\(\int (d x)^m (a x+b x^3+c x^5)^3 \, dx\) [69]

Optimal result

Integrand size = 22, antiderivative size = 156 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\frac {a^3 (d x)^{4+m}}{d^4 (4+m)}+\frac {3 a^2 b (d x)^{6+m}}{d^6 (6+m)}+\frac {3 a \left (b^2+a c\right ) (d x)^{8+m}}{d^8 (8+m)}+\frac {b \left (b^2+6 a c\right ) (d x)^{10+m}}{d^{10} (10+m)}+\frac {3 c \left (b^2+a c\right ) (d x)^{12+m}}{d^{12} (12+m)}+\frac {3 b c^2 (d x)^{14+m}}{d^{14} (14+m)}+\frac {c^3 (d x)^{16+m}}{d^{16} (16+m)} \] Output:

a^3*(d*x)^(4+m)/d^4/(4+m)+3*a^2*b*(d*x)^(6+m)/d^6/(6+m)+3*a*(a*c+b^2)*(d*x 
)^(8+m)/d^8/(8+m)+b*(6*a*c+b^2)*(d*x)^(10+m)/d^10/(10+m)+3*c*(a*c+b^2)*(d* 
x)^(12+m)/d^12/(12+m)+3*b*c^2*(d*x)^(14+m)/d^14/(14+m)+c^3*(d*x)^(16+m)/d^ 
16/(16+m)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=x^4 (d x)^m \left (\frac {a^3}{4+m}+\frac {3 a^2 b x^2}{6+m}+\frac {3 a \left (b^2+a c\right ) x^4}{8+m}+\frac {b \left (b^2+6 a c\right ) x^6}{10+m}+\frac {3 c \left (b^2+a c\right ) x^8}{12+m}+\frac {3 b c^2 x^{10}}{14+m}+\frac {c^3 x^{12}}{16+m}\right ) \] Input:

Integrate[(d*x)^m*(a*x + b*x^3 + c*x^5)^3,x]
 

Output:

x^4*(d*x)^m*(a^3/(4 + m) + (3*a^2*b*x^2)/(6 + m) + (3*a*(b^2 + a*c)*x^4)/( 
8 + m) + (b*(b^2 + 6*a*c)*x^6)/(10 + m) + (3*c*(b^2 + a*c)*x^8)/(12 + m) + 
 (3*b*c^2*x^10)/(14 + m) + (c^3*x^12)/(16 + m))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 1433, 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.

\(\displaystyle \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {\int (d x)^{m+3} \left (c x^4+b x^2+a\right )^3dx}{d^3}\)

\(\Big \downarrow \) 1433

\(\displaystyle \frac {\int \left (a^3 (d x)^{m+3}+\frac {3 a^2 b (d x)^{m+5}}{d^2}+\frac {3 a \left (b^2+a c\right ) (d x)^{m+7}}{d^4}+\frac {b \left (b^2+6 a c\right ) (d x)^{m+9}}{d^6}+\frac {3 c \left (b^2+a c\right ) (d x)^{m+11}}{d^8}+\frac {3 b c^2 (d x)^{m+13}}{d^{10}}+\frac {c^3 (d x)^{m+15}}{d^{12}}\right )dx}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^3 (d x)^{m+4}}{d (m+4)}+\frac {3 a^2 b (d x)^{m+6}}{d^3 (m+6)}+\frac {3 c \left (a c+b^2\right ) (d x)^{m+12}}{d^9 (m+12)}+\frac {b \left (6 a c+b^2\right ) (d x)^{m+10}}{d^7 (m+10)}+\frac {3 a \left (a c+b^2\right ) (d x)^{m+8}}{d^5 (m+8)}+\frac {3 b c^2 (d x)^{m+14}}{d^{11} (m+14)}+\frac {c^3 (d x)^{m+16}}{d^{13} (m+16)}}{d^3}\)

Input:

Int[(d*x)^m*(a*x + b*x^3 + c*x^5)^3,x]
 

Output:

((a^3*(d*x)^(4 + m))/(d*(4 + m)) + (3*a^2*b*(d*x)^(6 + m))/(d^3*(6 + m)) + 
 (3*a*(b^2 + a*c)*(d*x)^(8 + m))/(d^5*(8 + m)) + (b*(b^2 + 6*a*c)*(d*x)^(1 
0 + m))/(d^7*(10 + m)) + (3*c*(b^2 + a*c)*(d*x)^(12 + m))/(d^9*(12 + m)) + 
 (3*b*c^2*(d*x)^(14 + m))/(d^11*(14 + m)) + (c^3*(d*x)^(16 + m))/(d^13*(16 
 + m)))/d^3
 

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 1433

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, 
b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] ||  !IntegerQ[(m + 1)/2])
 

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. \(783\) vs. \(2(156)=312\).

