3.1.21 \(\int \frac {(e x)^m (A+B x^2) (c+d x^2)^3}{(a+b x^2)^3} \, dx\) [21]

3.1.21.1 Optimal result

Integrand size = 31, antiderivative size = 480 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=-\frac {d \left (A b \left (2 b^2 c^2 \left (3+2 m-m^2\right )+3 a b c d \left (3+4 m+m^2\right )-a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (2 b^2 c^2 (1+m)^2-3 a b c d \left (15+8 m+m^2\right )+a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m}}{8 a^2 b^4 e (1+m)}-\frac {d^2 \left (A b (3+m) (b c (3-m)+a d (5+m))+a B \left (b c \left (3+4 m+m^2\right )-a d \left (35+12 m+m^2\right )\right )\right ) (e x)^{3+m}}{8 a^2 b^3 e^3 (3+m)}+\frac {(A b (b c (3-m)+a d (3+m))+a B (b c (1+m)-a d (7+m))) (e x)^{1+m} \left (c+d x^2\right )^2}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^3}{4 a b e \left (a+b x^2\right )^2}+\frac {(b c-a d) \left (A b \left (2 a b c d \left (3-2 m-m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )+a^2 d^2 \left (15+8 m+m^2\right )\right )+a B \left (b^2 c^2 \left (1-m^2\right )+2 a b c d \left (5+6 m+m^2\right )-a^2 d^2 \left (35+12 m+m^2\right )\right )\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{8 a^3 b^4 e (1+m)} \]

output

-1/8*d*(A*b*(2*b^2*c^2*(-m^2+2*m+3)+3*a*b*c*d*(m^2+4*m+3)-a^2*d^2*(m^2+8*m 
+15))+a*B*(2*b^2*c^2*(1+m)^2-3*a*b*c*d*(m^2+8*m+15)+a^2*d^2*(m^2+12*m+35)) 
)*(e*x)^(1+m)/a^2/b^4/e/(1+m)-1/8*d^2*(A*b*(3+m)*(b*c*(3-m)+a*d*(5+m))+a*B 
*(b*c*(m^2+4*m+3)-a*d*(m^2+12*m+35)))*(e*x)^(3+m)/a^2/b^3/e^3/(3+m)+1/8*(A 
*b*(b*c*(3-m)+a*d*(3+m))+a*B*(b*c*(1+m)-a*d*(7+m)))*(e*x)^(1+m)*(d*x^2+c)^ 
2/a^2/b^2/e/(b*x^2+a)+1/4*(A*b-B*a)*(e*x)^(1+m)*(d*x^2+c)^3/a/b/e/(b*x^2+a 
)^2+1/8*(-a*d+b*c)*(A*b*(2*a*b*c*d*(-m^2-2*m+3)+b^2*c^2*(m^2-4*m+3)+a^2*d^ 
2*(m^2+8*m+15))+a*B*(b^2*c^2*(-m^2+1)+2*a*b*c*d*(m^2+6*m+5)-a^2*d^2*(m^2+1 
2*m+35)))*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^3/b 
^4/e/(1+m)
 

3.1.21.2 Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\frac {x (e x)^m \left (\frac {d^2 (3 b B c+A b d-3 a B d)}{1+m}+\frac {b B d^3 x^2}{3+m}+\frac {3 d (b c-a d) (b B c+A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a (1+m)}+\frac {(b c-a d)^2 (b B c+3 A b d-4 a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 (1+m)}+\frac {(-A b+a B) (-b c+a d)^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^3 (1+m)}\right )}{b^4} \]

input

Integrate[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^3,x]
 

output

(x*(e*x)^m*((d^2*(3*b*B*c + A*b*d - 3*a*B*d))/(1 + m) + (b*B*d^3*x^2)/(3 + 
 m) + (3*d*(b*c - a*d)*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 + 
 m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*(1 + m)) + ((b*c - a*d)^2*(b*B*c + 3*A 
*b*d - 4*a*B*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/ 
(a^2*(1 + m)) + ((-(A*b) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[3, (1 + 
 m)/2, (3 + m)/2, -((b*x^2)/a)])/(a^3*(1 + m))))/b^4
 

3.1.21.3 Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {439, 25, 439, 25, 437, 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (c+d x^2\right )^3 (e x)^m}{\left (a+b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 439

