\(\int x^m (A+B x^2) (b x^2+c x^4)^3 \, dx\) [273]

Optimal result

Integrand size = 24, antiderivative size = 96 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {A b^3 x^{7+m}}{7+m}+\frac {b^2 (b B+3 A c) x^{9+m}}{9+m}+\frac {3 b c (b B+A c) x^{11+m}}{11+m}+\frac {c^2 (3 b B+A c) x^{13+m}}{13+m}+\frac {B c^3 x^{15+m}}{15+m} \] Output:

A*b^3*x^(7+m)/(7+m)+b^2*(3*A*c+B*b)*x^(9+m)/(9+m)+3*b*c*(A*c+B*b)*x^(11+m) 
/(11+m)+c^2*(A*c+3*B*b)*x^(13+m)/(13+m)+B*c^3*x^(15+m)/(15+m)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=x^{7+m} \left (\frac {A b^3}{7+m}+\frac {b^2 (b B+3 A c) x^2}{9+m}+\frac {3 b c (b B+A c) x^4}{11+m}+\frac {c^2 (3 b B+A c) x^6}{13+m}+\frac {B c^3 x^8}{15+m}\right ) \] Input:

Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]
 

Output:

x^(7 + m)*((A*b^3)/(7 + m) + (b^2*(b*B + 3*A*c)*x^2)/(9 + m) + (3*b*c*(b*B 
 + A*c)*x^4)/(11 + m) + (c^2*(3*b*B + A*c)*x^6)/(13 + m) + (B*c^3*x^8)/(15 
 + m))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 96, 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.

\(\displaystyle \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int x^{m+6} \left (A+B x^2\right ) \left (b+c x^2\right )^3dx\)

\(\Big \downarrow \) 355

\(\displaystyle \int \left (A b^3 x^{m+6}+b^2 x^{m+8} (3 A c+b B)+c^2 x^{m+12} (A c+3 b B)+3 b c x^{m+10} (A c+b B)+B c^3 x^{m+14}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {A b^3 x^{m+7}}{m+7}+\frac {b^2 x^{m+9} (3 A c+b B)}{m+9}+\frac {c^2 x^{m+13} (A c+3 b B)}{m+13}+\frac {3 b c x^{m+11} (A c+b B)}{m+11}+\frac {B c^3 x^{m+15}}{m+15}\)

Input:

Int[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^3,x]
 

Output:

(A*b^3*x^(7 + m))/(7 + m) + (b^2*(b*B + 3*A*c)*x^(9 + m))/(9 + m) + (3*b*c 
*(b*B + A*c)*x^(11 + m))/(11 + m) + (c^2*(3*b*B + A*c)*x^(13 + m))/(13 + m 
) + (B*c^3*x^(15 + m))/(15 + m)
 

Defintions of rubi rules used

rule 9

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]
 

rule 355

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]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(473\) vs. \(2(96)=192\).

