Integrand size = 22, antiderivative size = 91 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {a^2 A (c x)^{1+m}}{c (1+m)}+\frac {a (2 A b+a B) (c x)^{3+m}}{c^3 (3+m)}+\frac {b (A b+2 a B) (c x)^{5+m}}{c^5 (5+m)}+\frac {b^2 B (c x)^{7+m}}{c^7 (7+m)} \] Output:
a^2*A*(c*x)^(1+m)/c/(1+m)+a*(2*A*b+B*a)*(c*x)^(3+m)/c^3/(3+m)+b*(A*b+2*B*a )*(c*x)^(5+m)/c^5/(5+m)+b^2*B*(c*x)^(7+m)/c^7/(7+m)
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x (c x)^m \left (\frac {a^2 A}{1+m}+\frac {a (2 A b+a B) x^2}{3+m}+\frac {b (A b+2 a B) x^4}{5+m}+\frac {b^2 B x^6}{7+m}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^2)^2*(A + B*x^2),x]
Output:
x*(c*x)^m*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(3 + m) + (b*(A*b + 2*a *B)*x^4)/(5 + m) + (b^2*B*x^6)/(7 + m))
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.
\(\displaystyle \int \left (a+b x^2\right )^2 \left (A+B x^2\right ) (c x)^m \, dx\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \int \left (a^2 A (c x)^m+\frac {b (c x)^{m+4} (2 a B+A b)}{c^4}+\frac {a (c x)^{m+2} (a B+2 A b)}{c^2}+\frac {b^2 B (c x)^{m+6}}{c^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 A (c x)^{m+1}}{c (m+1)}+\frac {b (c x)^{m+5} (2 a B+A b)}{c^5 (m+5)}+\frac {a (c x)^{m+3} (a B+2 A b)}{c^3 (m+3)}+\frac {b^2 B (c x)^{m+7}}{c^7 (m+7)}\) |
Input:
Int[(c*x)^m*(a + b*x^2)^2*(A + B*x^2),x]
Output:
(a^2*A*(c*x)^(1 + m))/(c*(1 + m)) + (a*(2*A*b + a*B)*(c*x)^(3 + m))/(c^3*( 3 + m)) + (b*(A*b + 2*a*B)*(c*x)^(5 + m))/(c^5*(5 + m)) + (b^2*B*(c*x)^(7 + m))/(c^7*(7 + m))
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]
Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {B \,b^{2} x^{7} {\mathrm e}^{m \ln \left (c x \right )}}{7+m}+\frac {a \left (2 A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (c x \right )}}{3+m}+\frac {a^{2} A x \,{\mathrm e}^{m \ln \left (c x \right )}}{1+m}+\frac {b \left (A b +2 B a \right ) x^{5} {\mathrm e}^{m \ln \left (c x \right )}}{5+m}\) | \(90\) |
gosper | \(\frac {x \left (B \,b^{2} m^{3} x^{6}+9 B \,b^{2} m^{2} x^{6}+A \,b^{2} m^{3} x^{4}+2 B a b \,m^{3} x^{4}+23 m \,x^{6} B \,b^{2}+11 A \,b^{2} m^{2} x^{4}+22 B a b \,m^{2} x^{4}+15 b^{2} B \,x^{6}+2 A a b \,m^{3} x^{2}+31 A \,b^{2} x^{4} m +B \,a^{2} m^{3} x^{2}+62 B a b \,x^{4} m +26 A a b \,m^{2} x^{2}+21 A \,b^{2} x^{4}+13 B \,a^{2} m^{2} x^{2}+42 B a b \,x^{4}+A \,a^{2} m^{3}+94 a A b \,x^{2} m +47 B \,a^{2} x^{2} m +15 A \,a^{2} m^{2}+70 a A b \,x^{2}+35 B \,a^{2} x^{2}+71 a^{2} A m +105 a^{2} A \right ) \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(263\) |