Time = 0.42 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.03

method	result	size
gosper	\(\frac {\left (d x \right )^{m} \left (c^{3} m^{6} x^{12}+54 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+1180 c^{3} m^{4} x^{12}+168 b \,c^{2} m^{5} x^{10}+13320 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+3780 b \,c^{2} m^{4} x^{10}+81664 c^{3} m^{2} x^{12}+174 a \,c^{2} m^{5} x^{8}+174 b^{2} c \,m^{5} x^{8}+43680 b \,c^{2} m^{3} x^{10}+256896 m \,x^{12} c^{3}+6 a b c \,m^{6} x^{6}+4044 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+4044 b^{2} c \,m^{4} x^{8}+272832 b \,c^{2} m^{2} x^{10}+322560 c^{3} x^{12}+360 a b c \,m^{5} x^{6}+48072 a \,c^{2} m^{3} x^{8}+60 b^{3} m^{5} x^{6}+48072 b^{2} c \,m^{3} x^{8}+870912 m \,x^{10} b \,c^{2}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+8664 a b c \,m^{4} x^{6}+307488 a \,c^{2} m^{2} x^{8}+1444 b^{3} m^{4} x^{6}+307488 b^{2} c \,m^{2} x^{8}+1105920 b \,c^{2} x^{10}+186 a^{2} c \,m^{5} x^{4}+186 a \,b^{2} m^{5} x^{4}+106560 a b c \,m^{3} x^{6}+1000704 a \,c^{2} m \,x^{8}+17760 b^{3} m^{3} x^{6}+1000704 b^{2} c m \,x^{8}+3 a^{2} b \,m^{6} x^{2}+4644 a^{2} c \,m^{4} x^{4}+4644 a \,b^{2} m^{4} x^{4}+703104 a b c \,m^{2} x^{6}+1290240 a \,c^{2} x^{8}+117184 b^{3} m^{2} x^{6}+1290240 b^{2} c \,x^{8}+192 a^{2} b \,m^{5} x^{2}+59448 a^{2} c \,m^{3} x^{4}+59448 a \,b^{2} m^{3} x^{4}+2350080 a b c m \,x^{6}+391680 b^{3} m \,x^{6}+a^{3} m^{6}+4980 a^{2} b \,m^{4} x^{2}+408768 a^{2} c \,m^{2} x^{4}+408768 a \,b^{2} m^{2} x^{4}+3096576 a b c \,x^{6}+516096 b^{3} x^{6}+66 a^{3} m^{5}+66720 a^{2} b \,m^{3} x^{2}+1420416 a^{2} c m \,x^{4}+1420416 a \,b^{2} m \,x^{4}+1780 a^{3} m^{4}+484032 a^{2} b \,m^{2} x^{2}+1935360 a^{2} c \,x^{4}+1935360 a \,b^{2} x^{4}+25080 a^{3} m^{3}+1786368 a^{2} b m \,x^{2}+194464 a^{3} m^{2}+2580480 b \,a^{2} x^{2}+785664 m \,a^{3}+1290240 a^{3}\right ) x^{4}}{\left (16+m \right ) \left (14+m \right ) \left (12+m \right ) \left (10+m \right ) \left (8+m \right ) \left (6+m \right ) \left (4+m \right )}\)	\(784\)
risch	\(\frac {\left (d x \right )^{m} \left (c^{3} m^{6} x^{12}+54 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+1180 c^{3} m^{4} x^{12}+168 b \,c^{2} m^{5} x^{10}+13320 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+3780 b \,c^{2} m^{4} x^{10}+81664 c^{3} m^{2} x^{12}+174 a \,c^{2} m^{5} x^{8}+174 b^{2} c \,m^{5} x^{8}+43680 b \,c^{2} m^{3} x^{10}+256896 m \,x^{12} c^{3}+6 a b c \,m^{6} x^{6}+4044 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+4044 b^{2} c \,m^{4} x^{8}+272832 b \,c^{2} m^{2} x^{10}+322560 c^{3} x^{12}+360 a b c \,m^{5} x^{6}+48072 a \,c^{2} m^{3} x^{8}+60 b^{3} m^{5} x^{6}+48072 b^{2} c \,m^{3} x^{8}+870912 m \,x^{10} b \,c^{2}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+8664 a b c \,m^{4} x^{6}+307488 a \,c^{2} m^{2} x^{8}+1444 b^{3} m^{4} x^{6}+307488 b^{2} c \,m^{2} x^{8}+1105920 b \,c^{2} x^{10}+186 a^{2} c \,m^{5} x^{4}+186 a \,b^{2} m^{5} x^{4}+106560 a b c \,m^{3} x^{6}+1000704 a \,c^{2} m \,x^{8}+17760 b^{3} m^{3} x^{6}+1000704 b^{2} c