\(\int (d x)^m (a^2+2 a b x^2+b^2 x^4)^3 \, dx\) [672]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [A] (verification not implemented)
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 26, antiderivative size = 150 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6 (d x)^{1+m}}{d (1+m)}+\frac {6 a^5 b (d x)^{3+m}}{d^3 (3+m)}+\frac {15 a^4 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {20 a^3 b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {15 a^2 b^4 (d x)^{9+m}}{d^9 (9+m)}+\frac {6 a b^5 (d x)^{11+m}}{d^{11} (11+m)}+\frac {b^6 (d x)^{13+m}}{d^{13} (13+m)} \] Output:

a^6*(d*x)^(1+m)/d/(1+m)+6*a^5*b*(d*x)^(3+m)/d^3/(3+m)+15*a^4*b^2*(d*x)^(5+ 
m)/d^5/(5+m)+20*a^3*b^3*(d*x)^(7+m)/d^7/(7+m)+15*a^2*b^4*(d*x)^(9+m)/d^9/( 
9+m)+6*a*b^5*(d*x)^(11+m)/d^11/(11+m)+b^6*(d*x)^(13+m)/d^13/(13+m)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=x (d x)^m \left (\frac {a^6}{1+m}+\frac {6 a^5 b x^2}{3+m}+\frac {15 a^4 b^2 x^4}{5+m}+\frac {20 a^3 b^3 x^6}{7+m}+\frac {15 a^2 b^4 x^8}{9+m}+\frac {6 a b^5 x^{10}}{11+m}+\frac {b^6 x^{12}}{13+m}\right ) \] Input:

Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

x*(d*x)^m*(a^6/(1 + m) + (6*a^5*b*x^2)/(3 + m) + (15*a^4*b^2*x^4)/(5 + m) 
+ (20*a^3*b^3*x^6)/(7 + m) + (15*a^2*b^4*x^8)/(9 + m) + (6*a*b^5*x^10)/(11 
 + m) + (b^6*x^12)/(13 + m))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1380, 27, 244, 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.

\(\displaystyle \int \left (a^2+2 a b x^2+b^2 x^4\right )^3 (d x)^m \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle \frac {\int b^6 (d x)^m \left (b x^2+a\right )^6dx}{b^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \left (a+b x^2\right )^6 (d x)^mdx\)

\(\Big \downarrow \) 244

\(\displaystyle \int \left (a^6 (d x)^m+\frac {6 a^5 b (d x)^{m+2}}{d^2}+\frac {15 a^4 b^2 (d x)^{m+4}}{d^4}+\frac {20 a^3 b^3 (d x)^{m+6}}{d^6}+\frac {15 a^2 b^4 (d x)^{m+8}}{d^8}+\frac {6 a b^5 (d x)^{m+10}}{d^{10}}+\frac {b^6 (d x)^{m+12}}{d^{12}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^6 (d x)^{m+1}}{d (m+1)}+\frac {6 a^5 b (d x)^{m+3}}{d^3 (m+3)}+\frac {15 a^4 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {20 a^3 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {15 a^2 b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac {6 a b^5 (d x)^{m+11}}{d^{11} (m+11)}+\frac {b^6 (d x)^{m+13}}{d^{13} (m+13)}\)

Input:

Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
 

Output:

(a^6*(d*x)^(1 + m))/(d*(1 + m)) + (6*a^5*b*(d*x)^(3 + m))/(d^3*(3 + m)) + 
(15*a^4*b^2*(d*x)^(5 + m))/(d^5*(5 + m)) + (20*a^3*b^3*(d*x)^(7 + m))/(d^7 
*(7 + m)) + (15*a^2*b^4*(d*x)^(9 + m))/(d^9*(9 + m)) + (6*a*b^5*(d*x)^(11 
+ m))/(d^11*(11 + m)) + (b^6*(d*x)^(13 + m))/(d^13*(13 + m))
 

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 1380
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

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. \(601\) vs. \(2(150)=300\).

