{33} D x^{4}+\frac {4}{9} C \,x^{2}+B \right ) a^{3} b^{2}}{7}+\frac {16 \left (\frac {5 D x^{2}}{11}+C \right ) a^{4} b}{21}-\frac {160 D a^{5}}{231}\right ) \sqrt {b \,x^{2}+a}}{15 b^{6}}\) | \(142\) |
| gosper | \(\frac {\sqrt {b \,x^{2}+a}\, \left (315 D x^{10} b^{5}+385 C \,b^{5} x^{8}-350 D a \,b^{4} x^{8}+495 B \,b^{5} x^{6}-440 C a \,b^{4} x^{6}+400 D a^{2} b^{3} x^{6}+693 A \,b^{5} x^{4}-594 B a \,b^{4} x^{4}+528 C \,a^{2} b^{3} x^{4}-480 D a^{3} b^{2} x^{4}-924 A a \,b^{4} x^{2}+792 B \,a^{2} b^{3} x^{2}-704 C \,a^{3} b^{2} x^{2}+640 D a^{4} b \,x^{2}+1848 a^{2} b^{3} A -1584 a^{3} b^{2} B +1408 C \,a^{4} b -1280 D a^{5}\right )}{3465 b^{6}}\) | \(193\) |
| trager | \(\frac {\sqrt {b \,x^{2}+a}\, \left (315 D x^{10} b^{5}+385 C \,b^{5} x^{8}-350 D a \,b^{4} x^{8}+495 B \,b^{5} x^{6}-440 C a \,b^{4} x^{6}+400 D a^{2} b^{3} x^{6}+693 A \,b^{5} x^{4}-594 B a \,b^{4} x^{4}+528 C \,a^{2} b^{3} x^{4}-480 D a^{3} b^{2} x^{4}-924 A a \,b^{4} x^{2}+792 B \,a^{2} b^{3} x^{2}-704 C \,a^{3} b^{2} x^{2}+640 D a^{4} b \,x^{2}+1848 a^{2} b^{3} A -1584 a^{3} b^{2} B +1408 C \,a^{4} b -1280 D a^{5}\right )}{3465 b^{6}}\) | \(193\) |
| orering | \(\frac {\sqrt {b \,x^{2}+a}\, \left (315 D x^{10} b^{5}+385 C \,b^{5} x^{8}-350 D a \,b^{4} x^{8}+495 B \,b^{5} x^{6}-440 C a \,b^{4} x^{6}+400 D a^{2} b^{3} x^{6}+693 A \,b^{5} x^{4}-594 B a \,b^{4} x^{4}+528 C \,a^{2} b^{3} x^{4}-480 D a^{3} b^{2} x^{4}-924 A a \,b^{4} x^{2}+792 B \,a^{2} b^{3} x^{2}-704 C \,a^{3} b^{2} x^{2}+640 D a^{4} b \,x^{2}+1848 a^{2} b^{3} A -1584 a^{3} b^{2} B +1408 C \,a^{4} b -1280 D a^{5}\right )}{3465 b^{6}}\) | \(193\) |
| default | \(A \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )+B \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )+C \left (\frac {x^{8} \sqrt {b \,x^{2}+a}}{9 b}-\frac {8 a \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )+D \left (\frac {x^{10} \sqrt {b \,x^{2}+a}}{11 b}-\frac {10 a \left (\frac {x^{8} \sqrt {b \,x^{2}+a}}{9 b}-\frac {8 a \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )\) | \(382\) |
Input:
int(x^5*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
8/15*(3/8*(5/11*D*x^6+5/9*C*x^4+5/7*x^2*B+A)*x^4*b^5-1/2*(25/66*D*x^6+10/2 1*C*x^4+9/14*x^2*B+A)*x^2*a*b^4+a^2*(50/231*D*x^6+2/7*C*x^4+3/7*x^2*B+A)*b ^3-6/7*(10/33*D*x^4+4/9*C*x^2+B)*a^3*b^2+16/21*(5/11*D*x^2+C)*a^4*b-160/23 1*D*a^5)*(b*x^2+a)^(1/2)/b^6
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {{\left (315 \, D b^{5} x^{10} - 35 \, {\left (10 \, D a b^{4} - 11 \, C b^{5}\right )} x^{8} + 5 \, {\left (80 \, D a^{2} b^{3} - 88 \, C a b^{4} + 99 \, B b^{5}\right )} x^{6} - 1280 \, D a^{5} + 1408 \, C a^{4} b - 1584 \, B a^{3} b^{2} + 1848 \, A a^{2} b^{3} - 3 \, {\left (160 \, D a^{3} b^{2} - 176 \, C a^{2} b^{3} + 198 \, B a b^{4} - 231 \, A b^{5}\right )} x^{4} + 4 \, {\left (160 \, D a^{4} b - 176 \, C a^{3} b^{2} + 198 \, B a^{2} b^{3} - 231 \, A a b^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, b^{6}} \] Input:
integrate(x^5*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
1/3465*(315*D*b^5*x^10 - 35*(10*D*a*b^4 - 11*C*b^5)*x^8 + 5*(80*D*a^2*b^3 - 88*C*a*b^4 + 99*B*b^5)*x^6 - 1280*D*a^5 + 1408*C*a^4*b - 1584*B*a^3*b^2 + 1848*A*a^2*b^3 - 3*(160*D*a^3*b^2 - 176*C*a^2*b^3 + 198*B*a*b^4 - 231*A* b^5)*x^4 + 4*(160*D*a^4*b - 176*C*a^3*b^2 + 198*B*a^2*b^3 - 231*A*a*b^4)*x ^2)*sqrt(b*x^2 + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (207) = 414\).