m \,x^{8}+3 a^{2} b \,m^{6} x^{2}+4644 a^{2} c \,m^{4} x^{4}+4644 a \,b^{2} m^{4} x^{4}+703104 a b c \,m^{2} x^{6}+1290240 a \,c^{2} x^{8}+117184 b^{3} m^{2} x^{6}+1290240 b^{2} c \,x^{8}+192 a^{2} b \,m^{5} x^{2}+59448 a^{2} c \,m^{3} x^{4}+59448 a \,b^{2} m^{3} x^{4}+2350080 a b c m \,x^{6}+391680 b^{3} m \,x^{6}+a^{3} m^{6}+4980 a^{2} b \,m^{4} x^{2}+408768 a^{2} c \,m^{2} x^{4}+408768 a \,b^{2} m^{2} x^{4}+3096576 a b c \,x^{6}+516096 b^{3} x^{6}+66 a^{3} m^{5}+66720 a^{2} b \,m^{3} x^{2}+1420416 a^{2} c m \,x^{4}+1420416 a \,b^{2} m \,x^{4}+1780 a^{3} m^{4}+484032 a^{2} b \,m^{2} x^{2}+1935360 a^{2} c \,x^{4}+1935360 a \,b^{2} x^{4}+25080 a^{3} m^{3}+1786368 a^{2} b m \,x^{2}+194464 a^{3} m^{2}+2580480 b \,a^{2} x^{2}+785664 m \,a^{3}+1290240 a^{3}\right ) x^{4}}{\left (16+m \right ) \left (14+m \right ) \left (12+m \right ) \left (10+m \right ) \left (8+m \right ) \left (6+m \right ) \left (4+m \right )}\)	\(784\)
orering	\(\frac {\left (c^{3} m^{6} x^{12}+54 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+1180 c^{3} m^{4} x^{12}+168 b \,c^{2} m^{5} x^{10}+13320 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+3780 b \,c^{2} m^{4} x^{10}+81664 c^{3} m^{2} x^{12}+174 a \,c^{2} m^{5} x^{8}+174 b^{2} c \,m^{5} x^{8}+43680 b \,c^{2} m^{3} x^{10}+256896 m \,x^{12} c^{3}+6 a b c \,m^{6} x^{6}+4044 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+4044 b^{2} c \,m^{4} x^{8}+272832 b \,c^{2} m^{2} x^{10}+322560 c^{3} x^{12}+360 a b c \,m^{5} x^{6}+48072 a \,c^{2} m^{3} x^{8}+60 b^{3} m^{5} x^{6}+48072 b^{2} c \,m^{3} x^{8}+870912 m \,x^{10} b \,c^{2}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+8664 a b c \,m^{4} x^{6}+307488 a \,c^{2} m^{2} x^{8}+1444 b^{3} m^{4} x^{6}+307488 b^{2} c \,m^{2} x^{8}+1105920 b \,c^{2} x^{10}+186 a^{2} c \,m^{5} x^{4}+186 a \,b^{2} m^{5} x^{4}+106560 a b c \,m^{3} x^{6}+1000704 a \,c^{2} m \,x^{8}+17760 b^{3} m^{3} x^{6}+1000704 b^{2} c m \,x^{8}+3 a^{2} b \,m^{6} x^{2}+4644 a^{2} c \,m^{4} x^{4}+4644 a \,b^{2} m^{4} x^{4}+703104 a b c \,m^{2} x^{6}+1290240 a \,c^{2} x^{8}+117184 b^{3} m^{2} x^{6}+1290240 b^{2} c \,x^{8}+192 a^{2} b \,m^{5} x^{2}+59448 a^{2} c \,m^{3} x^{4}+59448 a \,b^{2} m^{3} x^{4}+2350080 a b c m \,x^{6}+391680 b^{3} m \,x^{6}+a^{3} m^{6}+4980 a^{2} b \,m^{4} x^{2}+408768 a^{2} c \,m^{2} x^{4}+408768 a \,b^{2} m^{2} x^{4}+3096576 a b c \,x^{6}+516096 b^{3} x^{6}+66 a^{3} m^{5}+66720 a^{2} b \,m^{3} x^{2}+1420416 a^{2} c m \,x^{4}+1420416 a \,b^{2} m \,x^{4}+1780 a^{3} m^{4}+484032 a^{2} b \,m^{2} x^{2}+1935360 a^{2} c \,x^{4}+1935360 a \,b^{2} x^{4}+25080 a^{3} m^{3}+1786368 a^{2} b m \,x^{2}+194464 a^{3} m^{2}+2580480 b \,a^{2} x^{2}+785664 m \,a^{3}+1290240 a^{3}\right ) x \left (d x \right )^{m} \left (c \,x^{5}+b \,x^{3}+x a \right )^{3}}{\left (16+m \right ) \left (14+m \right ) \left (12+m \right ) \left (10+m \right ) \left (8+m \right ) \left (6+m \right ) \left (4+m \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{3}}\)	\(812\)
parallelrisch	\(\text {Expression too large to display}\)	\(1147\)