Time = 0.45 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.01

method result size
gosper \(\frac {\left (d x \right )^{m} \left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 m \,x^{12} b^{6}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 m \,x^{10} b^{5} a +15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 m \,x^{8} a^{2} b^{4}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 m \,x^{6} a^{3} b^{3}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 m \,x^{4} a^{4} b^{2}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 m \,x^{2} a^{5} b +48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 m \,a^{6}+135135 a^{6}\right ) x}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) \(602\)
risch \(\frac {\left (d x \right )^{m} \left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 m \,x^{12} b^{6}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 m \,x^{10} b^{5} a +15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 m \,x^{8} a^{2} b^{4}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 m \,x^{6} a^{3} b^{3}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 m \,x^{4} a^{4} b^{2}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 m \,x^{2} a^{5} b +48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 m \,a^{6}+135135 a^{6}\right ) x}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) \(602\)
orering \(\frac {\left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 m \,x^{12} b^{6}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 m \,x^{10} b^{5} a +15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 m \,x^{8} a^{2} b^{4}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 m \,x^{6} a^{3} b^{3}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 m \,x^{4} a^{4} b^{2}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 m \,x^{2} a^{5} b +48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 m \,a^{6}+135135 a^{6}\right ) x \left (d x \right )^{m} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{6}}\) \(631\)
parallelrisch \(\frac {6 x^{11} \left (d x \right )^{m} a \,b^{5} m^{6}+228 x^{11} \left (d x \right )^{m} a \,b^{5} m^{5}+3330 x^{11} \left (d x \right )^{m} a \,b^{5} m^{4}+15 x^{9} \left (d x \right )^{m} a^{2} b^{4} m^{6}+23640 x^{11} \left (d x \right )^{m} a \,b^{5} m^{3}+600 x^{9} \left (d x \right )^{m} a^{2} b^{4} m^{5}+84234 x^{11} \left (d x \right )^{m} a \,b^{5} m^{2}+9195 x^{9} \left (d x \right )^{m} a^{2} b^{4} m^{4}+20 x^{7} \left (d x \right )^{m} a^{3} b^{3} m^{6}+137412 x^{11} \left (d x \right )^{m} a \,b^{5} m +67920 x^{9} \left (d x \right )^{m} a^{2} b^{4} m^{3}+840 x^{7} \left (d x \right )^{m} a^{3} b^{3} m^{5}+249405 x^{9} \left (d x \right )^{m} a^{2} b^{4} m^{2}+13580 x^{7} \left (d x \right )^{m} a^{3} b^{3} m^{4}+15 x^{5} \left (d x \right )^{m} a^{4} b^{2} m^{6}+415320 x^{9} \left (d x \right )^{m} a^{2} b^{4} m +699720 x^{7} \left (d x \right )^{m} a^{3} b^{3} m +94200 x^{5} \left (d x \right )^{m} a^{4} b^{2} m^{3}+276 x^{3} \left (d x \right )^{m} a^{5} b \,m^{5}+389685 x^{5} \left (d x \right )^{m} a^{4} b^{2} m^{2}+5010 x^{3} \left (d x \right )^{m} a^{5} b \,m^{4}+711540 x^{5} \left (d x \right )^{m} a^{4} b^{2} m +45240 x^{3} \left (d x \right )^{m} a^{5} b \,m^{3}+208554 x^{3} \left (d x \right )^{m} a^{5} b \,m^{2}+438324 x^{3} \left (d x \right )^{m} a^{5} b m +105840 x^{7} \left (d x \right )^{m} a^{3} b^{3} m^{3}+660 x^{5} \left (d x \right )^{m} a^{4} b^{2} m^{5}+406700 x^{7} \left (d x \right )^{m} a^{3} b^{3} m^{2}+11295 x^{5} \left (d x \right )^{m} a^{4} b^{2} m^{4}+6 x^{3} \left (d x \right )^{m} a^{5} b \,m^{6}+x^{13} \left (d x \right )^{m} b^{6} m^{6}+36 x^{13} \left (d x \right )^{m} b^{6} m^{5}+505 x^{13} \left (d x \right )^{m} b^{6} m^{4}+3480 x^{13} \left (d x \right )^{m} b^{6} m^{3}+12139 x^{13} \left (d x \right )^{m} b^{6} m^{2}+19524 x^{13} \left (d x \right )^{m} b^{6} m +73710 x^{11} \left (d x \right )^{m} a \,b^{5}+225225 x^{9} \left (d x \right )^{m} a^{2} b^{4}+386100 x^{7} \left (d x \right )^{m} a^{3} b^{3}+x \left (d x \right )^{m} a^{6} m^{6}+48 x \left (d x \right )^{m} a^{6} m^{5}+405405 x^{5} \left (d x \right )^{m} a^{4} b^{2}+925 x \left (d x \right )^{m} a^{6} m^{4}+9120 x \left (d x \right )^{m} a^{6} m^{3}+270270 x^{3} \left (d x \right )^{m} a^{5} b +48259 x \left (d x \right )^{m} a^{6} m^{2}+129072 x \left (d x \right )^{m} a^{6} m +135135 x \left (d x \right )^{m} a^{6}+10395 x^{13} \left (d x \right )^{m} b^{6}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) \(848\)