Time = 0.43 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.05 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {8 A a^{2} \sqrt {a + b x^{2}}}{15 b^{3}} - \frac {4 A a x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {A x^{4} \sqrt {a + b x^{2}}}{5 b} - \frac {16 B a^{3} \sqrt {a + b x^{2}}}{35 b^{4}} + \frac {8 B a^{2} x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} - \frac {6 B a x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} + \frac {B x^{6} \sqrt {a + b x^{2}}}{7 b} + \frac {128 C a^{4} \sqrt {a + b x^{2}}}{315 b^{5}} - \frac {64 C a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{4}} + \frac {16 C a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {8 C a x^{6} \sqrt {a + b x^{2}}}{63 b^{2}} + \frac {C x^{8} \sqrt {a + b x^{2}}}{9 b} - \frac {256 D a^{5} \sqrt {a + b x^{2}}}{693 b^{6}} + \frac {128 D a^{4} x^{2} \sqrt {a + b x^{2}}}{693 b^{5}} - \frac {32 D a^{3} x^{4} \sqrt {a + b x^{2}}}{231 b^{4}} + \frac {80 D a^{2} x^{6} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {10 D a x^{8} \sqrt {a + b x^{2}}}{99 b^{2}} + \frac {D x^{10} \sqrt {a + b x^{2}}}{11 b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{6}}{6} + \frac {B x^{8}}{8} + \frac {C x^{10}}{10} + \frac {D x^{12}}{12}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate(x**5*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(1/2),x)
Output:
Piecewise((8*A*a**2*sqrt(a + b*x**2)/(15*b**3) - 4*A*a*x**2*sqrt(a + b*x** 2)/(15*b**2) + A*x**4*sqrt(a + b*x**2)/(5*b) - 16*B*a**3*sqrt(a + b*x**2)/ (35*b**4) + 8*B*a**2*x**2*sqrt(a + b*x**2)/(35*b**3) - 6*B*a*x**4*sqrt(a + b*x**2)/(35*b**2) + B*x**6*sqrt(a + b*x**2)/(7*b) + 128*C*a**4*sqrt(a + b *x**2)/(315*b**5) - 64*C*a**3*x**2*sqrt(a + b*x**2)/(315*b**4) + 16*C*a**2 *x**4*sqrt(a + b*x**2)/(105*b**3) - 8*C*a*x**6*sqrt(a + b*x**2)/(63*b**2) + C*x**8*sqrt(a + b*x**2)/(9*b) - 256*D*a**5*sqrt(a + b*x**2)/(693*b**6) + 128*D*a**4*x**2*sqrt(a + b*x**2)/(693*b**5) - 32*D*a**3*x**4*sqrt(a + b*x **2)/(231*b**4) + 80*D*a**2*x**6*sqrt(a + b*x**2)/(693*b**3) - 10*D*a*x**8 *sqrt(a + b*x**2)/(99*b**2) + D*x**10*sqrt(a + b*x**2)/(11*b), Ne(b, 0)), ((A*x**6/6 + B*x**8/8 + C*x**10/10 + D*x**12/12)/sqrt(a), True))
Time = 0.04 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.61 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} D x^{10}}{11 \, b} - \frac {10 \, \sqrt {b x^{2} + a} D a x^{8}}{99 \, b^{2}} + \frac {\sqrt {b x^{2} + a} C x^{8}}{9 \, b} + \frac {80 \, \sqrt {b x^{2} + a} D a^{2} x^{6}}{693 \, b^{3}} - \frac {8 \, \sqrt {b x^{2} + a} C a x^{6}}{63 \, b^{2}} + \frac {\sqrt {b x^{2} + a} B x^{6}}{7 \, b} - \frac {32 \, \sqrt {b x^{2} + a} D a^{3} x^{4}}{231 \, b^{4}} + \frac {16 \, \sqrt {b x^{2} + a} C a^{2} x^{4}}{105 \, b^{3}} - \frac {6 \, \sqrt {b x^{2} + a} B a x^{4}}{35 \, b^{2}} + \frac {\sqrt {b x^{2} + a} A x^{4}}{5 \, b} + \frac {128 \, \sqrt {b x^{2} + a} D a^{4} x^{2}}{693 \, b^{5}} - \frac {64 \, \sqrt {b x^{2} + a} C a^{3} x^{2}}{315 \, b^{4}} + \frac {8 \, \sqrt {b x^{2} + a} B a^{2} x^{2}}{35 \, b^{3}} - \frac {4 \, \sqrt {b x^{2} + a} A a x^{2}}{15 \, b^{2}} - \frac {256 \, \sqrt {b x^{2} + a} D a^{5}}{693 \, b^{6}} + \frac {128 \, \sqrt {b x^{2} + a} C a^{4}}{315 \, b^{5}} - \frac {16 \, \sqrt {b x^{2} + a} B a^{3}}{35 \, b^{4}} + \frac {8 \, \sqrt {b x^{2} + a} A a^{2}}{15 \, b^{3}} \] Input:
integrate(x^5*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
1/11*sqrt(b*x^2 + a)*D*x^10/b - 10/99*sqrt(b*x^2 + a)*D*a*x^8/b^2 + 1/9*sq rt(b*x^2 + a)*C*x^8/b + 80/693*sqrt(b*x^2 + a)*D*a^2*x^6/b^3 - 8/63*sqrt(b *x^2 + a)*C*a*x^6/b^2 + 1/7*sqrt(b*x^2 + a)*B*x^6/b - 32/231*sqrt(b*x^2 + a)*D*a^3*x^4/b^4 + 16/105*sqrt(b*x^2 + a)*C*a^2*x^4/b^3 - 6/35*sqrt(b*x^2 + a)*B*a*x^4/b^2 + 1/5*sqrt(b*x^2 + a)*A*x^4/b + 128/693*sqrt(b*x^2 + a)*D *a^4*x^2/b^5 - 64/315*sqrt(b*x^2 + a)*C*a^3*x^2/b^4 + 8/35*sqrt(b*x^2 + a) *B*a^2*x^2/b^3 - 4/15*sqrt(b*x^2 + a)*A*a*x^2/b^2 - 256/693*sqrt(b*x^2 + a )*D*a^5/b^6 + 128/315*sqrt(b*x^2 + a)*C*a^4/b^5 - 16/35*sqrt(b*x^2 + a)*B* a^3/b^4 + 8/15*sqrt(b*x^2 + a)*A*a^2/b^3
Time = 0.13 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.20 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=-\frac {{\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} \sqrt {b x^{2} + a}}{b^{6}} + \frac {315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} D - 1925 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} D a + 4950 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} D a^{2} - 6930 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{3} + 5775 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D a^{4} + 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} C b - 1980 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} C a b + 4158 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{2} b - 4620 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} C a^{3} b + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{2} - 2079 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{2} + 3465 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{2} + 693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3} - 2310 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{3}}{3465 \, b^{6}} \] Input:
integrate(x^5*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
