\(\int (c x)^m (a+b x^2) (A+B x+C x^2+D x^3) \, dx\) [137]

Optimal result

Integrand size = 28, antiderivative size = 119 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a A (c x)^{1+m}}{c (1+m)}+\frac {a B (c x)^{2+m}}{c^2 (2+m)}+\frac {(A b+a C) (c x)^{3+m}}{c^3 (3+m)}+\frac {(b B+a D) (c x)^{4+m}}{c^4 (4+m)}+\frac {b C (c x)^{5+m}}{c^5 (5+m)}+\frac {b D (c x)^{6+m}}{c^6 (6+m)} \] Output:

a*A*(c*x)^(1+m)/c/(1+m)+a*B*(c*x)^(2+m)/c^2/(2+m)+(A*b+C*a)*(c*x)^(3+m)/c^ 
3/(3+m)+(B*b+D*a)*(c*x)^(4+m)/c^4/(4+m)+b*C*(c*x)^(5+m)/c^5/(5+m)+b*D*(c*x 
)^(6+m)/c^6/(6+m)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.66 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=x (c x)^m \left (\frac {a A}{1+m}+\frac {a B x}{2+m}+\frac {(A b+a C) x^2}{3+m}+\frac {(b B+a D) x^3}{4+m}+\frac {b C x^4}{5+m}+\frac {b D x^5}{6+m}\right ) \] Input:

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

Output:

x*(c*x)^m*((a*A)/(1 + m) + (a*B*x)/(2 + m) + ((A*b + a*C)*x^2)/(3 + m) + ( 
(b*B + a*D)*x^3)/(4 + m) + (b*C*x^4)/(5 + m) + (b*D*x^5)/(6 + m))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2333, 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.

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

\(\Big \downarrow \) 2333

\(\displaystyle \int \left (\frac {(c x)^{m+2} (a C+A b)}{c^2}+a A (c x)^m+\frac {(c x)^{m+3} (a D+b B)}{c^3}+\frac {a B (c x)^{m+1}}{c}+\frac {b D (c x)^{m+5}}{c^5}+\frac {b C (c x)^{m+4}}{c^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c x)^{m+3} (a C+A b)}{c^3 (m+3)}+\frac {a A (c x)^{m+1}}{c (m+1)}+\frac {(c x)^{m+4} (a D+b B)}{c^4 (m+4)}+\frac {a B (c x)^{m+2}}{c^2 (m+2)}+\frac {b D (c x)^{m+6}}{c^6 (m+6)}+\frac {b C (c x)^{m+5}}{c^5 (m+5)}\)

Input:

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

Output:

(a*A*(c*x)^(1 + m))/(c*(1 + m)) + (a*B*(c*x)^(2 + m))/(c^2*(2 + m)) + ((A* 
b + a*C)*(c*x)^(3 + m))/(c^3*(3 + m)) + ((b*B + a*D)*(c*x)^(4 + m))/(c^4*( 
4 + m)) + (b*C*(c*x)^(5 + m))/(c^5*(5 + m)) + (b*D*(c*x)^(6 + m))/(c^6*(6 
+ m))
 

