\(\int (d x)^m (a x^q+b x^r+c x^s)^2 \, dx\) [2]

Optimal result

Integrand size = 24, antiderivative size = 1 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 106.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=x (d x)^m \left (\frac {a^2 x^{2 q}}{1+m+2 q}+\frac {b^2 x^{2 r}}{1+m+2 r}+\frac {c^2 x^{2 s}}{1+m+2 s}+\frac {2 b c x^{r+s}}{1+m+r+s}+2 a x^q \left (\frac {b x^r}{1+m+q+r}+\frac {c x^s}{1+m+q+s}\right )\right ) \] Input:

Integrate[(d*x)^m*(a*x^q + b*x^r + c*x^s)^2,x]
 

Output:

x*(d*x)^m*((a^2*x^(2*q))/(1 + m + 2*q) + (b^2*x^(2*r))/(1 + m + 2*r) + (c^ 
2*x^(2*s))/(1 + m + 2*s) + (2*b*c*x^(r + s))/(1 + m + r + s) + 2*a*x^q*((b 
*x^r)/(1 + m + q + r) + (c*x^s)/(1 + m + q + s)))
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 0.53 (sec) , antiderivative size = 126, normalized size of antiderivative = 126.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2028, 30, 7293, 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.

\(\displaystyle \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx\)

\(\Big \downarrow \) 2028

\(\displaystyle \int x^{2 s} (d x)^m \left (a x^{q-s}+b x^{r-s}+c\right )^2dx\)

\(\Big \downarrow \) 30

\(\displaystyle x^{-m} (d x)^m \int x^{m+2 s} \left (a x^{q-s}+b x^{r-s}+c\right )^2dx\)

\(\Big \downarrow \) 7293

\(\displaystyle x^{-m} (d x)^m \int \left (a^2 x^{m+2 q}+2 a b x^{m+q+r}+b^2 x^{m+2 r}+2 a c x^{m+q+s}+2 b c x^{m+r+s}+c^2 x^{m+2 s}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x^{-m} (d x)^m \left (\frac {a^2 x^{m+2 q+1}}{m+2 q+1}+\frac {2 a b x^{m+q+r+1}}{m+q+r+1}+\frac {2 a c x^{m+q+s+1}}{m+q+s+1}+\frac {b^2 x^{m+2 r+1}}{m+2 r+1}+\frac {2 b c x^{m+r+s+1}}{m+r+s+1}+\frac {c^2 x^{m+2 s+1}}{m+2 s+1}\right )\)

Input:

Int[(d*x)^m*(a*x^q + b*x^r + c*x^s)^2,x]
 

Output:

((d*x)^m*((a^2*x^(1 + m + 2*q))/(1 + m + 2*q) + (2*a*b*x^(1 + m + q + r))/ 
(1 + m + q + r) + (b^2*x^(1 + m + 2*r))/(1 + m + 2*r) + (2*a*c*x^(1 + m + 
q + s))/(1 + m + q + s) + (2*b*c*x^(1 + m + r + s))/(1 + m + r + s) + (c^2 
*x^(1 + m + 2*s))/(1 + m + 2*s)))/x^m
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 2009

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

rule 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 4.39 (sec) , antiderivative size = 8255, normalized size of antiderivative = 8255.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(8255\)
parallelrisch	\(\text {Expression too large to display}\)	\(11592\)
orering	\(\text {Expression too large to display}\)	\(12502\)

Input:

int((d*x)^m*(a*x^q+b*x^r+c*x^s)^2,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.45 (sec) , antiderivative size = 4635, normalized size of antiderivative = 4635.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((d*x)^m*(a*x^q+b*x^r+c*x^s)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 771647 vs. \(2 (0) = 0\).