\(\displaystyle \frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}-\frac {\int -\frac {(e x)^m \left (d x^2+c\right )^2 \left (c (A b (3-m)+a B (m+1))-d (A b (m+3)-a B (m+7)) x^2\right )}{\left (b x^2+a\right )^2}dx}{4 a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(e x)^m \left (d x^2+c\right )^2 \left (c (A b (3-m)+a B (m+1))-d (A b (m+3)-a B (m+7)) x^2\right )}{\left (b x^2+a\right )^2}dx}{4 a b}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 439

\(\displaystyle \frac {\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{2 a b e \left (a+b x^2\right )}-\frac {\int -\frac {(e x)^m \left (d x^2+c\right ) \left (c \left (a B (m+1) (a d (m+7)+b (c-c m))+A b \left (b c \left (m^2-4 m+3\right )-a d \left (m^2+4 m+3\right )\right )\right )-d \left (A b (m+3) (b c (3-m)+a d (m+5))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right ) x^2\right )}{b x^2+a}dx}{2 a b}}{4 a b}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {(e x)^m \left (d x^2+c\right ) \left (c \left (a B (m+1) (a d (m+7)+b (c-c m))+A b \left (b c \left (m^2-4 m+3\right )-a d \left (m^2+4 m+3\right )\right )\right )-d \left (A b (m+3) (b c (3-m)+a d (m+5))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right ) x^2\right )}{b x^2+a}dx}{2 a b}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{2 a b e \left (a+b x^2\right )}}{4 a b}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 437

\(\displaystyle \frac {\frac {\int \left (-\frac {d \left (A b \left (2 b^2 \left (-m^2+2 m+3\right ) c^2+3 a b d \left (m^2+4 m+3\right ) c-a^2 d^2 \left (m^2+8 m+15\right )\right )+a B \left (a^2 \left (m^2+12 m+35\right ) d^2-3 a b c \left (m^2+8 m+15\right ) d+2 b^2 c^2 (m+1)^2\right )\right ) (e x)^m}{b^2}+\frac {\left (35 B d^3 a^4+B d^3 m^2 a^4+12 B d^3 m a^4-15 A b d^3 a^3-45 b B c d^2 a^3-A b d^3 m^2 a^3-3 b B c d^2 m^2 a^3-8 A b d^3 m a^3-24 b B c d^2 m a^3+9 A b^2 c d^2 a^2+3 A b^2 c d^2 m^2 a^2+3 b^2 B c^2 d m^2 a^2+9 b^2 B c^2 d a^2+12 A b^2 c d^2 m a^2+12 b^2 B c^2 d m a^2+b^3 B c^3 a-b^3 B c^3 m^2 a-3 A b^3 c^2 d m^2 a+3 A b^3 c^2 d a+3 A b^4 c^3+A b^4 c^3 m^2-4 A b^4 c^3 m\right ) (e x)^m}{b^2 \left (b x^2+a\right )}-\frac {d^2 \left (A b (m+3) (b c (3-m)+a d (m+5))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right ) (e x)^{m+2}}{b e^2}\right )dx}{2 a b}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{2 a b e \left (a+b x^2\right )}}{4 a b}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\frac {(e x)^{m+1} (b c-a d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+8 m+15\right )+2 a b c d \left (-m^2-2 m+3\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2+12 m+35\right )+2 a b c d \left (m^2+6 m+5\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{a b^2 e (m+1)}-\frac {d (e x)^{m+1} \left (A b \left (-a^2 d^2 \left (m^2+8 m+15\right )+3 a b c d \left (m^2+4 m+3\right )+2 b^2 c^2 \left (-m^2+2 m+3\right )\right )+a B \left (a^2 d^2 \left (m^2+12 m+35\right )-3 a b c d \left (m^2+8 m+15\right )+2 b^2 c^2 (m+1)^2\right )\right )}{b^2 e (m+1)}-\frac {d^2 (e x)^{m+3} \left (A b (m+3) (a d (m+5)+b c (3-m))+a B \left (b c \left (m^2+4 m+3\right )-a d \left (m^2+12 m+35\right )\right )\right )}{b e^3 (m+3)}}{2 a b}+\frac {\left (c+d x^2\right )^2 (e x)^{m+1} (A b (a d (m+3)+b c (3-m))+a B (b c (m+1)-a d (m+7)))}{2 a b e \left (a+b x^2\right )}}{4 a b}+\frac {\left (c+d x^2\right )^3 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2}\)