Input:

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

(d*x)^m*(b^6*m^6*x^12+36*b^6*m^5*x^12+6*a*b^5*m^6*x^10+505*b^6*m^4*x^12+22 
8*a*b^5*m^5*x^10+3480*b^6*m^3*x^12+15*a^2*b^4*m^6*x^8+3330*a*b^5*m^4*x^10+ 
12139*b^6*m^2*x^12+600*a^2*b^4*m^5*x^8+23640*a*b^5*m^3*x^10+19524*b^6*m*x^ 
12+20*a^3*b^3*m^6*x^6+9195*a^2*b^4*m^4*x^8+84234*a*b^5*m^2*x^10+10395*b^6* 
x^12+840*a^3*b^3*m^5*x^6+67920*a^2*b^4*m^3*x^8+137412*a*b^5*m*x^10+15*a^4* 
b^2*m^6*x^4+13580*a^3*b^3*m^4*x^6+249405*a^2*b^4*m^2*x^8+73710*a*b^5*x^10+ 
660*a^4*b^2*m^5*x^4+105840*a^3*b^3*m^3*x^6+415320*a^2*b^4*m*x^8+6*a^5*b*m^ 
6*x^2+11295*a^4*b^2*m^4*x^4+406700*a^3*b^3*m^2*x^6+225225*a^2*b^4*x^8+276* 
a^5*b*m^5*x^2+94200*a^4*b^2*m^3*x^4+699720*a^3*b^3*m*x^6+a^6*m^6+5010*a^5* 
b*m^4*x^2+389685*a^4*b^2*m^2*x^4+386100*a^3*b^3*x^6+48*a^6*m^5+45240*a^5*b 
*m^3*x^2+711540*a^4*b^2*m*x^4+925*a^6*m^4+208554*a^5*b*m^2*x^2+405405*a^4* 
b^2*x^4+9120*a^6*m^3+438324*a^5*b*m*x^2+48259*a^6*m^2+270270*a^5*b*x^2+129 
072*a^6*m+135135*a^6)*x/(13+m)/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (150) = 300\).