Time = 36.49 (sec) , antiderivative size = 771647, normalized size of antiderivative = 771647.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((d*x)**m*(a*x**q+b*x**r+c*x**s)**2,x)
 

Output:

Piecewise((a**2*x*x**(2*s)*(d*x)**(-2*s - 1)*log(x) + 2*a*b*x*x**(2*s)*(d* 
x)**(-2*s - 1)*log(x) + 2*a*c*x*x**(2*s)*(d*x)**(-2*s - 1)*log(x) + b**2*x 
*x**(2*s)*(d*x)**(-2*s - 1)*log(x) + 2*b*c*x*x**(2*s)*(d*x)**(-2*s - 1)*lo 
g(x) + c**2*x*x**(2*s)*(d*x)**(-2*s - 1)*log(x), Eq(q, s) & Eq(r, s) & Eq( 
m, -2*s - 1)), (a**2*x*x**(2*q)*(d*x)**(-2*q - 1)*log(x) + 2*a*b*Piecewise 
((x*x**q*x**r*(d*x)**(-2*q - 1)/(-q + r), Ne(q - r, 0)), (x*x**q*x**r*(d*x 
)**(-2*q - 1)*log(x), True)) + 2*a*c*Piecewise((x*x**q*x**s*(d*x)**(-2*q - 
 1)/(-q + s), Ne(q - s, 0)), (x*x**q*x**s*(d*x)**(-2*q - 1)*log(x), True)) 
 + b**2*Piecewise((x*x**(2*r)*(d*x)**(-2*q - 1)/(-2*q + 2*r), Ne(2*q - 2*r 
, 0)), (x*x**(2*r)*(d*x)**(-2*q - 1)*log(x), True)) + 2*b*c*Piecewise((x*x 
**r*x**s*(d*x)**(-2*q - 1)/(-2*q + r + s), Ne(-2*q + r + s, 0)), (x*x**r*x 
**s*(d*x)**(-2*q - 1)*log(x), True)) + c**2*Piecewise((x*x**(2*s)*(d*x)**( 
-2*q - 1)/(-2*q + 2*s), Ne(2*q - 2*s, 0)), (x*x**(2*s)*(d*x)**(-2*q - 1)*l 
og(x), True)), Eq(m, -2*q - 1)), (a**2*Piecewise((x*x**(2*q)*(d*x)**(-2*r 
- 1)/(2*q - 2*r), Ne(2*q - 2*r, 0)), (x*x**(2*q)*(d*x)**(-2*r - 1)*log(x), 
 True)) + 2*a*b*Piecewise((x*x**q*x**r*(d*x)**(-2*r - 1)/(q - r), Ne(q - r 
, 0)), (x*x**q*x**r*(d*x)**(-2*r - 1)*log(x), True)) + 2*a*c*Piecewise((x* 
x**q*x**s*(d*x)**(-2*r - 1)/(q - 2*r + s), Ne(q - 2*r + s, 0)), (x*x**q*x* 
*s*(d*x)**(-2*r - 1)*log(x), True)) + b**2*x*x**(2*r)*(d*x)**(-2*r - 1)*lo 
g(x) + 2*b*c*Piecewise((x*x**r*x**s*(d*x)**(-2*r - 1)/(-r + s), Ne(r - ...
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 169.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=\frac {a^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, q \log \left (x\right )\right )}}{m + 2 \, q + 1} + \frac {2 \, a b d^{m} x e^{\left (m \log \left (x\right ) + q \log \left (x\right ) + r \log \left (x\right )\right )}}{m + q + r + 1} + \frac {2 \, a c d^{m} x e^{\left (m \log \left (x\right ) + q \log \left (x\right ) + s \log \left (x\right )\right )}}{m + q + s + 1} + \frac {b^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, r \log \left (x\right )\right )}}{m + 2 \, r + 1} + \frac {2 \, b c d^{m} x e^{\left (m \log \left (x\right ) + r \log \left (x\right ) + s \log \left (x\right )\right )}}{m + r + s + 1} + \frac {c^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, s \log \left (x\right )\right )}}{m + 2 \, s + 1} \] Input:

integrate((d*x)^m*(a*x^q+b*x^r+c*x^s)^2,x, algorithm="maxima")
 