input

Int[((e*x)^m*(A + B*x^2)*(c + d*x^2)^3)/(a + b*x^2)^3,x]
 

output

((A*b - a*B)*(e*x)^(1 + m)*(c + d*x^2)^3)/(4*a*b*e*(a + b*x^2)^2) + (((A*b 
*(b*c*(3 - m) + a*d*(3 + m)) + a*B*(b*c*(1 + m) - a*d*(7 + m)))*(e*x)^(1 + 
 m)*(c + d*x^2)^2)/(2*a*b*e*(a + b*x^2)) + (-((d*(A*b*(2*b^2*c^2*(3 + 2*m 
- m^2) + 3*a*b*c*d*(3 + 4*m + m^2) - a^2*d^2*(15 + 8*m + m^2)) + a*B*(2*b^ 
2*c^2*(1 + m)^2 - 3*a*b*c*d*(15 + 8*m + m^2) + a^2*d^2*(35 + 12*m + m^2))) 
*(e*x)^(1 + m))/(b^2*e*(1 + m))) - (d^2*(A*b*(3 + m)*(b*c*(3 - m) + a*d*(5 
 + m)) + a*B*(b*c*(3 + 4*m + m^2) - a*d*(35 + 12*m + m^2)))*(e*x)^(3 + m)) 
/(b*e^3*(3 + m)) + ((b*c - a*d)*(A*b*(2*a*b*c*d*(3 - 2*m - m^2) + b^2*c^2* 
(3 - 4*m + m^2) + a^2*d^2*(15 + 8*m + m^2)) + a*B*(b^2*c^2*(1 - m^2) + 2*a 
*b*c*d*(5 + 6*m + m^2) - a^2*d^2*(35 + 12*m + m^2)))*(e*x)^(1 + m)*Hyperge 
ometric2F1[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(a*b^2*e*(1 + m)))/(2*a 
*b))/(4*a*b)
 

3.1.21.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 437

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( 
a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
 

rule 439

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.1.21.4 Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (x^{2} B +A \right ) \left (d \,x^{2}+c \right )^{3}}{\left (b \,x^{2}+a \right )^{3}}d x\]

input

int((e*x)^m*(B*x^2+A)*(d*x^2+c)^3/(b*x^2+a)^3,x)
 

output

int((e*x)^m*(B*x^2+A)*(d*x^2+c)^3/(b*x^2+a)^3,x)
 

3.1.21.5 Fricas [F]

\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]

input

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="fricas")
 

output

integral((B*d^3*x^8 + (3*B*c*d^2 + A*d^3)*x^6 + 3*(B*c^2*d + A*c*d^2)*x^4 
+ A*c^3 + (B*c^3 + 3*A*c^2*d)*x^2)*(e*x)^m/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2* 
b*x^2 + a^3), x)
 

3.1.21.6 Sympy [F]

\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int \frac {\left (e x\right )^{m} \left (A + B x^{2}\right ) \left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{3}}\, dx \]

input

integrate((e*x)**m*(B*x**2+A)*(d*x**2+c)**3/(b*x**2+a)**3,x)
 

output

Integral((e*x)**m*(A + B*x**2)*(c + d*x**2)**3/(a + b*x**2)**3, x)
 

3.1.21.7 Maxima [F]

\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]

input

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="maxima")
 

output

integrate((B*x^2 + A)*(d*x^2 + c)^3*(e*x)^m/(b*x^2 + a)^3, x)
 

3.1.21.8 Giac [F]

\[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{3} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]

input

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^3/(b*x^2+a)^3,x, algorithm="giac")
 

output

integrate((B*x^2 + A)*(d*x^2 + c)^3*(e*x)^m/(b*x^2 + a)^3, x)
 

3.1.21.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^3}{\left (a+b x^2\right )^3} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^3} \,d x \]

input

int(((A + B*x^2)*(e*x)^m*(c + d*x^2)^3)/(a + b*x^2)^3,x)
 

output

int(((A + B*x^2)*(e*x)^m*(c + d*x^2)^3)/(a + b*x^2)^3, x)