Defintions of rubi rules used

rule 2009

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

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {\left (A b +C a \right ) x^{3} {\mathrm e}^{m \ln \left (c x \right )}}{3+m}+\frac {\left (B b +D a \right ) x^{4} {\mathrm e}^{m \ln \left (c x \right )}}{4+m}+\frac {A a x \,{\mathrm e}^{m \ln \left (c x \right )}}{1+m}+\frac {B a \,x^{2} {\mathrm e}^{m \ln \left (c x \right )}}{2+m}+\frac {C b \,x^{5} {\mathrm e}^{m \ln \left (c x \right )}}{5+m}+\frac {D b \,x^{6} {\mathrm e}^{m \ln \left (c x \right )}}{6+m}\)	\(118\)
gosper	\(\frac {x \left (D b \,m^{5} x^{5}+C b \,m^{5} x^{4}+15 D b \,m^{4} x^{5}+B b \,m^{5} x^{3}+16 C b \,m^{4} x^{4}+D a \,m^{5} x^{3}+85 D b \,m^{3} x^{5}+A b \,m^{5} x^{2}+17 B b \,m^{4} x^{3}+C a \,m^{5} x^{2}+95 C b \,m^{3} x^{4}+17 D a \,m^{4} x^{3}+225 D b \,m^{2} x^{5}+18 A b \,m^{4} x^{2}+B a \,m^{5} x +107 B b \,m^{3} x^{3}+18 C a \,m^{4} x^{2}+260 C b \,m^{2} x^{4}+107 D a \,m^{3} x^{3}+274 D b m \,x^{5}+A a \,m^{5}+121 A b \,m^{3} x^{2}+19 B a \,m^{4} x +307 B b \,m^{2} x^{3}+121 C a \,m^{3} x^{2}+324 C b m \,x^{4}+307 D a \,m^{2} x^{3}+120 D b \,x^{5}+20 A a \,m^{4}+372 A b \,m^{2} x^{2}+137 B a \,m^{3} x +396 B b m \,x^{3}+372 C a \,m^{2} x^{2}+144 C b \,x^{4}+396 D a m \,x^{3}+155 A a \,m^{3}+508 A b m \,x^{2}+461 B a \,m^{2} x +180 b B \,x^{3}+508 C a m \,x^{2}+180 D a \,x^{3}+580 A a \,m^{2}+240 A b \,x^{2}+702 B a m x +240 C a \,x^{2}+1044 A a m +360 B a x +720 A a \right ) \left (c x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(441\)
orering	\(\frac {x \left (D b \,m^{5} x^{5}+C b \,m^{5} x^{4}+15 D b \,m^{4} x^{5}+B b \,m^{5} x^{3}+16 C b \,m^{4} x^{4}+D a \,m^{5} x^{3}+85 D b \,m^{3} x^{5}+A b \,m^{5} x^{2}+17 B b \,m^{4} x^{3}+C a \,m^{5} x^{2}+95 C b \,m^{3} x^{4}+17 D a \,m^{4} x^{3}+225 D b \,m^{2} x^{5}+18 A b \,m^{4} x^{2}+B a \,m^{5} x +107 B b \,m^{3} x^{3}+18 C a \,m^{4} x^{2}+260 C b \,m^{2} x^{4}+107 D a \,m^{3} x^{3}+274 D b m \,x^{5}+A a \,m^{5}+121 A b \,m^{3} x^{2}+19 B a \,m^{4} x +307 B b \,m^{2} x^{3}+121 C a \,m^{3} x^{2}+324 C b m \,x^{4}+307 D a \,m^{2} x^{3}+120 D b \,x^{5}+20 A a \,m^{4}+372 A b \,m^{2} x^{2}+137 B a \,m^{3} x +396 B b m \,x^{3}+372 C a \,m^{2} x^{2}+144 C b \,x^{4}+396 D a m \,x^{3}+155 A a \,m^{3}+508 A b m \,x^{2}+461 B a \,m^{2} x +180 b B \,x^{3}+508 C a m \,x^{2}+180 D a \,x^{3}+580 A a \,m^{2}+240 A b \,x^{2}+702 B a m x +240 C a \,x^{2}+1044 A a m +360 B a x +720 A a \right ) \left (c x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(441\)
parallelrisch	\(\frac {307 D x^{4} \left (c x \right )^{m} a \,m^{2}+225 D x^{6} \left (c x \right )^{m} b \,m^{2}+15 D x^{6} \left (c x \right )^{m} b \,m^{4}+B \,x^{4} \left (c x \right )^{m} b \,m^{5}+16 C \,x^{5} \left (c x \right )^{m} b \,m^{4}+85 D x^{6} \left (c x \right )^{m} b \,m^{3}+17 B \,x^{4} \left (c x \right )^{m} b \,m^{4}+95 C \,x^{5} \left (c x \right )^{m} b \,m^{3}+A x \left (c x \right )^{m} a \,m^{5}+307 B \,x^{4} \left (c x \right )^{m} b \,m^{2}+19 B \,x^{2} \left (c x \right )^{m} a \,m^{4}+324 C \,x^{5} \left (c x \right )^{m} b m +121 C \,x^{3} \left (c x \right )^{m} a \,m^{3}+372 A \,x^{3} \left (c x \right )^{m} b \,m^{2}+20 A x \left (c x \right )^{m} a \,m^{4}+461 B \,x^{2} \left (c x \right )^{m} a \,m^{2}+120 D x^{6} \left (c x \right )^{m} b +144 C \,x^{5} \left (c x \right )^{m} b +180 B \,x^{4} \left (c x \right )^{m} b +180 D x^{4} \left (c x \right )^{m} a +240 A \,x^{3} \left (c x \right )^{m} b +396 B \,x^{4} \left (c x \right )^{m} b m +C \,x^{3} \left (c x \right )^{m} a \,m^{5}+B \,x^{2} \left (c x \right )^{m} a \,m^{5}+1044 A x \left (c x \right )^{m} a m +D x^{6} \left (c x \right )^{m} b \,m^{5}+C \,x^{5} \left (c x \right )^{m} b \,m^{5}+508 A \,x^{3} \left (c x \right )^{m} b m +155 A x \left (c x \right )^{m} a \,m^{3}+D x^{4} \left (c x \right )^{m} a \,m^{5}+A \,x^{3} \left (c x \right )^{m} b \,m^{5}+240 C \,x^{3} \left (c x \right )^{m} a +360 B \,x^{2} \left (c x \right )^{m} a +720 A x \left (c x \right )^{m} a +260 C \,x^{5} \left (c x \right )^{m} b \,m^{2}+18 C \,x^{3} \left (c x \right )^{m} a \,m^{4}+274 D x^{6} \left (c x \right )^{m} b m +107 D x^{4} \left (c x \right )^{m} a \,m^{3}+121 A \,x^{3} \left (c x \right )^{m} b \,m^{3}+508 C \,x^{3} \left (c x \right )^{m} a m +137 B \,x^{2} \left (c x \right )^{m} a \,m^{3}+372 C \,x^{3} \left (c x \right )^{m} a \,m^{2}+396 D x^{4} \left (c x \right )^{m} a m +580 A x \left (c x \right )^{m} a \,m^{2}+702 B \,x^{2} \left (c x \right )^{m} a m +17 D x^{4} \left (c x \right )^{m} a \,m^{4}+18 A \,x^{3} \left (c x \right )^{m} b \,m^{4}+107 B \,x^{4} \left (c x \right )^{m} b \,m^{3}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(693\)

