Integrand size = 29, antiderivative size = 209 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {(A b (a d (1-m)-b c (3-m))-a B (b c (1+m)-a d (3+m))) (e x)^{1+m}}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )}{4 a b e \left (a+b x^2\right )^2}+\frac {(A b (1-m) (b c (3-m)+a d (1+m))+a B (1+m) (a d (3+m)+b (c-c m))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{8 a^3 b^2 e (1+m)} \]
-1/8*(A*b*(a*d*(1-m)-b*c*(3-m))-a*B*(b*c*(1+m)-a*d*(3+m)))*(e*x)^(1+m)/a^2 /b^2/e/(b*x^2+a)+1/4*(A*b-B*a)*(e*x)^(1+m)*(d*x^2+c)/a/b/e/(b*x^2+a)^2+1/8 *(A*b*(1-m)*(b*c*(3-m)+a*d*(1+m))+a*B*(1+m)*(a*d*(3+m)+b*(-c*m+c)))*(e*x)^ (1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^3/b^2/e/(1+m)
Time = 0.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.64 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {x (e x)^m \left (a^2 B d \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+a (b B c+A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+(A b-a B) (b c-a d) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{a^3 b^2 (1+m)} \]
(x*(e*x)^m*(a^2*B*d*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a )] + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2 , -((b*x^2)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^3*b^2*(1 + m))
Time = 0.39 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {439, 25, 362, 278}
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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (c+d x^2\right ) (e x)^m}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}-\frac {\int -\frac {(e x)^m \left (d (a B (m+3)+A (b-b m)) x^2+c (A b (3-m)+a B (m+1))\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(e x)^m \left (d (a B (m+3)+A (b-b m)) x^2+c (A b (3-m)+a B (m+1))\right )}{\left (b x^2+a\right )^2}dx}{4 a b}+\frac {\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {(A b (1-m) (a d (m+1)+b c (3-m))+a B (m+1) (a d (m+3)+b (c-c m))) \int \frac {(e x)^m}{b x^2+a}dx}{2 a b}-\frac {(e x)^{m+1} (A b (a d (1-m)-b c (3-m))-a B (b c (m+1)-a d (m+3)))}{2 a b e \left (a+b x^2\right )}}{4 a b}+\frac {\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right ) (A b (1-m) (a d (m+1)+b c (3-m))+a B (m+1) (a d (m+3)+b (c-c m)))}{2 a^2 b e (m+1)}-\frac {(e x)^{m+1} (A b (a d (1-m)-b c (3-m))-a B (b c (m+1)-a d (m+3)))}{2 a b e \left (a+b x^2\right )}}{4 a b}+\frac {\left (c+d x^2\right ) (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\) |
((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2))/(4*a*b*e*(a + b*x^2)^2) + (-1/2*(( A*b*(a*d*(1 - m) - b*c*(3 - m)) - a*B*(b*c*(1 + m) - a*d*(3 + m)))*(e*x)^( 1 + m))/(a*b*e*(a + b*x^2)) + ((A*b*(1 - m)*(b*c*(3 - m) + a*d*(1 + m)) + a*B*(1 + m)*(a*d*(3 + m) + b*(c - c*m)))*(e*x)^(1 + m)*Hypergeometric2F1[1 , (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(2*a^2*b*e*(1 + m)))/(4*a*b)
3.1.7.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
\[\int \frac {\left (e x \right )^{m} \left (x^{2} B +A \right ) \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{3}}d x\]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
integral((B*d*x^4 + (B*c + A*d)*x^2 + A*c)*(e*x)^m/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3), x)
Result contains complex when optimal does not.
Time = 96.01 (sec) , antiderivative size = 6411, normalized size of antiderivative = 30.67 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Too large to display} \]
A*c*(a**2*e**m*m**3*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/ 2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 3*a**2*e**m*m**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a **5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4 *gamma(m/2 + 3/2)) - 2*a**2*e**m*m**2*x**(m + 1)*gamma(m/2 + 1/2)/(32*a**5 *gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*ga mma(m/2 + 3/2)) - a**2*e**m*m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a , 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x** 2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 8*a**2*e**m*m*x **(m + 1)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamm a(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 3*a**2*e**m*x**(m + 1 )*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a* *5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4* gamma(m/2 + 3/2)) + 10*a**2*e**m*x**(m + 1)*gamma(m/2 + 1/2)/(32*a**5*gamm a(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m /2 + 3/2)) + 2*a*b*e**m*m**3*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*p i)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b *x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 6*a*b*e**m* m**2*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*g...
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )}{\left (a+b x^2\right )^3} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]