Integrand size = 22, antiderivative size = 103 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {a^2 (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {B \left (a+b x^2\right )^{13/2}}{13 b^4} \]
1/7*a^2*(A*b-B*a)*(b*x^2+a)^(7/2)/b^4-1/9*a*(2*A*b-3*B*a)*(b*x^2+a)^(9/2)/ b^4+1/11*(A*b-3*B*a)*(b*x^2+a)^(11/2)/b^4+1/13*B*(b*x^2+a)^(13/2)/b^4
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (104 a^2 A b-48 a^3 B-364 a A b^2 x^2+168 a^2 b B x^2+819 A b^3 x^4-378 a b^2 B x^4+693 b^3 B x^6\right )}{9009 b^4} \]
((a + b*x^2)^(7/2)*(104*a^2*A*b - 48*a^3*B - 364*a*A*b^2*x^2 + 168*a^2*b*B *x^2 + 819*A*b^3*x^4 - 378*a*b^2*B*x^4 + 693*b^3*B*x^6))/(9009*b^4)
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 86, 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.
\(\displaystyle \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int x^4 \left (b x^2+a\right )^{5/2} \left (B x^2+A\right )dx^2\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{2} \int \left (\frac {B \left (b x^2+a\right )^{11/2}}{b^3}+\frac {(A b-3 a B) \left (b x^2+a\right )^{9/2}}{b^3}+\frac {a (3 a B-2 A b) \left (b x^2+a\right )^{7/2}}{b^3}-\frac {a^2 (a B-A b) \left (b x^2+a\right )^{5/2}}{b^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {2 a^2 \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^4}+\frac {2 \left (a+b x^2\right )^{11/2} (A b-3 a B)}{11 b^4}-\frac {2 a \left (a+b x^2\right )^{9/2} (2 A b-3 a B)}{9 b^4}+\frac {2 B \left (a+b x^2\right )^{13/2}}{13 b^4}\right )\) |
((2*a^2*(A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x^2)^(9/2))/(9*b^4) + (2*(A*b - 3*a*B)*(a + b*x^2)^(11/2))/(11*b^4) + ( 2*B*(a + b*x^2)^(13/2))/(13*b^4))/2
3.6.38.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.82 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {8 \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (\frac {63 x^{4} \left (\frac {11 x^{2} B}{13}+A \right ) b^{3}}{8}-\frac {7 x^{2} \left (\frac {27 x^{2} B}{26}+A \right ) a \,b^{2}}{2}+a^{2} \left (\frac {21 x^{2} B}{13}+A \right ) b -\frac {6 a^{3} B}{13}\right )}{693 b^{4}}\) | \(68\) |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (693 b^{3} B \,x^{6}+819 A \,b^{3} x^{4}-378 B a \,b^{2} x^{4}-364 a A \,b^{2} x^{2}+168 B \,a^{2} b \,x^{2}+104 a^{2} b A -48 a^{3} B \right )}{9009 b^{4}}\) | \(77\) |
default | \(B \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )}{13 b}\right )+A \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )\) | \(144\) |
trager | \(\frac {\left (693 B \,b^{6} x^{12}+819 A \,b^{6} x^{10}+1701 B a \,b^{5} x^{10}+2093 A a \,b^{5} x^{8}+1113 B \,a^{2} b^{4} x^{8}+1469 A \,a^{2} b^{4} x^{6}+15 B \,a^{3} b^{3} x^{6}+39 A \,a^{3} b^{3} x^{4}-18 B \,a^{4} b^{2} x^{4}-52 A \,a^{4} b^{2} x^{2}+24 B \,a^{5} b \,x^{2}+104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b \,x^{2}+a}}{9009 b^{4}}\) | \(149\) |
risch | \(\frac {\left (693 B \,b^{6} x^{12}+819 A \,b^{6} x^{10}+1701 B a \,b^{5} x^{10}+2093 A a \,b^{5} x^{8}+1113 B \,a^{2} b^{4} x^{8}+1469 A \,a^{2} b^{4} x^{6}+15 B \,a^{3} b^{3} x^{6}+39 A \,a^{3} b^{3} x^{4}-18 B \,a^{4} b^{2} x^{4}-52 A \,a^{4} b^{2} x^{2}+24 B \,a^{5} b \,x^{2}+104 A \,a^{5} b -48 B \,a^{6}\right ) \sqrt {b \,x^{2}+a}}{9009 b^{4}}\) | \(149\) |
8/693*(b*x^2+a)^(7/2)*(63/8*x^4*(11/13*x^2*B+A)*b^3-7/2*x^2*(27/26*x^2*B+A )*a*b^2+a^2*(21/13*x^2*B+A)*b-6/13*a^3*B)/b^4
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.43 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {{\left (693 \, B b^{6} x^{12} + 63 \, {\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{10} + 7 \, {\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{8} - 48 \, B a^{6} + 104 \, A a^{5} b + {\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{6} - 3 \, {\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 4 \, {\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{9009 \, b^{4}} \]
1/9009*(693*B*b^6*x^12 + 63*(27*B*a*b^5 + 13*A*b^6)*x^10 + 7*(159*B*a^2*b^ 4 + 299*A*a*b^5)*x^8 - 48*B*a^6 + 104*A*a^5*b + (15*B*a^3*b^3 + 1469*A*a^2 *b^4)*x^6 - 3*(6*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 4*(6*B*a^5*b - 13*A*a^4*b ^2)*x^2)*sqrt(b*x^2 + a)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (94) = 188\).