Input:

int((d*x)^m*(c*x^5+b*x^3+a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

(d*x)^m*(c^3*m^6*x^12+54*c^3*m^5*x^12+3*b*c^2*m^6*x^10+1180*c^3*m^4*x^12+1 
68*b*c^2*m^5*x^10+13320*c^3*m^3*x^12+3*a*c^2*m^6*x^8+3*b^2*c*m^6*x^8+3780* 
b*c^2*m^4*x^10+81664*c^3*m^2*x^12+174*a*c^2*m^5*x^8+174*b^2*c*m^5*x^8+4368 
0*b*c^2*m^3*x^10+256896*c^3*m*x^12+6*a*b*c*m^6*x^6+4044*a*c^2*m^4*x^8+b^3* 
m^6*x^6+4044*b^2*c*m^4*x^8+272832*b*c^2*m^2*x^10+322560*c^3*x^12+360*a*b*c 
*m^5*x^6+48072*a*c^2*m^3*x^8+60*b^3*m^5*x^6+48072*b^2*c*m^3*x^8+870912*b*c 
^2*m*x^10+3*a^2*c*m^6*x^4+3*a*b^2*m^6*x^4+8664*a*b*c*m^4*x^6+307488*a*c^2* 
m^2*x^8+1444*b^3*m^4*x^6+307488*b^2*c*m^2*x^8+1105920*b*c^2*x^10+186*a^2*c 
*m^5*x^4+186*a*b^2*m^5*x^4+106560*a*b*c*m^3*x^6+1000704*a*c^2*m*x^8+17760* 
b^3*m^3*x^6+1000704*b^2*c*m*x^8+3*a^2*b*m^6*x^2+4644*a^2*c*m^4*x^4+4644*a* 
b^2*m^4*x^4+703104*a*b*c*m^2*x^6+1290240*a*c^2*x^8+117184*b^3*m^2*x^6+1290 
240*b^2*c*x^8+192*a^2*b*m^5*x^2+59448*a^2*c*m^3*x^4+59448*a*b^2*m^3*x^4+23 
50080*a*b*c*m*x^6+391680*b^3*m*x^6+a^3*m^6+4980*a^2*b*m^4*x^2+408768*a^2*c 
*m^2*x^4+408768*a*b^2*m^2*x^4+3096576*a*b*c*x^6+516096*b^3*x^6+66*a^3*m^5+ 
66720*a^2*b*m^3*x^2+1420416*a^2*c*m*x^4+1420416*a*b^2*m*x^4+1780*a^3*m^4+4 
84032*a^2*b*m^2*x^2+1935360*a^2*c*x^4+1935360*a*b^2*x^4+25080*a^3*m^3+1786 
368*a^2*b*m*x^2+194464*a^3*m^2+2580480*a^2*b*x^2+785664*a^3*m+1290240*a^3) 
*x^4/(16+m)/(14+m)/(12+m)/(10+m)/(8+m)/(6+m)/(4+m)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (156) = 312\).

Time = 0.08 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.82 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\frac {{\left ({\left (c^{3} m^{6} + 54 \, c^{3} m^{5} + 1180 \, c^{3} m^{4} + 13320 \, c^{3} m^{3} + 81664 \, c^{3} m^{2} + 256896 \, c^{3} m + 322560 \, c^{3}\right )} x^{16} + 3 \, {\left (b c^{2} m^{6} + 56 \, b c^{2} m^{5} + 1260 \, b c^{2} m^{4} + 14560 \, b c^{2} m^{3} + 90944 \, b c^{2} m^{2} + 290304 \, b c^{2} m + 368640 \, b c^{2}\right )} x^{14} + 3 \, {\left ({\left (b^{2} c + a c^{2}\right )} m^{6} + 58 \, {\left (b^{2} c + a c^{2}\right )} m^{5} + 1348 \, {\left (b^{2} c + a c^{2}\right )} m^{4} + 16024 \, {\left (b^{2} c + a c^{2}\right )} m^{3} + 430080 \, b^{2} c + 430080 \, a c^{2} + 102496 \, {\left (b^{2} c + a c^{2}\right )} m^{2} + 333568 \, {\left (b^{2} c + a c^{2}\right )} m\right )} x^{12} + {\left ({\left (b^{3} + 6 \, a b c\right )} m^{6} + 60 \, {\left (b^{3} + 6 \, a b c\right )} m^{5} + 1444 \, {\left (b^{3} + 6 \, a b c\right )} m^{4} + 17760 \, {\left (b^{3} + 6 \, a b c\right )} m^{3} + 516096 \, b^{3} + 3096576 \, a b c + 117184 \, {\left (b^{3} + 6 \, a b c\right )} m^{2} + 391680 \, {\left (b^{3} + 6 \, a b c\right )} m\right )} x^{10} + 3 \, {\left ({\left (a b^{2} + a^{2} c\right )} m^{6} + 62 \, {\left (a b^{2} + a^{2} c\right )} m^{5} + 1548 \, {\left (a b^{2} + a^{2} c\right )} m^{4} + 19816 \, {\left (a b^{2} + a^{2} c\right )} m^{3} + 645120 \, a b^{2} + 645120 \, a^{2} c + 136256 \, {\left (a b^{2} + a^{2} c\right )} m^{2} + 473472 \, {\left (a b^{2} + a^{2} c\right )} m\right )} x^{8} + 3 \, {\left (a^{2} b m^{6} + 64 \, a^{2} b m^{5} + 1660 \, a^{2} b m^{4} + 22240 \, a^{2} b m^{3} + 161344 \, a^{2} b m^{2} + 595456 \, a^{2} b m + 860160 \, a^{2} b\right )} x^{6} + {\left (a^{3} m^{6} + 66 \, a^{3} m^{5} + 1780 \, a^{3} m^{4} + 25080 \, a^{3} m^{3} + 194464 \, a^{3} m^{2} + 785664 \, a^{3} m + 1290240 \, a^{3}\right )} x^{4}\right )} \left (d x\right )^{m}}{m^{7} + 70 \, m^{6} + 2044 \, m^{5} + 32200 \, m^{4} + 294784 \, m^{3} + 1563520 \, m^{2} + 4432896 \, m + 5160960} \] Input:

integrate((d*x)^m*(c*x^5+b*x^3+a*x)^3,x, algorithm="fricas")
 

Output:

((c^3*m^6 + 54*c^3*m^5 + 1180*c^3*m^4 + 13320*c^3*m^3 + 81664*c^3*m^2 + 25 
6896*c^3*m + 322560*c^3)*x^16 + 3*(b*c^2*m^6 + 56*b*c^2*m^5 + 1260*b*c^2*m 
^4 + 14560*b*c^2*m^3 + 90944*b*c^2*m^2 + 290304*b*c^2*m + 368640*b*c^2)*x^ 
14 + 3*((b^2*c + a*c^2)*m^6 + 58*(b^2*c + a*c^2)*m^5 + 1348*(b^2*c + a*c^2 
)*m^4 + 16024*(b^2*c + a*c^2)*m^3 + 430080*b^2*c + 430080*a*c^2 + 102496*( 
b^2*c + a*c^2)*m^2 + 333568*(b^2*c + a*c^2)*m)*x^12 + ((b^3 + 6*a*b*c)*m^6 
 + 60*(b^3 + 6*a*b*c)*m^5 + 1444*(b^3 + 6*a*b*c)*m^4 + 17760*(b^3 + 6*a*b* 
c)*m^3 + 516096*b^3 + 3096576*a*b*c + 117184*(b^3 + 6*a*b*c)*m^2 + 391680* 
(b^3 + 6*a*b*c)*m)*x^10 + 3*((a*b^2 + a^2*c)*m^6 + 62*(a*b^2 + a^2*c)*m^5 
+ 1548*(a*b^2 + a^2*c)*m^4 + 19816*(a*b^2 + a^2*c)*m^3 + 645120*a*b^2 + 64 
5120*a^2*c + 136256*(a*b^2 + a^2*c)*m^2 + 473472*(a*b^2 + a^2*c)*m)*x^8 + 
3*(a^2*b*m^6 + 64*a^2*b*m^5 + 1660*a^2*b*m^4 + 22240*a^2*b*m^3 + 161344*a^ 
2*b*m^2 + 595456*a^2*b*m + 860160*a^2*b)*x^6 + (a^3*m^6 + 66*a^3*m^5 + 178 
0*a^3*m^4 + 25080*a^3*m^3 + 194464*a^3*m^2 + 785664*a^3*m + 1290240*a^3)*x 
^4)*(d*x)^m/(m^7 + 70*m^6 + 2044*m^5 + 32200*m^4 + 294784*m^3 + 1563520*m^ 
2 + 4432896*m + 5160960)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4345 vs. \(2 (144) = 288\).

Time = 1.42 (sec) , antiderivative size = 4345, normalized size of antiderivative = 27.85 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x)**m*(c*x**5+b*x**3+a*x)**3,x)
 

Output:

Piecewise(((-a**3/(12*x**12) - 3*a**2*b/(10*x**10) - 3*a**2*c/(8*x**8) - 3 
*a*b**2/(8*x**8) - a*b*c/x**6 - 3*a*c**2/(4*x**4) - b**3/(6*x**6) - 3*b**2 
*c/(4*x**4) - 3*b*c**2/(2*x**2) + c**3*log(x))/d**16, Eq(m, -16)), ((-a**3 
/(10*x**10) - 3*a**2*b/(8*x**8) - a**2*c/(2*x**6) - a*b**2/(2*x**6) - 3*a* 
b*c/(2*x**4) - 3*a*c**2/(2*x**2) - b**3/(4*x**4) - 3*b**2*c/(2*x**2) + 3*b 
*c**2*log(x) + c**3*x**2/2)/d**14, Eq(m, -14)), ((-a**3/(8*x**8) - a**2*b/ 
(2*x**6) - 3*a**2*c/(4*x**4) - 3*a*b**2/(4*x**4) - 3*a*b*c/x**2 + 3*a*c**2 
*log(x) - b**3/(2*x**2) + 3*b**2*c*log(x) + 3*b*c**2*x**2/2 + c**3*x**4/4) 
/d**12, Eq(m, -12)), ((-a**3/(6*x**6) - 3*a**2*b/(4*x**4) - 3*a**2*c/(2*x* 
*2) - 3*a*b**2/(2*x**2) + 6*a*b*c*log(x) + 3*a*c**2*x**2/2 + b**3*log(x) + 
 3*b**2*c*x**2/2 + 3*b*c**2*x**4/4 + c**3*x**6/6)/d**10, Eq(m, -10)), ((-a 
**3/(4*x**4) - 3*a**2*b/(2*x**2) + 3*a**2*c*log(x) + 3*a*b**2*log(x) + 3*a 
*b*c*x**2 + 3*a*c**2*x**4/4 + b**3*x**2/2 + 3*b**2*c*x**4/4 + b*c**2*x**6/ 
2 + c**3*x**8/8)/d**8, Eq(m, -8)), ((-a**3/(2*x**2) + 3*a**2*b*log(x) + 3* 
a**2*c*x**2/2 + 3*a*b**2*x**2/2 + 3*a*b*c*x**4/2 + a*c**2*x**6/2 + b**3*x* 
*4/4 + b**2*c*x**6/2 + 3*b*c**2*x**8/8 + c**3*x**10/10)/d**6, Eq(m, -6)), 
((a**3*log(x) + 3*a**2*b*x**2/2 + 3*a**2*c*x**4/4 + 3*a*b**2*x**4/4 + a*b* 
c*x**6 + 3*a*c**2*x**8/8 + b**3*x**6/6 + 3*b**2*c*x**8/8 + 3*b*c**2*x**10/ 
10 + c**3*x**12/12)/d**4, Eq(m, -4)), (a**3*m**6*x**4*(d*x)**m/(m**7 + 70* 
m**6 + 2044*m**5 + 32200*m**4 + 294784*m**3 + 1563520*m**2 + 4432896*m ...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.24 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\frac {c^{3} d^{m} x^{16} x^{m}}{m + 16} + \frac {3 \, b c^{2} d^{m} x^{14} x^{m}}{m + 14} + \frac {3 \, b^{2} c d^{m} x^{12} x^{m}}{m + 12} + \frac {3 \, a c^{2} d^{m} x^{12} x^{m}}{m + 12} + \frac {b^{3} d^{m} x^{10} x^{m}}{m + 10} + \frac {6 \, a b c d^{m} x^{10} x^{m}}{m + 10} + \frac {3 \, a b^{2} d^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, a^{2} c d^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, a^{2} b d^{m} x^{6} x^{m}}{m + 6} + \frac {a^{3} d^{m} x^{4} x^{m}}{m + 4} \] Input:

integrate((d*x)^m*(c*x^5+b*x^3+a*x)^3,x, algorithm="maxima")
 

Output:

c^3*d^m*x^16*x^m/(m + 16) + 3*b*c^2*d^m*x^14*x^m/(m + 14) + 3*b^2*c*d^m*x^ 
12*x^m/(m + 12) + 3*a*c^2*d^m*x^12*x^m/(m + 12) + b^3*d^m*x^10*x^m/(m + 10 
) + 6*a*b*c*d^m*x^10*x^m/(m + 10) + 3*a*b^2*d^m*x^8*x^m/(m + 8) + 3*a^2*c* 
d^m*x^8*x^m/(m + 8) + 3*a^2*b*d^m*x^6*x^m/(m + 6) + a^3*d^m*x^4*x^m/(m + 4 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1146 vs. \(2 (156) = 312\).

Time = 0.15 (sec) , antiderivative size = 1146, normalized size of antiderivative = 7.35 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x)^m*(c*x^5+b*x^3+a*x)^3,x, algorithm="giac")
 

Output:

((d*x)^m*c^3*m^6*x^16 + 54*(d*x)^m*c^3*m^5*x^16 + 3*(d*x)^m*b*c^2*m^6*x^14 
 + 1180*(d*x)^m*c^3*m^4*x^16 + 168*(d*x)^m*b*c^2*m^5*x^14 + 13320*(d*x)^m* 
c^3*m^3*x^16 + 3*(d*x)^m*b^2*c*m^6*x^12 + 3*(d*x)^m*a*c^2*m^6*x^12 + 3780* 
(d*x)^m*b*c^2*m^4*x^14 + 81664*(d*x)^m*c^3*m^2*x^16 + 174*(d*x)^m*b^2*c*m^ 
5*x^12 + 174*(d*x)^m*a*c^2*m^5*x^12 + 43680*(d*x)^m*b*c^2*m^3*x^14 + 25689 
6*(d*x)^m*c^3*m*x^16 + (d*x)^m*b^3*m^6*x^10 + 6*(d*x)^m*a*b*c*m^6*x^10 + 4 
044*(d*x)^m*b^2*c*m^4*x^12 + 4044*(d*x)^m*a*c^2*m^4*x^12 + 272832*(d*x)^m* 
b*c^2*m^2*x^14 + 322560*(d*x)^m*c^3*x^16 + 60*(d*x)^m*b^3*m^5*x^10 + 360*( 
d*x)^m*a*b*c*m^5*x^10 + 48072*(d*x)^m*b^2*c*m^3*x^12 + 48072*(d*x)^m*a*c^2 
*m^3*x^12 + 870912*(d*x)^m*b*c^2*m*x^14 + 3*(d*x)^m*a*b^2*m^6*x^8 + 3*(d*x 
)^m*a^2*c*m^6*x^8 + 1444*(d*x)^m*b^3*m^4*x^10 + 8664*(d*x)^m*a*b*c*m^4*x^1 
0 + 307488*(d*x)^m*b^2*c*m^2*x^12 + 307488*(d*x)^m*a*c^2*m^2*x^12 + 110592 
0*(d*x)^m*b*c^2*x^14 + 186*(d*x)^m*a*b^2*m^5*x^8 + 186*(d*x)^m*a^2*c*m^5*x 
^8 + 17760*(d*x)^m*b^3*m^3*x^10 + 106560*(d*x)^m*a*b*c*m^3*x^10 + 1000704* 
(d*x)^m*b^2*c*m*x^12 + 1000704*(d*x)^m*a*c^2*m*x^12 + 3*(d*x)^m*a^2*b*m^6* 
x^6 + 4644*(d*x)^m*a*b^2*m^4*x^8 + 4644*(d*x)^m*a^2*c*m^4*x^8 + 117184*(d* 
x)^m*b^3*m^2*x^10 + 703104*(d*x)^m*a*b*c*m^2*x^10 + 1290240*(d*x)^m*b^2*c* 
x^12 + 1290240*(d*x)^m*a*c^2*x^12 + 192*(d*x)^m*a^2*b*m^5*x^6 + 59448*(d*x 
)^m*a*b^2*m^3*x^8 + 59448*(d*x)^m*a^2*c*m^3*x^8 + 391680*(d*x)^m*b^3*m*x^1 
0 + 2350080*(d*x)^m*a*b*c*m*x^10 + (d*x)^m*a^3*m^6*x^4 + 4980*(d*x)^m*a...
 