-(D*a^5 - C*a^4*b + B*a^3*b^2 - A*a^2*b^3)*sqrt(b*x^2 + a)/b^6 + 1/3465*(3 15*(b*x^2 + a)^(11/2)*D - 1925*(b*x^2 + a)^(9/2)*D*a + 4950*(b*x^2 + a)^(7 /2)*D*a^2 - 6930*(b*x^2 + a)^(5/2)*D*a^3 + 5775*(b*x^2 + a)^(3/2)*D*a^4 + 385*(b*x^2 + a)^(9/2)*C*b - 1980*(b*x^2 + a)^(7/2)*C*a*b + 4158*(b*x^2 + a )^(5/2)*C*a^2*b - 4620*(b*x^2 + a)^(3/2)*C*a^3*b + 495*(b*x^2 + a)^(7/2)*B *b^2 - 2079*(b*x^2 + a)^(5/2)*B*a*b^2 + 3465*(b*x^2 + a)^(3/2)*B*a^2*b^2 + 693*(b*x^2 + a)^(5/2)*A*b^3 - 2310*(b*x^2 + a)^(3/2)*A*a*b^3)/b^6
Time = 2.04 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\sqrt {b\,x^2+a}\,\left (\frac {128\,C\,a^4}{315\,b^5}+\frac {C\,x^8}{9\,b}-\frac {8\,C\,a\,x^6}{63\,b^2}+\frac {16\,C\,a^2\,x^4}{105\,b^3}-\frac {64\,C\,a^3\,x^2}{315\,b^4}\right )-\sqrt {b\,x^2+a}\,\left (\frac {16\,B\,a^3}{35\,b^4}-\frac {B\,x^6}{7\,b}+\frac {6\,B\,a\,x^4}{35\,b^2}-\frac {8\,B\,a^2\,x^2}{35\,b^3}\right )+\sqrt {b\,x^2+a}\,\left (\frac {8\,A\,a^2}{15\,b^3}+\frac {A\,x^4}{5\,b}-\frac {4\,A\,a\,x^2}{15\,b^2}\right )+\frac {\frac {{\left (b\,x^2+a\right )}^{11/2}\,D}{11}-\frac {5\,a\,{\left (b\,x^2+a\right )}^{9/2}\,D}{9}-a^5\,\sqrt {b\,x^2+a}\,D+\frac {5\,a^4\,{\left (b\,x^2+a\right )}^{3/2}\,D}{3}-2\,a^3\,{\left (b\,x^2+a\right )}^{5/2}\,D+\frac {10\,a^2\,{\left (b\,x^2+a\right )}^{7/2}\,D}{7}}{b^6} \] Input:
int((x^5*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2)^(1/2),x)
Output:
(a + b*x^2)^(1/2)*((128*C*a^4)/(315*b^5) + (C*x^8)/(9*b) - (8*C*a*x^6)/(63 *b^2) + (16*C*a^2*x^4)/(105*b^3) - (64*C*a^3*x^2)/(315*b^4)) - (a + b*x^2) ^(1/2)*((16*B*a^3)/(35*b^4) - (B*x^6)/(7*b) + (6*B*a*x^4)/(35*b^2) - (8*B* a^2*x^2)/(35*b^3)) + (a + b*x^2)^(1/2)*((8*A*a^2)/(15*b^3) + (A*x^4)/(5*b) - (4*A*a*x^2)/(15*b^2)) + (((a + b*x^2)^(11/2)*D)/11 - (5*a*(a + b*x^2)^( 9/2)*D)/9 - a^5*(a + b*x^2)^(1/2)*D + (5*a^4*(a + b*x^2)^(3/2)*D)/3 - 2*a^ 3*(a + b*x^2)^(5/2)*D + (10*a^2*(a + b*x^2)^(7/2)*D)/7)/b^6
Time = 0.16 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.74 \[ \int \frac {x^5 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b \,x^{2}+a}\, \left (315 b^{5} d \,x^{10}-350 a \,b^{4} d \,x^{8}+385 b^{5} c \,x^{8}+400 a^{2} b^{3} d \,x^{6}-440 a \,b^{4} c \,x^{6}+495 b^{6} x^{6}-480 a^{3} b^{2} d \,x^{4}+528 a^{2} b^{3} c \,x^{4}+99 a \,b^{5} x^{4}+640 a^{4} b d \,x^{2}-704 a^{3} b^{2} c \,x^{2}-132 a^{2} b^{4} x^{2}-1280 a^{5} d +1408 a^{4} b c +264 a^{3} b^{3}\right )}{3465 b^{6}} \] Input:
int(x^5*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(1/2),x)
Output:
(sqrt(a + b*x**2)*( - 1280*a**5*d + 1408*a**4*b*c + 640*a**4*b*d*x**2 + 26 4*a**3*b**3 - 704*a**3*b**2*c*x**2 - 480*a**3*b**2*d*x**4 - 132*a**2*b**4* x**2 + 528*a**2*b**3*c*x**4 + 400*a**2*b**3*d*x**6 + 99*a*b**5*x**4 - 440* a*b**4*c*x**6 - 350*a*b**4*d*x**8 + 495*b**6*x**6 + 385*b**5*c*x**8 + 315* b**5*d*x**10))/(3465*b**6)