Time = 0.52 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.94

method	result	size
gosper	\(\frac {x^{7+m} \left (B \,c^{3} m^{4} x^{8}+40 B \,c^{3} m^{3} x^{8}+A \,c^{3} m^{4} x^{6}+3 B b \,c^{2} m^{4} x^{6}+590 B \,c^{3} m^{2} x^{8}+42 A \,c^{3} m^{3} x^{6}+126 B b \,c^{2} m^{3} x^{6}+3800 m \,x^{8} B \,c^{3}+3 A b \,c^{2} m^{4} x^{4}+644 A \,c^{3} m^{2} x^{6}+3 B \,b^{2} c \,m^{4} x^{4}+1932 B b \,c^{2} m^{2} x^{6}+9009 B \,c^{3} x^{8}+132 A b \,c^{2} m^{3} x^{4}+4278 A \,c^{3} m \,x^{6}+132 B \,b^{2} c \,m^{3} x^{4}+12834 B b \,c^{2} m \,x^{6}+3 A \,b^{2} c \,m^{4} x^{2}+2118 A b \,c^{2} m^{2} x^{4}+10395 A \,c^{3} x^{6}+B \,b^{3} m^{4} x^{2}+2118 B \,b^{2} c \,m^{2} x^{4}+31185 B b \,c^{2} x^{6}+138 A \,b^{2} c \,m^{3} x^{2}+14652 A b \,c^{2} m \,x^{4}+46 B \,b^{3} m^{3} x^{2}+14652 B \,b^{2} c m \,x^{4}+A \,b^{3} m^{4}+2328 A \,b^{2} c \,m^{2} x^{2}+36855 A b \,c^{2} x^{4}+776 B \,b^{3} m^{2} x^{2}+36855 x^{4} B \,b^{2} c +48 A \,b^{3} m^{3}+16998 A \,b^{2} c m \,x^{2}+5666 B \,b^{3} m \,x^{2}+854 A \,b^{3} m^{2}+45045 A \,b^{2} c \,x^{2}+15015 x^{2} B \,b^{3}+6672 A \,b^{3} m +19305 A \,b^{3}\right )}{\left (7+m \right ) \left (9+m \right ) \left (11+m \right ) \left (13+m \right ) \left (15+m \right )}\)	\(474\)
risch	\(\frac {x^{m} \left (B \,c^{3} m^{4} x^{8}+40 B \,c^{3} m^{3} x^{8}+A \,c^{3} m^{4} x^{6}+3 B b \,c^{2} m^{4} x^{6}+590 B \,c^{3} m^{2} x^{8}+42 A \,c^{3} m^{3} x^{6}+126 B b \,c^{2} m^{3} x^{6}+3800 m \,x^{8} B \,c^{3}+3 A b \,c^{2} m^{4} x^{4}+644 A \,c^{3} m^{2} x^{6}+3 B \,b^{2} c \,m^{4} x^{4}+1932 B b \,c^{2} m^{2} x^{6}+9009 B \,c^{3} x^{8}+132 A b \,c^{2} m^{3} x^{4}+4278 A \,c^{3} m \,x^{6}+132 B \,b^{2} c \,m^{3} x^{4}+12834 B b \,c^{2} m \,x^{6}+3 A \,b^{2} c \,m^{4} x^{2}+2118 A b \,c^{2} m^{2} x^{4}+10395 A \,c^{3} x^{6}+B \,b^{3} m^{4} x^{2}+2118 B \,b^{2} c \,m^{2} x^{4}+31185 B b \,c^{2} x^{6}+138 A \,b^{2} c \,m^{3} x^{2}+14652 A b \,c^{2} m \,x^{4}+46 B \,b^{3} m^{3} x^{2}+14652 B \,b^{2} c m \,x^{4}+A \,b^{3} m^{4}+2328 A \,b^{2} c \,m^{2} x^{2}+36855 A b \,c^{2} x^{4}+776 B \,b^{3} m^{2} x^{2}+36855 x^{4} B \,b^{2} c +48 A \,b^{3} m^{3}+16998 A \,b^{2} c m \,x^{2}+5666 B \,b^{3} m \,x^{2}+854 A \,b^{3} m^{2}+45045 A \,b^{2} c \,x^{2}+15015 x^{2} B \,b^{3}+6672 A \,b^{3} m +19305 A \,b^{3}\right ) x^{7}}{\left (15+m \right ) \left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\)	\(475\)
