Integrand size = 24, antiderivative size = 71 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=\frac {A b^2 x^{5+m}}{5+m}+\frac {b (b B+2 A c) x^{7+m}}{7+m}+\frac {c (2 b B+A c) x^{9+m}}{9+m}+\frac {B c^2 x^{11+m}}{11+m} \]
A*b^2*x^(5+m)/(5+m)+b*(2*A*c+B*b)*x^(7+m)/(7+m)+c*(A*c+2*B*b)*x^(9+m)/(9+m )+B*c^2*x^(11+m)/(11+m)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=x^{5+m} \left (\frac {A b^2}{5+m}+\frac {b (b B+2 A c) x^2}{7+m}+\frac {c (2 b B+A c) x^4}{9+m}+\frac {B c^2 x^6}{11+m}\right ) \]
x^(5 + m)*((A*b^2)/(5 + m) + (b*(b*B + 2*A*c)*x^2)/(7 + m) + (c*(2*b*B + A *c)*x^4)/(9 + m) + (B*c^2*x^6)/(11 + m))
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {9, 355, 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.
\(\displaystyle \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int x^{m+4} \left (A+B x^2\right ) \left (b+c x^2\right )^2dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (A b^2 x^{m+4}+b x^{m+6} (2 A c+b B)+c x^{m+8} (A c+2 b B)+B c^2 x^{m+10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {A b^2 x^{m+5}}{m+5}+\frac {b x^{m+7} (2 A c+b B)}{m+7}+\frac {c x^{m+9} (A c+2 b B)}{m+9}+\frac {B c^2 x^{m+11}}{m+11}\) |
(A*b^2*x^(5 + m))/(5 + m) + (b*(b*B + 2*A*c)*x^(7 + m))/(7 + m) + (c*(2*b* B + A*c)*x^(9 + m))/(9 + m) + (B*c^2*x^(11 + m))/(11 + m)
3.3.70.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(71)=142\).
Time = 2.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.69
method | result | size |
gosper | \(\frac {x^{5+m} \left (B \,c^{2} m^{3} x^{6}+21 B \,c^{2} m^{2} x^{6}+A \,c^{2} m^{3} x^{4}+2 B b c \,m^{3} x^{4}+143 m \,x^{6} B \,c^{2}+23 A \,c^{2} m^{2} x^{4}+46 B b c \,m^{2} x^{4}+315 B \,c^{2} x^{6}+2 A b c \,m^{3} x^{2}+167 A \,c^{2} m \,x^{4}+B \,b^{2} m^{3} x^{2}+334 B b c m \,x^{4}+50 A b c \,m^{2} x^{2}+385 A \,c^{2} x^{4}+25 B \,b^{2} m^{2} x^{2}+770 x^{4} B b c +A \,b^{2} m^{3}+398 A b c m \,x^{2}+199 B \,b^{2} m \,x^{2}+27 A \,b^{2} m^{2}+990 A b c \,x^{2}+495 b^{2} B \,x^{2}+239 A \,b^{2} m +693 b^{2} A \right )}{\left (5+m \right ) \left (7+m \right ) \left (9+m \right ) \left (11+m \right )}\) | \(262\) |
risch | \(\frac {x^{m} \left (B \,c^{2} m^{3} x^{6}+21 B \,c^{2} m^{2} x^{6}+A \,c^{2} m^{3} x^{4}+2 B b c \,m^{3} x^{4}+143 m \,x^{6} B \,c^{2}+23 A \,c^{2} m^{2} x^{4}+46 B b c \,m^{2} x^{4}+315 B \,c^{2} x^{6}+2 A b c \,m^{3} x^{2}+167 A \,c^{2} m \,x^{4}+B \,b^{2} m^{3} x^{2}+334 B b c m \,x^{4}+50 A b c \,m^{2} x^{2}+385 A \,c^{2} x^{4}+25 B \,b^{2} m^{2} x^{2}+770 x^{4} B b c +A \,b^{2} m^{3}+398 A b c m \,x^{2}+199 B \,b^{2} m \,x^{2}+27 A \,b^{2} m^{2}+990 A b c \,x^{2}+495 b^{2} B \,x^{2}+239 A \,b^{2} m +693 b^{2} A \right ) x^{5}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right )}\) | \(263\) |