Time = 0.10 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.38 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {{\left ({\left (b^{6} m^{6} + 36 \, b^{6} m^{5} + 505 \, b^{6} m^{4} + 3480 \, b^{6} m^{3} + 12139 \, b^{6} m^{2} + 19524 \, b^{6} m + 10395 \, b^{6}\right )} x^{13} + 6 \, {\left (a b^{5} m^{6} + 38 \, a b^{5} m^{5} + 555 \, a b^{5} m^{4} + 3940 \, a b^{5} m^{3} + 14039 \, a b^{5} m^{2} + 22902 \, a b^{5} m + 12285 \, a b^{5}\right )} x^{11} + 15 \, {\left (a^{2} b^{4} m^{6} + 40 \, a^{2} b^{4} m^{5} + 613 \, a^{2} b^{4} m^{4} + 4528 \, a^{2} b^{4} m^{3} + 16627 \, a^{2} b^{4} m^{2} + 27688 \, a^{2} b^{4} m + 15015 \, a^{2} b^{4}\right )} x^{9} + 20 \, {\left (a^{3} b^{3} m^{6} + 42 \, a^{3} b^{3} m^{5} + 679 \, a^{3} b^{3} m^{4} + 5292 \, a^{3} b^{3} m^{3} + 20335 \, a^{3} b^{3} m^{2} + 34986 \, a^{3} b^{3} m + 19305 \, a^{3} b^{3}\right )} x^{7} + 15 \, {\left (a^{4} b^{2} m^{6} + 44 \, a^{4} b^{2} m^{5} + 753 \, a^{4} b^{2} m^{4} + 6280 \, a^{4} b^{2} m^{3} + 25979 \, a^{4} b^{2} m^{2} + 47436 \, a^{4} b^{2} m + 27027 \, a^{4} b^{2}\right )} x^{5} + 6 \, {\left (a^{5} b m^{6} + 46 \, a^{5} b m^{5} + 835 \, a^{5} b m^{4} + 7540 \, a^{5} b m^{3} + 34759 \, a^{5} b m^{2} + 73054 \, a^{5} b m + 45045 \, a^{5} b\right )} x^{3} + {\left (a^{6} m^{6} + 48 \, a^{6} m^{5} + 925 \, a^{6} m^{4} + 9120 \, a^{6} m^{3} + 48259 \, a^{6} m^{2} + 129072 \, a^{6} m + 135135 \, a^{6}\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
 

Output:

((b^6*m^6 + 36*b^6*m^5 + 505*b^6*m^4 + 3480*b^6*m^3 + 12139*b^6*m^2 + 1952 
4*b^6*m + 10395*b^6)*x^13 + 6*(a*b^5*m^6 + 38*a*b^5*m^5 + 555*a*b^5*m^4 + 
3940*a*b^5*m^3 + 14039*a*b^5*m^2 + 22902*a*b^5*m + 12285*a*b^5)*x^11 + 15* 
(a^2*b^4*m^6 + 40*a^2*b^4*m^5 + 613*a^2*b^4*m^4 + 4528*a^2*b^4*m^3 + 16627 
*a^2*b^4*m^2 + 27688*a^2*b^4*m + 15015*a^2*b^4)*x^9 + 20*(a^3*b^3*m^6 + 42 
*a^3*b^3*m^5 + 679*a^3*b^3*m^4 + 5292*a^3*b^3*m^3 + 20335*a^3*b^3*m^2 + 34 
986*a^3*b^3*m + 19305*a^3*b^3)*x^7 + 15*(a^4*b^2*m^6 + 44*a^4*b^2*m^5 + 75 
3*a^4*b^2*m^4 + 6280*a^4*b^2*m^3 + 25979*a^4*b^2*m^2 + 47436*a^4*b^2*m + 2 
7027*a^4*b^2)*x^5 + 6*(a^5*b*m^6 + 46*a^5*b*m^5 + 835*a^5*b*m^4 + 7540*a^5 
*b*m^3 + 34759*a^5*b*m^2 + 73054*a^5*b*m + 45045*a^5*b)*x^3 + (a^6*m^6 + 4 
8*a^6*m^5 + 925*a^6*m^4 + 9120*a^6*m^3 + 48259*a^6*m^2 + 129072*a^6*m + 13 
5135*a^6)*x)*(d*x)^m/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177 
331*m^2 + 264207*m + 135135)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3104 vs. \(2 (138) = 276\).

Time = 0.82 (sec) , antiderivative size = 3104, normalized size of antiderivative = 20.69 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
 

Output:

Piecewise(((-a**6/(12*x**12) - 3*a**5*b/(5*x**10) - 15*a**4*b**2/(8*x**8) 
- 10*a**3*b**3/(3*x**6) - 15*a**2*b**4/(4*x**4) - 3*a*b**5/x**2 + b**6*log 
(x))/d**13, Eq(m, -13)), ((-a**6/(10*x**10) - 3*a**5*b/(4*x**8) - 5*a**4*b 
**2/(2*x**6) - 5*a**3*b**3/x**4 - 15*a**2*b**4/(2*x**2) + 6*a*b**5*log(x) 
+ b**6*x**2/2)/d**11, Eq(m, -11)), ((-a**6/(8*x**8) - a**5*b/x**6 - 15*a** 
4*b**2/(4*x**4) - 10*a**3*b**3/x**2 + 15*a**2*b**4*log(x) + 3*a*b**5*x**2 
+ b**6*x**4/4)/d**9, Eq(m, -9)), ((-a**6/(6*x**6) - 3*a**5*b/(2*x**4) - 15 
*a**4*b**2/(2*x**2) + 20*a**3*b**3*log(x) + 15*a**2*b**4*x**2/2 + 3*a*b**5 
*x**4/2 + b**6*x**6/6)/d**7, Eq(m, -7)), ((-a**6/(4*x**4) - 3*a**5*b/x**2 
+ 15*a**4*b**2*log(x) + 10*a**3*b**3*x**2 + 15*a**2*b**4*x**4/4 + a*b**5*x 
**6 + b**6*x**8/8)/d**5, Eq(m, -5)), ((-a**6/(2*x**2) + 6*a**5*b*log(x) + 
15*a**4*b**2*x**2/2 + 5*a**3*b**3*x**4 + 5*a**2*b**4*x**6/2 + 3*a*b**5*x** 
8/4 + b**6*x**10/10)/d**3, Eq(m, -3)), ((a**6*log(x) + 3*a**5*b*x**2 + 15* 
a**4*b**2*x**4/4 + 10*a**3*b**3*x**6/3 + 15*a**2*b**4*x**8/8 + 3*a*b**5*x* 
*10/5 + b**6*x**12/12)/d, Eq(m, -1)), (a**6*m**6*x*(d*x)**m/(m**7 + 49*m** 
6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) 
+ 48*a**6*m**5*x*(d*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379* 
m**3 + 177331*m**2 + 264207*m + 135135) + 925*a**6*m**4*x*(d*x)**m/(m**7 + 
 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 1 
35135) + 9120*a**6*m**3*x*(d*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {b^{6} d^{m} x^{13} x^{m}}{m + 13} + \frac {6 \, a b^{5} d^{m} x^{11} x^{m}}{m + 11} + \frac {15 \, a^{2} b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {20 \, a^{3} b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {15 \, a^{4} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, a^{5} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{6}}{d {\left (m + 1\right )}} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
 

Output:

b^6*d^m*x^13*x^m/(m + 13) + 6*a*b^5*d^m*x^11*x^m/(m + 11) + 15*a^2*b^4*d^m 
*x^9*x^m/(m + 9) + 20*a^3*b^3*d^m*x^7*x^m/(m + 7) + 15*a^4*b^2*d^m*x^5*x^m 
/(m + 5) + 6*a^5*b*d^m*x^3*x^m/(m + 3) + (d*x)^(m + 1)*a^6/(d*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (150) = 300\).

Time = 0.15 (sec) , antiderivative size = 847, normalized size of antiderivative = 5.65 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx =\text {Too large to display} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
 

Output:

((d*x)^m*b^6*m^6*x^13 + 36*(d*x)^m*b^6*m^5*x^13 + 6*(d*x)^m*a*b^5*m^6*x^11 
 + 505*(d*x)^m*b^6*m^4*x^13 + 228*(d*x)^m*a*b^5*m^5*x^11 + 3480*(d*x)^m*b^ 
6*m^3*x^13 + 15*(d*x)^m*a^2*b^4*m^6*x^9 + 3330*(d*x)^m*a*b^5*m^4*x^11 + 12 
139*(d*x)^m*b^6*m^2*x^13 + 600*(d*x)^m*a^2*b^4*m^5*x^9 + 23640*(d*x)^m*a*b 
^5*m^3*x^11 + 19524*(d*x)^m*b^6*m*x^13 + 20*(d*x)^m*a^3*b^3*m^6*x^7 + 9195 
*(d*x)^m*a^2*b^4*m^4*x^9 + 84234*(d*x)^m*a*b^5*m^2*x^11 + 10395*(d*x)^m*b^ 
6*x^13 + 840*(d*x)^m*a^3*b^3*m^5*x^7 + 67920*(d*x)^m*a^2*b^4*m^3*x^9 + 137 
412*(d*x)^m*a*b^5*m*x^11 + 15*(d*x)^m*a^4*b^2*m^6*x^5 + 13580*(d*x)^m*a^3* 
b^3*m^4*x^7 + 249405*(d*x)^m*a^2*b^4*m^2*x^9 + 73710*(d*x)^m*a*b^5*x^11 + 
660*(d*x)^m*a^4*b^2*m^5*x^5 + 105840*(d*x)^m*a^3*b^3*m^3*x^7 + 415320*(d*x 
)^m*a^2*b^4*m*x^9 + 6*(d*x)^m*a^5*b*m^6*x^3 + 11295*(d*x)^m*a^4*b^2*m^4*x^ 
5 + 406700*(d*x)^m*a^3*b^3*m^2*x^7 + 225225*(d*x)^m*a^2*b^4*x^9 + 276*(d*x 
)^m*a^5*b*m^5*x^3 + 94200*(d*x)^m*a^4*b^2*m^3*x^5 + 699720*(d*x)^m*a^3*b^3 
*m*x^7 + (d*x)^m*a^6*m^6*x + 5010*(d*x)^m*a^5*b*m^4*x^3 + 389685*(d*x)^m*a 
^4*b^2*m^2*x^5 + 386100*(d*x)^m*a^3*b^3*x^7 + 48*(d*x)^m*a^6*m^5*x + 45240 
*(d*x)^m*a^5*b*m^3*x^3 + 711540*(d*x)^m*a^4*b^2*m*x^5 + 925*(d*x)^m*a^6*m^ 
4*x + 208554*(d*x)^m*a^5*b*m^2*x^3 + 405405*(d*x)^m*a^4*b^2*x^5 + 9120*(d* 
x)^m*a^6*m^3*x + 438324*(d*x)^m*a^5*b*m*x^3 + 48259*(d*x)^m*a^6*m^2*x + 27 
0270*(d*x)^m*a^5*b*x^3 + 129072*(d*x)^m*a^6*m*x + 135135*(d*x)^m*a^6*x)/(m 
^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177331*m^2 + 264207*m +...
 

Mupad [B] (verification not implemented)

Time = 18.45 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.60 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {a^6\,x\,{\left (d\,x\right )}^m\,\left (m^6+48\,m^5+925\,m^4+9120\,m^3+48259\,m^2+129072\,m+135135\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b^6\,x^{13}\,{\left (d\,x\right )}^m\,\left (m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {6\,a\,b^5\,x^{11}\,{\left (d\,x\right )}^m\,\left (m^6+38\,m^5+555\,m^4+3940\,m^3+14039\,m^2+22902\,m+12285\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {6\,a^5\,b\,x^3\,{\left (d\,x\right )}^m\,\left (m^6+46\,m^5+835\,m^4+7540\,m^3+34759\,m^2+73054\,m+45045\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {15\,a^2\,b^4\,x^9\,{\left (d\,x\right )}^m\,\left (m^6+40\,m^5+613\,m^4+4528\,m^3+16627\,m^2+27688\,m+15015\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {20\,a^3\,b^3\,x^7\,{\left (d\,x\right )}^m\,\left (m^6+42\,m^5+679\,m^4+5292\,m^3+20335\,m^2+34986\,m+19305\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {15\,a^4\,b^2\,x^5\,{\left (d\,x\right )}^m\,\left (m^6+44\,m^5+753\,m^4+6280\,m^3+25979\,m^2+47436\,m+27027\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135} \] Input:

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
 

Output:

(a^6*x*(d*x)^m*(129072*m + 48259*m^2 + 9120*m^3 + 925*m^4 + 48*m^5 + m^6 + 
 135135))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^ 
6 + m^7 + 135135) + (b^6*x^13*(d*x)^m*(19524*m + 12139*m^2 + 3480*m^3 + 50 
5*m^4 + 36*m^5 + m^6 + 10395))/(264207*m + 177331*m^2 + 57379*m^3 + 10045* 
m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (6*a*b^5*x^11*(d*x)^m*(22902*m + 
14039*m^2 + 3940*m^3 + 555*m^4 + 38*m^5 + m^6 + 12285))/(264207*m + 177331 
*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (6*a^5*b 
*x^3*(d*x)^m*(73054*m + 34759*m^2 + 7540*m^3 + 835*m^4 + 46*m^5 + m^6 + 45 
045))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + 
m^7 + 135135) + (15*a^2*b^4*x^9*(d*x)^m*(27688*m + 16627*m^2 + 4528*m^3 + 
613*m^4 + 40*m^5 + m^6 + 15015))/(264207*m + 177331*m^2 + 57379*m^3 + 1004 
5*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (20*a^3*b^3*x^7*(d*x)^m*(34986* 
m + 20335*m^2 + 5292*m^3 + 679*m^4 + 42*m^5 + m^6 + 19305))/(264207*m + 17 
7331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (15* 
a^4*b^2*x^5*(d*x)^m*(47436*m + 25979*m^2 + 6280*m^3 + 753*m^4 + 44*m^5 + m 
^6 + 27027))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49 
*m^6 + m^7 + 135135)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.01 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx=\frac {x^{m} d^{m} x \left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 b^{6} m \,x^{12}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 a \,b^{5} m \,x^{10}+15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 a^{2} b^{4} m \,x^{8}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 a^{3} b^{3} m \,x^{6}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 a^{4} b^{2} m \,x^{4}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 a^{5} b m \,x^{2}+48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 a^{6} m +135135 a^{6}\right )}{m^{7}+49 m^{6}+973 m^{5}+10045 m^{4}+57379 m^{3}+177331 m^{2}+264207 m +135135} \] Input:

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x)
 

Output:

(x**m*d**m*x*(a**6*m**6 + 48*a**6*m**5 + 925*a**6*m**4 + 9120*a**6*m**3 + 
48259*a**6*m**2 + 129072*a**6*m + 135135*a**6 + 6*a**5*b*m**6*x**2 + 276*a 
**5*b*m**5*x**2 + 5010*a**5*b*m**4*x**2 + 45240*a**5*b*m**3*x**2 + 208554* 
a**5*b*m**2*x**2 + 438324*a**5*b*m*x**2 + 270270*a**5*b*x**2 + 15*a**4*b** 
2*m**6*x**4 + 660*a**4*b**2*m**5*x**4 + 11295*a**4*b**2*m**4*x**4 + 94200* 
a**4*b**2*m**3*x**4 + 389685*a**4*b**2*m**2*x**4 + 711540*a**4*b**2*m*x**4 
 + 405405*a**4*b**2*x**4 + 20*a**3*b**3*m**6*x**6 + 840*a**3*b**3*m**5*x** 
6 + 13580*a**3*b**3*m**4*x**6 + 105840*a**3*b**3*m**3*x**6 + 406700*a**3*b 
**3*m**2*x**6 + 699720*a**3*b**3*m*x**6 + 386100*a**3*b**3*x**6 + 15*a**2* 
b**4*m**6*x**8 + 600*a**2*b**4*m**5*x**8 + 9195*a**2*b**4*m**4*x**8 + 6792 
0*a**2*b**4*m**3*x**8 + 249405*a**2*b**4*m**2*x**8 + 415320*a**2*b**4*m*x* 
*8 + 225225*a**2*b**4*x**8 + 6*a*b**5*m**6*x**10 + 228*a*b**5*m**5*x**10 + 
 3330*a*b**5*m**4*x**10 + 23640*a*b**5*m**3*x**10 + 84234*a*b**5*m**2*x**1 
0 + 137412*a*b**5*m*x**10 + 73710*a*b**5*x**10 + b**6*m**6*x**12 + 36*b**6 
*m**5*x**12 + 505*b**6*m**4*x**12 + 3480*b**6*m**3*x**12 + 12139*b**6*m**2 
*x**12 + 19524*b**6*m*x**12 + 10395*b**6*x**12))/(m**7 + 49*m**6 + 973*m** 
5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)