orering	\(\frac {\left (B \,c^{3} m^{4} x^{8}+40 B \,c^{3} m^{3} x^{8}+A \,c^{3} m^{4} x^{6}+3 B b \,c^{2} m^{4} x^{6}+590 B \,c^{3} m^{2} x^{8}+42 A \,c^{3} m^{3} x^{6}+126 B b \,c^{2} m^{3} x^{6}+3800 m \,x^{8} B \,c^{3}+3 A b \,c^{2} m^{4} x^{4}+644 A \,c^{3} m^{2} x^{6}+3 B \,b^{2} c \,m^{4} x^{4}+1932 B b \,c^{2} m^{2} x^{6}+9009 B \,c^{3} x^{8}+132 A b \,c^{2} m^{3} x^{4}+4278 A \,c^{3} m \,x^{6}+132 B \,b^{2} c \,m^{3} x^{4}+12834 B b \,c^{2} m \,x^{6}+3 A \,b^{2} c \,m^{4} x^{2}+2118 A b \,c^{2} m^{2} x^{4}+10395 A \,c^{3} x^{6}+B \,b^{3} m^{4} x^{2}+2118 B \,b^{2} c \,m^{2} x^{4}+31185 B b \,c^{2} x^{6}+138 A \,b^{2} c \,m^{3} x^{2}+14652 A b \,c^{2} m \,x^{4}+46 B \,b^{3} m^{3} x^{2}+14652 B \,b^{2} c m \,x^{4}+A \,b^{3} m^{4}+2328 A \,b^{2} c \,m^{2} x^{2}+36855 A b \,c^{2} x^{4}+776 B \,b^{3} m^{2} x^{2}+36855 x^{4} B \,b^{2} c +48 A \,b^{3} m^{3}+16998 A \,b^{2} c m \,x^{2}+5666 B \,b^{3} m \,x^{2}+854 A \,b^{3} m^{2}+45045 A \,b^{2} c \,x^{2}+15015 x^{2} B \,b^{3}+6672 A \,b^{3} m +19305 A \,b^{3}\right ) x \,x^{m} \left (c \,x^{4}+b \,x^{2}\right )^{3}}{\left (15+m \right ) \left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (c \,x^{2}+b \right )^{3}}\)	\(495\)
parallelrisch	\(\frac {2328 A \,x^{9} x^{m} b^{2} c \,m^{2}+3 B \,x^{11} x^{m} b^{2} c \,m^{4}+132 A \,x^{11} x^{m} b \,c^{2} m^{3}+12834 B \,x^{13} x^{m} b \,c^{2} m +132 B \,x^{11} x^{m} b^{2} c \,m^{3}+2118 A \,x^{11} x^{m} b \,c^{2} m^{2}+3 B \,x^{13} x^{m} b \,c^{2} m^{4}+4278 A \,x^{13} x^{m} c^{3} m +126 B \,x^{13} x^{m} b \,c^{2} m^{3}+3 A \,x^{11} x^{m} b \,c^{2} m^{4}+1932 B \,x^{13} x^{m} b \,c^{2} m^{2}+2118 B \,x^{11} x^{m} b^{2} c \,m^{2}+776 B \,x^{9} x^{m} b^{3} m^{2}+48 A \,x^{7} x^{m} b^{3} m^{3}+5666 B \,x^{9} x^{m} b^{3} m +45045 A \,x^{9} x^{m} b^{2} c +854 A \,x^{7} x^{m} b^{3} m^{2}+6672 A \,x^{7} x^{m} b^{3} m +3 A \,x^{9} x^{m} b^{2} c \,m^{4}+B \,x^{9} x^{m} b^{3} m^{4}+644 A \,x^{13} x^{m} c^{3} m^{2}+46 B \,x^{9} x^{m} b^{3} m^{3}+36855 A \,x^{11} x^{m} b \,c^{2}+A \,x^{7} x^{m} b^{3} m^{4}+36855 B \,x^{11} x^{m} b^{2} c +16998 A \,x^{9} x^{m} b^{2} c m +138 A \,x^{9} x^{m} b^{2} c \,m^{3}+14652 A \,x^{11} x^{m} b \,c^{2} m +14652 B \,x^{11} x^{m} b^{2} c m +B \,x^{15} x^{m} c^{3} m^{4}+40 B \,x^{15} x^{m} c^{3} m^{3}+A \,x^{13} x^{m} c^{3} m^{4}+590 B \,x^{15} x^{m} c^{3} m^{2}+42 A \,x^{13} x^{m} c^{3} m^{3}+3800 B \,x^{15} x^{m} c^{3} m +31185 B \,x^{13} x^{m} b \,c^{2}+9009 B \,x^{15} x^{m} c^{3}+10395 A \,x^{13} x^{m} c^{3}+15015 B \,x^{9} x^{m} b^{3}+19305 A \,x^{7} x^{m} b^{3}}{\left (15+m \right ) \left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\)	\(604\)