Mupad [B] (verification not implemented)

Time = 12.78 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.51 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\frac {a^3\,x^4\,{\left (d\,x\right )}^m\,\left (m^6+66\,m^5+1780\,m^4+25080\,m^3+194464\,m^2+785664\,m+1290240\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {c^3\,x^{16}\,{\left (d\,x\right )}^m\,\left (m^6+54\,m^5+1180\,m^4+13320\,m^3+81664\,m^2+256896\,m+322560\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {3\,a^2\,b\,x^6\,{\left (d\,x\right )}^m\,\left (m^6+64\,m^5+1660\,m^4+22240\,m^3+161344\,m^2+595456\,m+860160\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {3\,b\,c^2\,x^{14}\,{\left (d\,x\right )}^m\,\left (m^6+56\,m^5+1260\,m^4+14560\,m^3+90944\,m^2+290304\,m+368640\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {3\,a\,x^8\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+62\,m^5+1548\,m^4+19816\,m^3+136256\,m^2+473472\,m+645120\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {b\,x^{10}\,{\left (d\,x\right )}^m\,\left (b^2+6\,a\,c\right )\,\left (m^6+60\,m^5+1444\,m^4+17760\,m^3+117184\,m^2+391680\,m+516096\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960}+\frac {3\,c\,x^{12}\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+58\,m^5+1348\,m^4+16024\,m^3+102496\,m^2+333568\,m+430080\right )}{m^7+70\,m^6+2044\,m^5+32200\,m^4+294784\,m^3+1563520\,m^2+4432896\,m+5160960} \] Input:

int((d*x)^m*(a*x + b*x^3 + c*x^5)^3,x)
 

Output:

(a^3*x^4*(d*x)^m*(785664*m + 194464*m^2 + 25080*m^3 + 1780*m^4 + 66*m^5 + 
m^6 + 1290240))/(4432896*m + 1563520*m^2 + 294784*m^3 + 32200*m^4 + 2044*m 
^5 + 70*m^6 + m^7 + 5160960) + (c^3*x^16*(d*x)^m*(256896*m + 81664*m^2 + 1 
3320*m^3 + 1180*m^4 + 54*m^5 + m^6 + 322560))/(4432896*m + 1563520*m^2 + 2 
94784*m^3 + 32200*m^4 + 2044*m^5 + 70*m^6 + m^7 + 5160960) + (3*a^2*b*x^6* 
(d*x)^m*(595456*m + 161344*m^2 + 22240*m^3 + 1660*m^4 + 64*m^5 + m^6 + 860 
160))/(4432896*m + 1563520*m^2 + 294784*m^3 + 32200*m^4 + 2044*m^5 + 70*m^ 
6 + m^7 + 5160960) + (3*b*c^2*x^14*(d*x)^m*(290304*m + 90944*m^2 + 14560*m 
^3 + 1260*m^4 + 56*m^5 + m^6 + 368640))/(4432896*m + 1563520*m^2 + 294784* 
m^3 + 32200*m^4 + 2044*m^5 + 70*m^6 + m^7 + 5160960) + (3*a*x^8*(d*x)^m*(a 
*c + b^2)*(473472*m + 136256*m^2 + 19816*m^3 + 1548*m^4 + 62*m^5 + m^6 + 6 
45120))/(4432896*m + 1563520*m^2 + 294784*m^3 + 32200*m^4 + 2044*m^5 + 70* 
m^6 + m^7 + 5160960) + (b*x^10*(d*x)^m*(6*a*c + b^2)*(391680*m + 117184*m^ 
2 + 17760*m^3 + 1444*m^4 + 60*m^5 + m^6 + 516096))/(4432896*m + 1563520*m^ 
2 + 294784*m^3 + 32200*m^4 + 2044*m^5 + 70*m^6 + m^7 + 5160960) + (3*c*x^1 
2*(d*x)^m*(a*c + b^2)*(333568*m + 102496*m^2 + 16024*m^3 + 1348*m^4 + 58*m 
^5 + m^6 + 430080))/(4432896*m + 1563520*m^2 + 294784*m^3 + 32200*m^4 + 20 
44*m^5 + 70*m^6 + m^7 + 5160960)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.03 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^3 \, dx=\frac {x^{m} d^{m} x^{4} \left (c^{3} m^{6} x^{12}+54 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+1180 c^{3} m^{4} x^{12}+168 b \,c^{2} m^{5} x^{10}+13320 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+3780 b \,c^{2} m^{4} x^{10}+81664 c^{3} m^{2} x^{12}+174 a \,c^{2} m^{5} x^{8}+174 b^{2} c \,m^{5} x^{8}+43680 b \,c^{2} m^{3} x^{10}+256896 c^{3} m \,x^{12}+6 a b c \,m^{6} x^{6}+4044 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+4044 b^{2} c \,m^{4} x^{8}+272832 b \,c^{2} m^{2} x^{10}+322560 c^{3} x^{12}+360 a b c \,m^{5} x^{6}+48072 a \,c^{2} m^{3} x^{8}+60 b^{3} m^{5} x^{6}+48072 b^{2} c \,m^{3} x^{8}+870912 b \,c^{2} m \,x^{10}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+8664 a b c \,m^{4} x^{6}+307488 a \,c^{2} m^{2} x^{8}+1444 b^{3} m^{4} x^{6}+307488 b^{2} c \,m^{2} x^{8}+1105920 b \,c^{2} x^{10}+186 a^{2} c \,m^{5} x^{4}+186 a \,b^{2} m^{5} x^{4}+106560 a b c \,m^{3} x^{6}+1000704 a \,c^{2} m \,x^{8}+17760 b^{3} m^{3} x^{6}+1000704 b^{2} c m \,x^{8}+3 a^{2} b \,m^{6} x^{2}+4644 a^{2} c \,m^{4} x^{4}+4644 a \,b^{2} m^{4} x^{4}+703104 a b c \,m^{2} x^{6}+1290240 a \,c^{2} x^{8}+117184 b^{3} m^{2} x^{6}+1290240 b^{2} c \,x^{8}+192 a^{2} b \,m^{5} x^{2}+59448 a^{2} c \,m^{3} x^{4}+59448 a \,b^{2} m^{3} x^{4}+2350080 a b c m \,x^{6}+391680 b^{3} m \,x^{6}+a^{3} m^{6}+4980 a^{2} b \,m^{4} x^{2}+408768 a^{2} c \,m^{2} x^{4}+408768 a \,b^{2} m^{2} x^{4}+3096576 a b c \,x^{6}+516096 b^{3} x^{6}+66 a^{3} m^{5}+66720 a^{2} b \,m^{3} x^{2}+1420416 a^{2} c m \,x^{4}+1420416 a \,b^{2} m \,x^{4}+1780 a^{3} m^{4}+484032 a^{2} b \,m^{2} x^{2}+1935360 a^{2} c \,x^{4}+1935360 a \,b^{2} x^{4}+25080 a^{3} m^{3}+1786368 a^{2} b m \,x^{2}+194464 a^{3} m^{2}+2580480 a^{2} b \,x^{2}+785664 a^{3} m +1290240 a^{3}\right )}{m^{7}+70 m^{6}+2044 m^{5}+32200 m^{4}+294784 m^{3}+1563520 m^{2}+4432896 m +5160960} \] Input:

int((d*x)^m*(c*x^5+b*x^3+a*x)^3,x)
 

Output:

(x**m*d**m*x**4*(a**3*m**6 + 66*a**3*m**5 + 1780*a**3*m**4 + 25080*a**3*m* 
*3 + 194464*a**3*m**2 + 785664*a**3*m + 1290240*a**3 + 3*a**2*b*m**6*x**2 
+ 192*a**2*b*m**5*x**2 + 4980*a**2*b*m**4*x**2 + 66720*a**2*b*m**3*x**2 + 
484032*a**2*b*m**2*x**2 + 1786368*a**2*b*m*x**2 + 2580480*a**2*b*x**2 + 3* 
a**2*c*m**6*x**4 + 186*a**2*c*m**5*x**4 + 4644*a**2*c*m**4*x**4 + 59448*a* 
*2*c*m**3*x**4 + 408768*a**2*c*m**2*x**4 + 1420416*a**2*c*m*x**4 + 1935360 
*a**2*c*x**4 + 3*a*b**2*m**6*x**4 + 186*a*b**2*m**5*x**4 + 4644*a*b**2*m** 
4*x**4 + 59448*a*b**2*m**3*x**4 + 408768*a*b**2*m**2*x**4 + 1420416*a*b**2 
*m*x**4 + 1935360*a*b**2*x**4 + 6*a*b*c*m**6*x**6 + 360*a*b*c*m**5*x**6 + 
8664*a*b*c*m**4*x**6 + 106560*a*b*c*m**3*x**6 + 703104*a*b*c*m**2*x**6 + 2 
350080*a*b*c*m*x**6 + 3096576*a*b*c*x**6 + 3*a*c**2*m**6*x**8 + 174*a*c**2 
*m**5*x**8 + 4044*a*c**2*m**4*x**8 + 48072*a*c**2*m**3*x**8 + 307488*a*c** 
2*m**2*x**8 + 1000704*a*c**2*m*x**8 + 1290240*a*c**2*x**8 + b**3*m**6*x**6 
 + 60*b**3*m**5*x**6 + 1444*b**3*m**4*x**6 + 17760*b**3*m**3*x**6 + 117184 
*b**3*m**2*x**6 + 391680*b**3*m*x**6 + 516096*b**3*x**6 + 3*b**2*c*m**6*x* 
*8 + 174*b**2*c*m**5*x**8 + 4044*b**2*c*m**4*x**8 + 48072*b**2*c*m**3*x**8 
 + 307488*b**2*c*m**2*x**8 + 1000704*b**2*c*m*x**8 + 1290240*b**2*c*x**8 + 
 3*b*c**2*m**6*x**10 + 168*b*c**2*m**5*x**10 + 3780*b*c**2*m**4*x**10 + 43 
680*b*c**2*m**3*x**10 + 272832*b*c**2*m**2*x**10 + 870912*b*c**2*m*x**10 + 
 1105920*b*c**2*x**10 + c**3*m**6*x**12 + 54*c**3*m**5*x**12 + 1180*c**...