parallelrisch | \(\frac {2 B \,x^{9} x^{m} b c \,m^{3}+46 B \,x^{9} x^{m} b c \,m^{2}+25 B \,x^{7} x^{m} b^{2} m^{2}+A \,x^{5} x^{m} b^{2} m^{3}+199 B \,x^{7} x^{m} b^{2} m +990 A \,x^{7} x^{m} b c +27 A \,x^{5} x^{m} b^{2} m^{2}+239 A \,x^{5} x^{m} b^{2} m +315 B \,x^{11} x^{m} c^{2}+385 A \,x^{9} x^{m} c^{2}+495 B \,x^{7} x^{m} b^{2}+693 A \,x^{5} x^{m} b^{2}+2 A \,x^{7} x^{m} b c \,m^{3}+334 B \,x^{9} x^{m} b c m +50 A \,x^{7} x^{m} b c \,m^{2}+398 A \,x^{7} x^{m} b c m +B \,x^{11} x^{m} c^{2} m^{3}+21 B \,x^{11} x^{m} c^{2} m^{2}+A \,x^{9} x^{m} c^{2} m^{3}+143 B \,x^{11} x^{m} c^{2} m +23 A \,x^{9} x^{m} c^{2} m^{2}+167 A \,x^{9} x^{m} c^{2} m +B \,x^{7} x^{m} b^{2} m^{3}+770 B \,x^{9} x^{m} b c}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right )}\) | \(341\) |
x^(5+m)/(5+m)/(7+m)/(9+m)/(11+m)*(B*c^2*m^3*x^6+21*B*c^2*m^2*x^6+A*c^2*m^3 *x^4+2*B*b*c*m^3*x^4+143*B*c^2*m*x^6+23*A*c^2*m^2*x^4+46*B*b*c*m^2*x^4+315 *B*c^2*x^6+2*A*b*c*m^3*x^2+167*A*c^2*m*x^4+B*b^2*m^3*x^2+334*B*b*c*m*x^4+5 0*A*b*c*m^2*x^2+385*A*c^2*x^4+25*B*b^2*m^2*x^2+770*B*b*c*x^4+A*b^2*m^3+398 *A*b*c*m*x^2+199*B*b^2*m*x^2+27*A*b^2*m^2+990*A*b*c*x^2+495*B*b^2*x^2+239* A*b^2*m+693*A*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (71) = 142\).
Time = 0.31 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.06 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=\frac {{\left ({\left (B c^{2} m^{3} + 21 \, B c^{2} m^{2} + 143 \, B c^{2} m + 315 \, B c^{2}\right )} x^{11} + {\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 770 \, B b c + 385 \, A c^{2} + 23 \, {\left (2 \, B b c + A c^{2}\right )} m^{2} + 167 \, {\left (2 \, B b c + A c^{2}\right )} m\right )} x^{9} + {\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 495 \, B b^{2} + 990 \, A b c + 25 \, {\left (B b^{2} + 2 \, A b c\right )} m^{2} + 199 \, {\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{7} + {\left (A b^{2} m^{3} + 27 \, A b^{2} m^{2} + 239 \, A b^{2} m + 693 \, A b^{2}\right )} x^{5}\right )} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]
((B*c^2*m^3 + 21*B*c^2*m^2 + 143*B*c^2*m + 315*B*c^2)*x^11 + ((2*B*b*c + A *c^2)*m^3 + 770*B*b*c + 385*A*c^2 + 23*(2*B*b*c + A*c^2)*m^2 + 167*(2*B*b* c + A*c^2)*m)*x^9 + ((B*b^2 + 2*A*b*c)*m^3 + 495*B*b^2 + 990*A*b*c + 25*(B *b^2 + 2*A*b*c)*m^2 + 199*(B*b^2 + 2*A*b*c)*m)*x^7 + (A*b^2*m^3 + 27*A*b^2 *m^2 + 239*A*b^2*m + 693*A*b^2)*x^5)*x^m/(m^4 + 32*m^3 + 374*m^2 + 1888*m + 3465)
Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (63) = 126\).