Input:

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)
 

Output:

x^(7+m)/(7+m)/(9+m)/(11+m)/(13+m)/(15+m)*(B*c^3*m^4*x^8+40*B*c^3*m^3*x^8+A 
*c^3*m^4*x^6+3*B*b*c^2*m^4*x^6+590*B*c^3*m^2*x^8+42*A*c^3*m^3*x^6+126*B*b* 
c^2*m^3*x^6+3800*B*c^3*m*x^8+3*A*b*c^2*m^4*x^4+644*A*c^3*m^2*x^6+3*B*b^2*c 
*m^4*x^4+1932*B*b*c^2*m^2*x^6+9009*B*c^3*x^8+132*A*b*c^2*m^3*x^4+4278*A*c^ 
3*m*x^6+132*B*b^2*c*m^3*x^4+12834*B*b*c^2*m*x^6+3*A*b^2*c*m^4*x^2+2118*A*b 
*c^2*m^2*x^4+10395*A*c^3*x^6+B*b^3*m^4*x^2+2118*B*b^2*c*m^2*x^4+31185*B*b* 
c^2*x^6+138*A*b^2*c*m^3*x^2+14652*A*b*c^2*m*x^4+46*B*b^3*m^3*x^2+14652*B*b 
^2*c*m*x^4+A*b^3*m^4+2328*A*b^2*c*m^2*x^2+36855*A*b*c^2*x^4+776*B*b^3*m^2* 
x^2+36855*B*b^2*c*x^4+48*A*b^3*m^3+16998*A*b^2*c*m*x^2+5666*B*b^3*m*x^2+85 
4*A*b^3*m^2+45045*A*b^2*c*x^2+15015*B*b^3*x^2+6672*A*b^3*m+19305*A*b^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (96) = 192\).

Time = 0.09 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.97 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {{\left ({\left (B c^{3} m^{4} + 40 \, B c^{3} m^{3} + 590 \, B c^{3} m^{2} + 3800 \, B c^{3} m + 9009 \, B c^{3}\right )} x^{15} + {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} m^{4} + 31185 \, B b c^{2} + 10395 \, A c^{3} + 42 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m^{3} + 644 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m^{2} + 4278 \, {\left (3 \, B b c^{2} + A c^{3}\right )} m\right )} x^{13} + 3 \, {\left ({\left (B b^{2} c + A b c^{2}\right )} m^{4} + 12285 \, B b^{2} c + 12285 \, A b c^{2} + 44 \, {\left (B b^{2} c + A b c^{2}\right )} m^{3} + 706 \, {\left (B b^{2} c + A b c^{2}\right )} m^{2} + 4884 \, {\left (B b^{2} c + A b c^{2}\right )} m\right )} x^{11} + {\left ({\left (B b^{3} + 3 \, A b^{2} c\right )} m^{4} + 15015 \, B b^{3} + 45045 \, A b^{2} c + 46 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m^{3} + 776 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m^{2} + 5666 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} m\right )} x^{9} + {\left (A b^{3} m^{4} + 48 \, A b^{3} m^{3} + 854 \, A b^{3} m^{2} + 6672 \, A b^{3} m + 19305 \, A b^{3}\right )} x^{7}\right )} x^{m}}{m^{5} + 55 \, m^{4} + 1190 \, m^{3} + 12650 \, m^{2} + 66009 \, m + 135135} \] Input:

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^3,x, algorithm="fricas")
 

Output:

((B*c^3*m^4 + 40*B*c^3*m^3 + 590*B*c^3*m^2 + 3800*B*c^3*m + 9009*B*c^3)*x^ 
15 + ((3*B*b*c^2 + A*c^3)*m^4 + 31185*B*b*c^2 + 10395*A*c^3 + 42*(3*B*b*c^ 
2 + A*c^3)*m^3 + 644*(3*B*b*c^2 + A*c^3)*m^2 + 4278*(3*B*b*c^2 + A*c^3)*m) 
*x^13 + 3*((B*b^2*c + A*b*c^2)*m^4 + 12285*B*b^2*c + 12285*A*b*c^2 + 44*(B 
*b^2*c + A*b*c^2)*m^3 + 706*(B*b^2*c + A*b*c^2)*m^2 + 4884*(B*b^2*c + A*b* 
c^2)*m)*x^11 + ((B*b^3 + 3*A*b^2*c)*m^4 + 15015*B*b^3 + 45045*A*b^2*c + 46 
*(B*b^3 + 3*A*b^2*c)*m^3 + 776*(B*b^3 + 3*A*b^2*c)*m^2 + 5666*(B*b^3 + 3*A 
*b^2*c)*m)*x^9 + (A*b^3*m^4 + 48*A*b^3*m^3 + 854*A*b^3*m^2 + 6672*A*b^3*m 
+ 19305*A*b^3)*x^7)*x^m/(m^5 + 55*m^4 + 1190*m^3 + 12650*m^2 + 66009*m + 1 
35135)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2077 vs. \(2 (87) = 174\).