risch | \(\frac {x \left (B \,b^{2} m^{3} x^{6}+9 B \,b^{2} m^{2} x^{6}+A \,b^{2} m^{3} x^{4}+2 B a b \,m^{3} x^{4}+23 m \,x^{6} B \,b^{2}+11 A \,b^{2} m^{2} x^{4}+22 B a b \,m^{2} x^{4}+15 b^{2} B \,x^{6}+2 A a b \,m^{3} x^{2}+31 A \,b^{2} x^{4} m +B \,a^{2} m^{3} x^{2}+62 B a b \,x^{4} m +26 A a b \,m^{2} x^{2}+21 A \,b^{2} x^{4}+13 B \,a^{2} m^{2} x^{2}+42 B a b \,x^{4}+A \,a^{2} m^{3}+94 a A b \,x^{2} m +47 B \,a^{2} x^{2} m +15 A \,a^{2} m^{2}+70 a A b \,x^{2}+35 B \,a^{2} x^{2}+71 a^{2} A m +105 a^{2} A \right ) \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(263\) |
orering | \(\frac {x \left (B \,b^{2} m^{3} x^{6}+9 B \,b^{2} m^{2} x^{6}+A \,b^{2} m^{3} x^{4}+2 B a b \,m^{3} x^{4}+23 m \,x^{6} B \,b^{2}+11 A \,b^{2} m^{2} x^{4}+22 B a b \,m^{2} x^{4}+15 b^{2} B \,x^{6}+2 A a b \,m^{3} x^{2}+31 A \,b^{2} x^{4} m +B \,a^{2} m^{3} x^{2}+62 B a b \,x^{4} m +26 A a b \,m^{2} x^{2}+21 A \,b^{2} x^{4}+13 B \,a^{2} m^{2} x^{2}+42 B a b \,x^{4}+A \,a^{2} m^{3}+94 a A b \,x^{2} m +47 B \,a^{2} x^{2} m +15 A \,a^{2} m^{2}+70 a A b \,x^{2}+35 B \,a^{2} x^{2}+71 a^{2} A m +105 a^{2} A \right ) \left (c x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(263\) |
parallelrisch | \(\frac {26 A \,x^{3} \left (c x \right )^{m} a b \,m^{2}+2 B \,x^{5} \left (c x \right )^{m} a b \,m^{3}+22 B \,x^{5} \left (c x \right )^{m} a b \,m^{2}+2 A \,x^{3} \left (c x \right )^{m} a b \,m^{3}+B \,x^{7} \left (c x \right )^{m} b^{2} m^{3}+9 B \,x^{7} \left (c x \right )^{m} b^{2} m^{2}+15 B \,x^{7} \left (c x \right )^{m} b^{2}+21 A \,x^{5} \left (c x \right )^{m} b^{2}+35 B \,x^{3} \left (c x \right )^{m} a^{2}+105 A x \left (c x \right )^{m} a^{2}+94 A \,x^{3} \left (c x \right )^{m} a b m +62 B \,x^{5} \left (c x \right )^{m} a b m +A \,x^{5} \left (c x \right )^{m} b^{2} m^{3}+23 B \,x^{7} \left (c x \right )^{m} b^{2} m +11 A \,x^{5} \left (c x \right )^{m} b^{2} m^{2}+31 A \,x^{5} \left (c x \right )^{m} b^{2} m +B \,x^{3} \left (c x \right )^{m} a^{2} m^{3}+42 B \,x^{5} \left (c x \right )^{m} a b +13 B \,x^{3} \left (c x \right )^{m} a^{2} m^{2}+A x \left (c x \right )^{m} a^{2} m^{3}+47 B \,x^{3} \left (c x \right )^{m} a^{2} m +70 A \,x^{3} \left (c x \right )^{m} a b +15 A x \left (c x \right )^{m} a^{2} m^{2}+71 A x \left (c x \right )^{m} a^{2} m}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(381\) |
Input:
int((c*x)^m*(b*x^2+a)^2*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
B*b^2/(7+m)*x^7*exp(m*ln(c*x))+a*(2*A*b+B*a)/(3+m)*x^3*exp(m*ln(c*x))+a^2* A/(1+m)*x*exp(m*ln(c*x))+b*(A*b+2*B*a)/(5+m)*x^5*exp(m*ln(c*x))
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (91) = 182\).
Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.38 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {{\left ({\left (B b^{2} m^{3} + 9 \, B b^{2} m^{2} + 23 \, B b^{2} m + 15 \, B b^{2}\right )} x^{7} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 42 \, B a b + 21 \, A b^{2} + 11 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 31 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 35 \, B a^{2} + 70 \, A a b + 13 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 47 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{3} + {\left (A a^{2} m^{3} + 15 \, A a^{2} m^{2} + 71 \, A a^{2} m + 105 \, A a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(B*x^2+A),x, algorithm="fricas")
Output:
((B*b^2*m^3 + 9*B*b^2*m^2 + 23*B*b^2*m + 15*B*b^2)*x^7 + ((2*B*a*b + A*b^2 )*m^3 + 42*B*a*b + 21*A*b^2 + 11*(2*B*a*b + A*b^2)*m^2 + 31*(2*B*a*b + A*b ^2)*m)*x^5 + ((B*a^2 + 2*A*a*b)*m^3 + 35*B*a^2 + 70*A*a*b + 13*(B*a^2 + 2* A*a*b)*m^2 + 47*(B*a^2 + 2*A*a*b)*m)*x^3 + (A*a^2*m^3 + 15*A*a^2*m^2 + 71* A*a^2*m + 105*A*a^2)*x)*(c*x)^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (82) = 164\).
Time = 0.41 (sec) , antiderivative size = 1096, normalized size of antiderivative = 12.04 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(b*x**2+a)**2*(B*x**2+A),x)
Output:
Piecewise(((-A*a**2/(6*x**6) - A*a*b/(2*x**4) - A*b**2/(2*x**2) - B*a**2/( 4*x**4) - B*a*b/x**2 + B*b**2*log(x))/c**7, Eq(m, -7)), ((-A*a**2/(4*x**4) - A*a*b/x**2 + A*b**2*log(x) - B*a**2/(2*x**2) + 2*B*a*b*log(x) + B*b**2* x**2/2)/c**5, Eq(m, -5)), ((-A*a**2/(2*x**2) + 2*A*a*b*log(x) + A*b**2*x** 2/2 + B*a**2*log(x) + B*a*b*x**2 + B*b**2*x**4/4)/c**3, Eq(m, -3)), ((A*a* *2*log(x) + A*a*b*x**2 + A*b**2*x**4/4 + B*a**2*x**2/2 + B*a*b*x**4/2 + B* b**2*x**6/6)/c, Eq(m, -1)), (A*a**2*m**3*x*(c*x)**m/(m**4 + 16*m**3 + 86*m **2 + 176*m + 105) + 15*A*a**2*m**2*x*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 71*A*a**2*m*x*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*A*a**2*x*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2 *A*a*b*m**3*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 26*A* a*b*m**2*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 94*A*a*b *m*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 70*A*a*b*x**3* (c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*b**2*m**3*x**5*(c*x) **m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*A*b**2*m**2*x**5*(c*x)** m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*A*b**2*m*x**5*(c*x)**m/(m* *4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*A*b**2*x**5*(c*x)**m/(m**4 + 16 *m**3 + 86*m**2 + 176*m + 105) + B*a**2*m**3*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*B*a**2*m**2*x**3*(c*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*B*a**2*m*x**3*(c*x)**m/(m**4 + 16*m**3 + 8...
Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {B b^{2} c^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B a b c^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} c^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} c^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b c^{m} x^{3} x^{m}}{m + 3} + \frac {\left (c x\right )^{m + 1} A a^{2}}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(B*x^2+A),x, algorithm="maxima")
Output:
B*b^2*c^m*x^7*x^m/(m + 7) + 2*B*a*b*c^m*x^5*x^m/(m + 5) + A*b^2*c^m*x^5*x^ m/(m + 5) + B*a^2*c^m*x^3*x^m/(m + 3) + 2*A*a*b*c^m*x^3*x^m/(m + 3) + (c*x )^(m + 1)*A*a^2/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (91) = 182\).