Time = 0.67 (sec) , antiderivative size = 1051, normalized size of antiderivative = 14.80 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=\begin {cases} - \frac {A b^{2}}{6 x^{6}} - \frac {A b c}{2 x^{4}} - \frac {A c^{2}}{2 x^{2}} - \frac {B b^{2}}{4 x^{4}} - \frac {B b c}{x^{2}} + B c^{2} \log {\left (x \right )} & \text {for}\: m = -11 \\- \frac {A b^{2}}{4 x^{4}} - \frac {A b c}{x^{2}} + A c^{2} \log {\left (x \right )} - \frac {B b^{2}}{2 x^{2}} + 2 B b c \log {\left (x \right )} + \frac {B c^{2} x^{2}}{2} & \text {for}\: m = -9 \\- \frac {A b^{2}}{2 x^{2}} + 2 A b c \log {\left (x \right )} + \frac {A c^{2} x^{2}}{2} + B b^{2} \log {\left (x \right )} + B b c x^{2} + \frac {B c^{2} x^{4}}{4} & \text {for}\: m = -7 \\A b^{2} \log {\left (x \right )} + A b c x^{2} + \frac {A c^{2} x^{4}}{4} + \frac {B b^{2} x^{2}}{2} + \frac {B b c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} & \text {for}\: m = -5 \\\frac {A b^{2} m^{3} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {27 A b^{2} m^{2} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {239 A b^{2} m x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {693 A b^{2} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {2 A b c m^{3} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {50 A b c m^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {398 A b c m x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {990 A b c x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {A c^{2} m^{3} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {23 A c^{2} m^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {167 A c^{2} m x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {385 A c^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {B b^{2} m^{3} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {25 B b^{2} m^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {199 B b^{2} m x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {495 B b^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {2 B b c m^{3} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {46 B b c m^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {334 B b c m x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {770 B b c x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {B c^{2} m^{3} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {21 B c^{2} m^{2} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {143 B c^{2} m x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {315 B c^{2} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} & \text {otherwise} \end {cases} \]
Piecewise((-A*b**2/(6*x**6) - A*b*c/(2*x**4) - A*c**2/(2*x**2) - B*b**2/(4 *x**4) - B*b*c/x**2 + B*c**2*log(x), Eq(m, -11)), (-A*b**2/(4*x**4) - A*b* c/x**2 + A*c**2*log(x) - B*b**2/(2*x**2) + 2*B*b*c*log(x) + B*c**2*x**2/2, Eq(m, -9)), (-A*b**2/(2*x**2) + 2*A*b*c*log(x) + A*c**2*x**2/2 + B*b**2*l og(x) + B*b*c*x**2 + B*c**2*x**4/4, Eq(m, -7)), (A*b**2*log(x) + A*b*c*x** 2 + A*c**2*x**4/4 + B*b**2*x**2/2 + B*b*c*x**4/2 + B*c**2*x**6/6, Eq(m, -5 )), (A*b**2*m**3*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 2 7*A*b**2*m**2*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 239* A*b**2*m*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 693*A*b** 2*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 2*A*b*c*m**3*x** 7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 50*A*b*c*m**2*x**7*x* *m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 398*A*b*c*m*x**7*x**m/(m* *4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 990*A*b*c*x**7*x**m/(m**4 + 32* m**3 + 374*m**2 + 1888*m + 3465) + A*c**2*m**3*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 23*A*c**2*m**2*x**9*x**m/(m**4 + 32*m**3 + 37 4*m**2 + 1888*m + 3465) + 167*A*c**2*m*x**9*x**m/(m**4 + 32*m**3 + 374*m** 2 + 1888*m + 3465) + 385*A*c**2*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 188 8*m + 3465) + B*b**2*m**3*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 25*B*b**2*m**2*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 346 5) + 199*B*b**2*m*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465)...