Time = 1.23 (sec) , antiderivative size = 2077, normalized size of antiderivative = 21.64 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2)**3,x)
 

Output:

Piecewise((-A*b**3/(8*x**8) - A*b**2*c/(2*x**6) - 3*A*b*c**2/(4*x**4) - A* 
c**3/(2*x**2) - B*b**3/(6*x**6) - 3*B*b**2*c/(4*x**4) - 3*B*b*c**2/(2*x**2 
) + B*c**3*log(x), Eq(m, -15)), (-A*b**3/(6*x**6) - 3*A*b**2*c/(4*x**4) - 
3*A*b*c**2/(2*x**2) + A*c**3*log(x) - B*b**3/(4*x**4) - 3*B*b**2*c/(2*x**2 
) + 3*B*b*c**2*log(x) + B*c**3*x**2/2, Eq(m, -13)), (-A*b**3/(4*x**4) - 3* 
A*b**2*c/(2*x**2) + 3*A*b*c**2*log(x) + A*c**3*x**2/2 - B*b**3/(2*x**2) + 
3*B*b**2*c*log(x) + 3*B*b*c**2*x**2/2 + B*c**3*x**4/4, Eq(m, -11)), (-A*b* 
*3/(2*x**2) + 3*A*b**2*c*log(x) + 3*A*b*c**2*x**2/2 + A*c**3*x**4/4 + B*b* 
*3*log(x) + 3*B*b**2*c*x**2/2 + 3*B*b*c**2*x**4/4 + B*c**3*x**6/6, Eq(m, - 
9)), (A*b**3*log(x) + 3*A*b**2*c*x**2/2 + 3*A*b*c**2*x**4/4 + A*c**3*x**6/ 
6 + B*b**3*x**2/2 + 3*B*b**2*c*x**4/4 + B*b*c**2*x**6/2 + B*c**3*x**8/8, E 
q(m, -7)), (A*b**3*m**4*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 
 + 66009*m + 135135) + 48*A*b**3*m**3*x**7*x**m/(m**5 + 55*m**4 + 1190*m** 
3 + 12650*m**2 + 66009*m + 135135) + 854*A*b**3*m**2*x**7*x**m/(m**5 + 55* 
m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 6672*A*b**3*m*x**7*x** 
m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 19305*A*b 
**3*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) 
 + 3*A*b**2*c*m**4*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66 
009*m + 135135) + 138*A*b**2*c*m**3*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 
+ 12650*m**2 + 66009*m + 135135) + 2328*A*b**2*c*m**2*x**9*x**m/(m**5 +...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {B c^{3} x^{m + 15}}{m + 15} + \frac {3 \, B b c^{2} x^{m + 13}}{m + 13} + \frac {A c^{3} x^{m + 13}}{m + 13} + \frac {3 \, B b^{2} c x^{m + 11}}{m + 11} + \frac {3 \, A b c^{2} x^{m + 11}}{m + 11} + \frac {B b^{3} x^{m + 9}}{m + 9} + \frac {3 \, A b^{2} c x^{m + 9}}{m + 9} + \frac {A b^{3} x^{m + 7}}{m + 7} \] Input:

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^3,x, algorithm="maxima")
 

Output:

B*c^3*x^(m + 15)/(m + 15) + 3*B*b*c^2*x^(m + 13)/(m + 13) + A*c^3*x^(m + 1 
3)/(m + 13) + 3*B*b^2*c*x^(m + 11)/(m + 11) + 3*A*b*c^2*x^(m + 11)/(m + 11 
) + B*b^3*x^(m + 9)/(m + 9) + 3*A*b^2*c*x^(m + 9)/(m + 9) + A*b^3*x^(m + 7 
)/(m + 7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (96) = 192\).