Time = 0.14 (sec) , antiderivative size = 380, normalized size of antiderivative = 4.18 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {\left (c x\right )^{m} B b^{2} m^{3} x^{7} + 9 \, \left (c x\right )^{m} B b^{2} m^{2} x^{7} + 2 \, \left (c x\right )^{m} B a b m^{3} x^{5} + \left (c x\right )^{m} A b^{2} m^{3} x^{5} + 23 \, \left (c x\right )^{m} B b^{2} m x^{7} + 22 \, \left (c x\right )^{m} B a b m^{2} x^{5} + 11 \, \left (c x\right )^{m} A b^{2} m^{2} x^{5} + 15 \, \left (c x\right )^{m} B b^{2} x^{7} + \left (c x\right )^{m} B a^{2} m^{3} x^{3} + 2 \, \left (c x\right )^{m} A a b m^{3} x^{3} + 62 \, \left (c x\right )^{m} B a b m x^{5} + 31 \, \left (c x\right )^{m} A b^{2} m x^{5} + 13 \, \left (c x\right )^{m} B a^{2} m^{2} x^{3} + 26 \, \left (c x\right )^{m} A a b m^{2} x^{3} + 42 \, \left (c x\right )^{m} B a b x^{5} + 21 \, \left (c x\right )^{m} A b^{2} x^{5} + \left (c x\right )^{m} A a^{2} m^{3} x + 47 \, \left (c x\right )^{m} B a^{2} m x^{3} + 94 \, \left (c x\right )^{m} A a b m x^{3} + 15 \, \left (c x\right )^{m} A a^{2} m^{2} x + 35 \, \left (c x\right )^{m} B a^{2} x^{3} + 70 \, \left (c x\right )^{m} A a b x^{3} + 71 \, \left (c x\right )^{m} A a^{2} m x + 105 \, \left (c x\right )^{m} A a^{2} x}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(B*x^2+A),x, algorithm="giac")
Output:
((c*x)^m*B*b^2*m^3*x^7 + 9*(c*x)^m*B*b^2*m^2*x^7 + 2*(c*x)^m*B*a*b*m^3*x^5 + (c*x)^m*A*b^2*m^3*x^5 + 23*(c*x)^m*B*b^2*m*x^7 + 22*(c*x)^m*B*a*b*m^2*x ^5 + 11*(c*x)^m*A*b^2*m^2*x^5 + 15*(c*x)^m*B*b^2*x^7 + (c*x)^m*B*a^2*m^3*x ^3 + 2*(c*x)^m*A*a*b*m^3*x^3 + 62*(c*x)^m*B*a*b*m*x^5 + 31*(c*x)^m*A*b^2*m *x^5 + 13*(c*x)^m*B*a^2*m^2*x^3 + 26*(c*x)^m*A*a*b*m^2*x^3 + 42*(c*x)^m*B* a*b*x^5 + 21*(c*x)^m*A*b^2*x^5 + (c*x)^m*A*a^2*m^3*x + 47*(c*x)^m*B*a^2*m* x^3 + 94*(c*x)^m*A*a*b*m*x^3 + 15*(c*x)^m*A*a^2*m^2*x + 35*(c*x)^m*B*a^2*x ^3 + 70*(c*x)^m*A*a*b*x^3 + 71*(c*x)^m*A*a^2*m*x + 105*(c*x)^m*A*a^2*x)/(m ^4 + 16*m^3 + 86*m^2 + 176*m + 105)
Time = 0.59 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx={\left (c\,x\right )}^m\,\left (\frac {B\,b^2\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,a^2\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a\,x^3\,\left (2\,A\,b+B\,a\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b\,x^5\,\left (A\,b+2\,B\,a\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \] Input:
int((A + B*x^2)*(c*x)^m*(a + b*x^2)^2,x)
Output:
(c*x)^m*((B*b^2*x^7*(23*m + 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (A*a^2*x*(71*m + 15*m^2 + m^3 + 105))/(176*m + 86*m^2 + 16*m^ 3 + m^4 + 105) + (a*x^3*(2*A*b + B*a)*(47*m + 13*m^2 + m^3 + 35))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (b*x^5*(A*b + 2*B*a)*(31*m + 11*m^2 + m^3 + 21))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105))
Time = 0.23 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {x^{m} c^{m} x \left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 b^{3} m \,x^{6}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 a \,b^{2} m \,x^{4}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 a^{2} b m \,x^{2}+15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 a^{3} m +105 a^{3}\right )}{m^{4}+16 m^{3}+86 m^{2}+176 m +105} \] Input:
int((c*x)^m*(b*x^2+a)^2*(B*x^2+A),x)
Output:
(x**m*c**m*x*(a**3*m**3 + 15*a**3*m**2 + 71*a**3*m + 105*a**3 + 3*a**2*b*m **3*x**2 + 39*a**2*b*m**2*x**2 + 141*a**2*b*m*x**2 + 105*a**2*b*x**2 + 3*a *b**2*m**3*x**4 + 33*a*b**2*m**2*x**4 + 93*a*b**2*m*x**4 + 63*a*b**2*x**4 + b**3*m**3*x**6 + 9*b**3*m**2*x**6 + 23*b**3*m*x**6 + 15*b**3*x**6))/(m** 4 + 16*m**3 + 86*m**2 + 176*m + 105)