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.28 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=\frac {B c^{2} x^{m + 11}}{m + 11} + \frac {2 \, B b c x^{m + 9}}{m + 9} + \frac {A c^{2} x^{m + 9}}{m + 9} + \frac {B b^{2} x^{m + 7}}{m + 7} + \frac {2 \, A b c x^{m + 7}}{m + 7} + \frac {A b^{2} x^{m + 5}}{m + 5} \]
B*c^2*x^(m + 11)/(m + 11) + 2*B*b*c*x^(m + 9)/(m + 9) + A*c^2*x^(m + 9)/(m + 9) + B*b^2*x^(m + 7)/(m + 7) + 2*A*b*c*x^(m + 7)/(m + 7) + A*b^2*x^(m + 5)/(m + 5)
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (71) = 142\).
Time = 0.27 (sec) , antiderivative size = 340, normalized size of antiderivative = 4.79 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=\frac {B c^{2} m^{3} x^{11} x^{m} + 21 \, B c^{2} m^{2} x^{11} x^{m} + 2 \, B b c m^{3} x^{9} x^{m} + A c^{2} m^{3} x^{9} x^{m} + 143 \, B c^{2} m x^{11} x^{m} + 46 \, B b c m^{2} x^{9} x^{m} + 23 \, A c^{2} m^{2} x^{9} x^{m} + 315 \, B c^{2} x^{11} x^{m} + B b^{2} m^{3} x^{7} x^{m} + 2 \, A b c m^{3} x^{7} x^{m} + 334 \, B b c m x^{9} x^{m} + 167 \, A c^{2} m x^{9} x^{m} + 25 \, B b^{2} m^{2} x^{7} x^{m} + 50 \, A b c m^{2} x^{7} x^{m} + 770 \, B b c x^{9} x^{m} + 385 \, A c^{2} x^{9} x^{m} + A b^{2} m^{3} x^{5} x^{m} + 199 \, B b^{2} m x^{7} x^{m} + 398 \, A b c m x^{7} x^{m} + 27 \, A b^{2} m^{2} x^{5} x^{m} + 495 \, B b^{2} x^{7} x^{m} + 990 \, A b c x^{7} x^{m} + 239 \, A b^{2} m x^{5} x^{m} + 693 \, A b^{2} x^{5} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]
(B*c^2*m^3*x^11*x^m + 21*B*c^2*m^2*x^11*x^m + 2*B*b*c*m^3*x^9*x^m + A*c^2* m^3*x^9*x^m + 143*B*c^2*m*x^11*x^m + 46*B*b*c*m^2*x^9*x^m + 23*A*c^2*m^2*x ^9*x^m + 315*B*c^2*x^11*x^m + B*b^2*m^3*x^7*x^m + 2*A*b*c*m^3*x^7*x^m + 33 4*B*b*c*m*x^9*x^m + 167*A*c^2*m*x^9*x^m + 25*B*b^2*m^2*x^7*x^m + 50*A*b*c* m^2*x^7*x^m + 770*B*b*c*x^9*x^m + 385*A*c^2*x^9*x^m + A*b^2*m^3*x^5*x^m + 199*B*b^2*m*x^7*x^m + 398*A*b*c*m*x^7*x^m + 27*A*b^2*m^2*x^5*x^m + 495*B*b ^2*x^7*x^m + 990*A*b*c*x^7*x^m + 239*A*b^2*m*x^5*x^m + 693*A*b^2*x^5*x^m)/ (m^4 + 32*m^3 + 374*m^2 + 1888*m + 3465)
Time = 9.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.52 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx=x^m\,\left (\frac {A\,b^2\,x^5\,\left (m^3+27\,m^2+239\,m+693\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {B\,c^2\,x^{11}\,\left (m^3+21\,m^2+143\,m+315\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {b\,x^7\,\left (2\,A\,c+B\,b\right )\,\left (m^3+25\,m^2+199\,m+495\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {c\,x^9\,\left (A\,c+2\,B\,b\right )\,\left (m^3+23\,m^2+167\,m+385\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}\right ) \]
x^m*((A*b^2*x^5*(239*m + 27*m^2 + m^3 + 693))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (B*c^2*x^11*(143*m + 21*m^2 + m^3 + 315))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (b*x^7*(2*A*c + B*b)*(199*m + 25*m^2 + m^3 + 495 ))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (c*x^9*(A*c + 2*B*b)*(167*m + 23*m^2 + m^3 + 385))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465))