Time = 0.76 (sec) , antiderivative size = 603, normalized size of antiderivative = 6.28 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {B c^{3} m^{4} x^{15} x^{m} + 40 \, B c^{3} m^{3} x^{15} x^{m} + 3 \, B b c^{2} m^{4} x^{13} x^{m} + A c^{3} m^{4} x^{13} x^{m} + 590 \, B c^{3} m^{2} x^{15} x^{m} + 126 \, B b c^{2} m^{3} x^{13} x^{m} + 42 \, A c^{3} m^{3} x^{13} x^{m} + 3800 \, B c^{3} m x^{15} x^{m} + 3 \, B b^{2} c m^{4} x^{11} x^{m} + 3 \, A b c^{2} m^{4} x^{11} x^{m} + 1932 \, B b c^{2} m^{2} x^{13} x^{m} + 644 \, A c^{3} m^{2} x^{13} x^{m} + 9009 \, B c^{3} x^{15} x^{m} + 132 \, B b^{2} c m^{3} x^{11} x^{m} + 132 \, A b c^{2} m^{3} x^{11} x^{m} + 12834 \, B b c^{2} m x^{13} x^{m} + 4278 \, A c^{3} m x^{13} x^{m} + B b^{3} m^{4} x^{9} x^{m} + 3 \, A b^{2} c m^{4} x^{9} x^{m} + 2118 \, B b^{2} c m^{2} x^{11} x^{m} + 2118 \, A b c^{2} m^{2} x^{11} x^{m} + 31185 \, B b c^{2} x^{13} x^{m} + 10395 \, A c^{3} x^{13} x^{m} + 46 \, B b^{3} m^{3} x^{9} x^{m} + 138 \, A b^{2} c m^{3} x^{9} x^{m} + 14652 \, B b^{2} c m x^{11} x^{m} + 14652 \, A b c^{2} m x^{11} x^{m} + A b^{3} m^{4} x^{7} x^{m} + 776 \, B b^{3} m^{2} x^{9} x^{m} + 2328 \, A b^{2} c m^{2} x^{9} x^{m} + 36855 \, B b^{2} c x^{11} x^{m} + 36855 \, A b c^{2} x^{11} x^{m} + 48 \, A b^{3} m^{3} x^{7} x^{m} + 5666 \, B b^{3} m x^{9} x^{m} + 16998 \, A b^{2} c m x^{9} x^{m} + 854 \, A b^{3} m^{2} x^{7} x^{m} + 15015 \, B b^{3} x^{9} x^{m} + 45045 \, A b^{2} c x^{9} x^{m} + 6672 \, A b^{3} m x^{7} x^{m} + 19305 \, A b^{3} x^{7} x^{m}}{m^{5} + 55 \, m^{4} + 1190 \, m^{3} + 12650 \, m^{2} + 66009 \, m + 135135} \] Input:

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^3,x, algorithm="giac")
 

Output:

(B*c^3*m^4*x^15*x^m + 40*B*c^3*m^3*x^15*x^m + 3*B*b*c^2*m^4*x^13*x^m + A*c 
^3*m^4*x^13*x^m + 590*B*c^3*m^2*x^15*x^m + 126*B*b*c^2*m^3*x^13*x^m + 42*A 
*c^3*m^3*x^13*x^m + 3800*B*c^3*m*x^15*x^m + 3*B*b^2*c*m^4*x^11*x^m + 3*A*b 
*c^2*m^4*x^11*x^m + 1932*B*b*c^2*m^2*x^13*x^m + 644*A*c^3*m^2*x^13*x^m + 9 
009*B*c^3*x^15*x^m + 132*B*b^2*c*m^3*x^11*x^m + 132*A*b*c^2*m^3*x^11*x^m + 
 12834*B*b*c^2*m*x^13*x^m + 4278*A*c^3*m*x^13*x^m + B*b^3*m^4*x^9*x^m + 3* 
A*b^2*c*m^4*x^9*x^m + 2118*B*b^2*c*m^2*x^11*x^m + 2118*A*b*c^2*m^2*x^11*x^ 
m + 31185*B*b*c^2*x^13*x^m + 10395*A*c^3*x^13*x^m + 46*B*b^3*m^3*x^9*x^m + 
 138*A*b^2*c*m^3*x^9*x^m + 14652*B*b^2*c*m*x^11*x^m + 14652*A*b*c^2*m*x^11 
*x^m + A*b^3*m^4*x^7*x^m + 776*B*b^3*m^2*x^9*x^m + 2328*A*b^2*c*m^2*x^9*x^ 
m + 36855*B*b^2*c*x^11*x^m + 36855*A*b*c^2*x^11*x^m + 48*A*b^3*m^3*x^7*x^m 
 + 5666*B*b^3*m*x^9*x^m + 16998*A*b^2*c*m*x^9*x^m + 854*A*b^3*m^2*x^7*x^m 
+ 15015*B*b^3*x^9*x^m + 45045*A*b^2*c*x^9*x^m + 6672*A*b^3*m*x^7*x^m + 193 
05*A*b^3*x^7*x^m)/(m^5 + 55*m^4 + 1190*m^3 + 12650*m^2 + 66009*m + 135135)
 

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.03 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {A\,b^3\,x^m\,x^7\,\left (m^4+48\,m^3+854\,m^2+6672\,m+19305\right )}{m^5+55\,m^4+1190\,m^3+12650\,m^2+66009\,m+135135}+\frac {B\,c^3\,x^m\,x^{15}\,\left (m^4+40\,m^3+590\,m^2+3800\,m+9009\right )}{m^5+55\,m^4+1190\,m^3+12650\,m^2+66009\,m+135135}+\frac {b^2\,x^m\,x^9\,\left (3\,A\,c+B\,b\right )\,\left (m^4+46\,m^3+776\,m^2+5666\,m+15015\right )}{m^5+55\,m^4+1190\,m^3+12650\,m^2+66009\,m+135135}+\frac {c^2\,x^m\,x^{13}\,\left (A\,c+3\,B\,b\right )\,\left (m^4+42\,m^3+644\,m^2+4278\,m+10395\right )}{m^5+55\,m^4+1190\,m^3+12650\,m^2+66009\,m+135135}+\frac {3\,b\,c\,x^m\,x^{11}\,\left (A\,c+B\,b\right )\,\left (m^4+44\,m^3+706\,m^2+4884\,m+12285\right )}{m^5+55\,m^4+1190\,m^3+12650\,m^2+66009\,m+135135} \] Input:

int(x^m*(A + B*x^2)*(b*x^2 + c*x^4)^3,x)
 

Output:

(A*b^3*x^m*x^7*(6672*m + 854*m^2 + 48*m^3 + m^4 + 19305))/(66009*m + 12650 
*m^2 + 1190*m^3 + 55*m^4 + m^5 + 135135) + (B*c^3*x^m*x^15*(3800*m + 590*m 
^2 + 40*m^3 + m^4 + 9009))/(66009*m + 12650*m^2 + 1190*m^3 + 55*m^4 + m^5 
+ 135135) + (b^2*x^m*x^9*(3*A*c + B*b)*(5666*m + 776*m^2 + 46*m^3 + m^4 + 
15015))/(66009*m + 12650*m^2 + 1190*m^3 + 55*m^4 + m^5 + 135135) + (c^2*x^ 
m*x^13*(A*c + 3*B*b)*(4278*m + 644*m^2 + 42*m^3 + m^4 + 10395))/(66009*m + 
 12650*m^2 + 1190*m^3 + 55*m^4 + m^5 + 135135) + (3*b*c*x^m*x^11*(A*c + B* 
b)*(4884*m + 706*m^2 + 44*m^3 + m^4 + 12285))/(66009*m + 12650*m^2 + 1190* 
m^3 + 55*m^4 + m^5 + 135135)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.89 \[ \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx=\frac {x^{m} x^{7} \left (b \,c^{3} m^{4} x^{8}+40 b \,c^{3} m^{3} x^{8}+a \,c^{3} m^{4} x^{6}+3 b^{2} c^{2} m^{4} x^{6}+590 b \,c^{3} m^{2} x^{8}+42 a \,c^{3} m^{3} x^{6}+126 b^{2} c^{2} m^{3} x^{6}+3800 b \,c^{3} m \,x^{8}+3 a b \,c^{2} m^{4} x^{4}+644 a \,c^{3} m^{2} x^{6}+3 b^{3} c \,m^{4} x^{4}+1932 b^{2} c^{2} m^{2} x^{6}+9009 b \,c^{3} x^{8}+132 a b \,c^{2} m^{3} x^{4}+4278 a \,c^{3} m \,x^{6}+132 b^{3} c \,m^{3} x^{4}+12834 b^{2} c^{2} m \,x^{6}+3 a \,b^{2} c \,m^{4} x^{2}+2118 a b \,c^{2} m^{2} x^{4}+10395 a \,c^{3} x^{6}+b^{4} m^{4} x^{2}+2118 b^{3} c \,m^{2} x^{4}+31185 b^{2} c^{2} x^{6}+138 a \,b^{2} c \,m^{3} x^{2}+14652 a b \,c^{2} m \,x^{4}+46 b^{4} m^{3} x^{2}+14652 b^{3} c m \,x^{4}+a \,b^{3} m^{4}+2328 a \,b^{2} c \,m^{2} x^{2}+36855 a b \,c^{2} x^{4}+776 b^{4} m^{2} x^{2}+36855 b^{3} c \,x^{4}+48 a \,b^{3} m^{3}+16998 a \,b^{2} c m \,x^{2}+5666 b^{4} m \,x^{2}+854 a \,b^{3} m^{2}+45045 a \,b^{2} c \,x^{2}+15015 b^{4} x^{2}+6672 a \,b^{3} m +19305 a \,b^{3}\right )}{m^{5}+55 m^{4}+1190 m^{3}+12650 m^{2}+66009 m +135135} \] Input:

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^3,x)
 

Output:

(x**m*x**7*(a*b**3*m**4 + 48*a*b**3*m**3 + 854*a*b**3*m**2 + 6672*a*b**3*m 
 + 19305*a*b**3 + 3*a*b**2*c*m**4*x**2 + 138*a*b**2*c*m**3*x**2 + 2328*a*b 
**2*c*m**2*x**2 + 16998*a*b**2*c*m*x**2 + 45045*a*b**2*c*x**2 + 3*a*b*c**2 
*m**4*x**4 + 132*a*b*c**2*m**3*x**4 + 2118*a*b*c**2*m**2*x**4 + 14652*a*b* 
c**2*m*x**4 + 36855*a*b*c**2*x**4 + a*c**3*m**4*x**6 + 42*a*c**3*m**3*x**6 
 + 644*a*c**3*m**2*x**6 + 4278*a*c**3*m*x**6 + 10395*a*c**3*x**6 + b**4*m* 
*4*x**2 + 46*b**4*m**3*x**2 + 776*b**4*m**2*x**2 + 5666*b**4*m*x**2 + 1501 
5*b**4*x**2 + 3*b**3*c*m**4*x**4 + 132*b**3*c*m**3*x**4 + 2118*b**3*c*m**2 
*x**4 + 14652*b**3*c*m*x**4 + 36855*b**3*c*x**4 + 3*b**2*c**2*m**4*x**6 + 
126*b**2*c**2*m**3*x**6 + 1932*b**2*c**2*m**2*x**6 + 12834*b**2*c**2*m*x** 
6 + 31185*b**2*c**2*x**6 + b*c**3*m**4*x**8 + 40*b*c**3*m**3*x**8 + 590*b* 
c**3*m**2*x**8 + 3800*b*c**3*m*x**8 + 9009*b*c**3*x**8))/(m**5 + 55*m**4 + 
 1190*m**3 + 12650*m**2 + 66009*m + 135135)