Time = 0.68 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.04 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\begin {cases} \frac {8 A a^{5} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {4 A a^{4} x^{2} \sqrt {a + b x^{2}}}{693 b^{2}} + \frac {A a^{3} x^{4} \sqrt {a + b x^{2}}}{231 b} + \frac {113 A a^{2} x^{6} \sqrt {a + b x^{2}}}{693} + \frac {23 A a b x^{8} \sqrt {a + b x^{2}}}{99} + \frac {A b^{2} x^{10} \sqrt {a + b x^{2}}}{11} - \frac {16 B a^{6} \sqrt {a + b x^{2}}}{3003 b^{4}} + \frac {8 B a^{5} x^{2} \sqrt {a + b x^{2}}}{3003 b^{3}} - \frac {2 B a^{4} x^{4} \sqrt {a + b x^{2}}}{1001 b^{2}} + \frac {5 B a^{3} x^{6} \sqrt {a + b x^{2}}}{3003 b} + \frac {53 B a^{2} x^{8} \sqrt {a + b x^{2}}}{429} + \frac {27 B a b x^{10} \sqrt {a + b x^{2}}}{143} + \frac {B b^{2} x^{12} \sqrt {a + b x^{2}}}{13} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{6}}{6} + \frac {B x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
Piecewise((8*A*a**5*sqrt(a + b*x**2)/(693*b**3) - 4*A*a**4*x**2*sqrt(a + b *x**2)/(693*b**2) + A*a**3*x**4*sqrt(a + b*x**2)/(231*b) + 113*A*a**2*x**6 *sqrt(a + b*x**2)/693 + 23*A*a*b*x**8*sqrt(a + b*x**2)/99 + A*b**2*x**10*s qrt(a + b*x**2)/11 - 16*B*a**6*sqrt(a + b*x**2)/(3003*b**4) + 8*B*a**5*x** 2*sqrt(a + b*x**2)/(3003*b**3) - 2*B*a**4*x**4*sqrt(a + b*x**2)/(1001*b**2 ) + 5*B*a**3*x**6*sqrt(a + b*x**2)/(3003*b) + 53*B*a**2*x**8*sqrt(a + b*x* *2)/429 + 27*B*a*b*x**10*sqrt(a + b*x**2)/143 + B*b**2*x**12*sqrt(a + b*x* *2)/13, Ne(b, 0)), (a**(5/2)*(A*x**6/6 + B*x**8/8), True))
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{6}}{13 \, b} - \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x^{4}}{143 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{4}}{11 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} x^{2}}{429 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a x^{2}}{99 \, b^{2}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{3}}{3003 \, b^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a^{2}}{693 \, b^{3}} \]
1/13*(b*x^2 + a)^(7/2)*B*x^6/b - 6/143*(b*x^2 + a)^(7/2)*B*a*x^4/b^2 + 1/1 1*(b*x^2 + a)^(7/2)*A*x^4/b + 8/429*(b*x^2 + a)^(7/2)*B*a^2*x^2/b^3 - 4/99 *(b*x^2 + a)^(7/2)*A*a*x^2/b^2 - 16/3003*(b*x^2 + a)^(7/2)*B*a^3/b^4 + 8/6 93*(b*x^2 + a)^(7/2)*A*a^2/b^3
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {693 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} B - 2457 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} B a + 3003 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a^{2} - 1287 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{3} + 819 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} A b - 2002 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A a b + 1287 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a^{2} b}{9009 \, b^{4}} \]
1/9009*(693*(b*x^2 + a)^(13/2)*B - 2457*(b*x^2 + a)^(11/2)*B*a + 3003*(b*x ^2 + a)^(9/2)*B*a^2 - 1287*(b*x^2 + a)^(7/2)*B*a^3 + 819*(b*x^2 + a)^(11/2 )*A*b - 2002*(b*x^2 + a)^(9/2)*A*a*b + 1287*(b*x^2 + a)^(7/2)*A*a^2*b)/b^4
Time = 5.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int x^5 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\sqrt {b\,x^2+a}\,\left (\frac {B\,b^2\,x^{12}}{13}-\frac {48\,B\,a^6-104\,A\,a^5\,b}{9009\,b^4}+\frac {x^{10}\,\left (819\,A\,b^6+1701\,B\,a\,b^5\right )}{9009\,b^4}+\frac {a\,x^8\,\left (299\,A\,b+159\,B\,a\right )}{1287}+\frac {a^3\,x^4\,\left (13\,A\,b-6\,B\,a\right )}{3003\,b^2}-\frac {4\,a^4\,x^2\,\left (13\,A\,b-6\,B\,a\right )}{9009\,b^3}+\frac {a^2\,x^6\,\left (1469\,A\,b+15\,B\,a\right )}{9009\,b}\right ) \]