Output:

a^2*d^m*x*e^(m*log(x) + 2*q*log(x))/(m + 2*q + 1) + 2*a*b*d^m*x*e^(m*log(x 
) + q*log(x) + r*log(x))/(m + q + r + 1) + 2*a*c*d^m*x*e^(m*log(x) + q*log 
(x) + s*log(x))/(m + q + s + 1) + b^2*d^m*x*e^(m*log(x) + 2*r*log(x))/(m + 
 2*r + 1) + 2*b*c*d^m*x*e^(m*log(x) + r*log(x) + s*log(x))/(m + r + s + 1) 
 + c^2*d^m*x*e^(m*log(x) + 2*s*log(x))/(m + 2*s + 1)
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.42 (sec) , antiderivative size = 33969, normalized size of antiderivative = 33969.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((d*x)^m*(a*x^q+b*x^r+c*x^s)^2,x, algorithm="giac")
 

Output:

(2*a*b*m^5*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 6*a*b*m^4*q*x*x^q*x^r*e^(m* 
log(d) + m*log(x)) + 4*a*b*m^3*q^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 6*a 
*b*m^4*r*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 18*a*b*m^3*q*r*x*x^q*x^r*e^(m 
*log(d) + m*log(x)) + 12*a*b*m^2*q^2*r*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 
 4*a*b*m^3*r^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 12*a*b*m^2*q*r^2*x*x^q* 
x^r*e^(m*log(d) + m*log(x)) + 8*a*b*m*q^2*r^2*x*x^q*x^r*e^(m*log(d) + m*lo 
g(x)) + 8*a*b*m^4*s*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 22*a*b*m^3*q*s*x*x 
^q*x^r*e^(m*log(d) + m*log(x)) + 12*a*b*m^2*q^2*s*x*x^q*x^r*e^(m*log(d) + 
m*log(x)) + 22*a*b*m^3*r*s*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 60*a*b*m^2* 
q*r*s*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 32*a*b*m*q^2*r*s*x*x^q*x^r*e^(m* 
log(d) + m*log(x)) + 12*a*b*m^2*r^2*s*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 
32*a*b*m*q*r^2*s*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 16*a*b*q^2*r^2*s*x*x^ 
q*x^r*e^(m*log(d) + m*log(x)) + 10*a*b*m^3*s^2*x*x^q*x^r*e^(m*log(d) + m*l 
og(x)) + 24*a*b*m^2*q*s^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 8*a*b*m*q^2* 
s^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 24*a*b*m^2*r*s^2*x*x^q*x^r*e^(m*lo 
g(d) + m*log(x)) + 56*a*b*m*q*r*s^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 16 
*a*b*q^2*r*s^2*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 8*a*b*m*r^2*s^2*x*x^q*x 
^r*e^(m*log(d) + m*log(x)) + 16*a*b*q*r^2*s^2*x*x^q*x^r*e^(m*log(d) + m*lo 
g(x)) + 4*a*b*m^2*s^3*x*x^q*x^r*e^(m*log(d) + m*log(x)) + 8*a*b*m*q*s^3*x* 
x^q*x^r*e^(m*log(d) + m*log(x)) + 8*a*b*m*r*s^3*x*x^q*x^r*e^(m*log(d) +...
 

Mupad [B] (verification not implemented)

Time = 12.88 (sec) , antiderivative size = 115, normalized size of antiderivative = 115.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {a^2\,x\,x^{2\,q}}{m+2\,q+1}+\frac {b^2\,x\,x^{2\,r}}{m+2\,r+1}+\frac {c^2\,x\,x^{2\,s}}{m+2\,s+1}+\frac {2\,a\,b\,x\,x^q\,x^r}{m+q+r+1}+\frac {2\,a\,c\,x\,x^q\,x^s}{m+q+s+1}+\frac {2\,b\,c\,x\,x^r\,x^s}{m+r+s+1}\right ) \] Input:

int((d*x)^m*(a*x^q + b*x^r + c*x^s)^2,x)
 

Output:

(d*x)^m*((a^2*x*x^(2*q))/(m + 2*q + 1) + (b^2*x*x^(2*r))/(m + 2*r + 1) + ( 
c^2*x*x^(2*s))/(m + 2*s + 1) + (2*a*b*x*x^q*x^r)/(m + q + r + 1) + (2*a*c* 
x*x^q*x^s)/(m + q + s + 1) + (2*b*c*x*x^r*x^s)/(m + r + s + 1))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 9074, normalized size of antiderivative = 9074.00 \[ \int (d x)^m \left (a x^q+b x^r+c x^s\right )^2 \, dx =\text {Too large to display} \] Input:

int((d*x)^m*(a*x^q+b*x^r+c*x^s)^2,x)
 

Output:

(x**m*d**m*x*(x**(2*q)*a**2*m**5 + 2*x**(2*q)*a**2*m**4*q + 4*x**(2*q)*a** 
2*m**4*r + 4*x**(2*q)*a**2*m**4*s + 5*x**(2*q)*a**2*m**4 + x**(2*q)*a**2*m 
**3*q**2 + 7*x**(2*q)*a**2*m**3*q*r + 7*x**(2*q)*a**2*m**3*q*s + 8*x**(2*q 
)*a**2*m**3*q + 5*x**(2*q)*a**2*m**3*r**2 + 15*x**(2*q)*a**2*m**3*r*s + 16 
*x**(2*q)*a**2*m**3*r + 5*x**(2*q)*a**2*m**3*s**2 + 16*x**(2*q)*a**2*m**3* 
s + 10*x**(2*q)*a**2*m**3 + 3*x**(2*q)*a**2*m**2*q**2*r + 3*x**(2*q)*a**2* 
m**2*q**2*s + 3*x**(2*q)*a**2*m**2*q**2 + 7*x**(2*q)*a**2*m**2*q*r**2 + 22 
*x**(2*q)*a**2*m**2*q*r*s + 21*x**(2*q)*a**2*m**2*q*r + 7*x**(2*q)*a**2*m* 
*2*q*s**2 + 21*x**(2*q)*a**2*m**2*q*s + 12*x**(2*q)*a**2*m**2*q + 2*x**(2* 
q)*a**2*m**2*r**3 + 17*x**(2*q)*a**2*m**2*r**2*s + 15*x**(2*q)*a**2*m**2*r 
**2 + 17*x**(2*q)*a**2*m**2*r*s**2 + 45*x**(2*q)*a**2*m**2*r*s + 24*x**(2* 
q)*a**2*m**2*r + 2*x**(2*q)*a**2*m**2*s**3 + 15*x**(2*q)*a**2*m**2*s**2 + 
24*x**(2*q)*a**2*m**2*s + 10*x**(2*q)*a**2*m**2 + 2*x**(2*q)*a**2*m*q**2*r 
**2 + 8*x**(2*q)*a**2*m*q**2*r*s + 6*x**(2*q)*a**2*m*q**2*r + 2*x**(2*q)*a 
**2*m*q**2*s**2 + 6*x**(2*q)*a**2*m*q**2*s + 3*x**(2*q)*a**2*m*q**2 + 2*x* 
*(2*q)*a**2*m*q*r**3 + 18*x**(2*q)*a**2*m*q*r**2*s + 14*x**(2*q)*a**2*m*q* 
r**2 + 18*x**(2*q)*a**2*m*q*r*s**2 + 44*x**(2*q)*a**2*m*q*r*s + 21*x**(2*q 
)*a**2*m*q*r + 2*x**(2*q)*a**2*m*q*s**3 + 14*x**(2*q)*a**2*m*q*s**2 + 21*x 
**(2*q)*a**2*m*q*s + 8*x**(2*q)*a**2*m*q + 6*x**(2*q)*a**2*m*r**3*s + 4*x* 
*(2*q)*a**2*m*r**3 + 16*x**(2*q)*a**2*m*r**2*s**2 + 34*x**(2*q)*a**2*m*...