Input:

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

Output:

(A*b+C*a)/(3+m)*x^3*exp(m*ln(c*x))+(B*b+D*a)/(4+m)*x^4*exp(m*ln(c*x))+A*a/ 
(1+m)*x*exp(m*ln(c*x))+B*a/(2+m)*x^2*exp(m*ln(c*x))+C*b/(5+m)*x^5*exp(m*ln 
(c*x))+D*b/(6+m)*x^6*exp(m*ln(c*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (119) = 238\).

Time = 0.09 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.85 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left ({\left (D b m^{5} + 15 \, D b m^{4} + 85 \, D b m^{3} + 225 \, D b m^{2} + 274 \, D b m + 120 \, D b\right )} x^{6} + {\left (C b m^{5} + 16 \, C b m^{4} + 95 \, C b m^{3} + 260 \, C b m^{2} + 324 \, C b m + 144 \, C b\right )} x^{5} + {\left ({\left (D a + B b\right )} m^{5} + 17 \, {\left (D a + B b\right )} m^{4} + 107 \, {\left (D a + B b\right )} m^{3} + 307 \, {\left (D a + B b\right )} m^{2} + 180 \, D a + 180 \, B b + 396 \, {\left (D a + B b\right )} m\right )} x^{4} + {\left ({\left (C a + A b\right )} m^{5} + 18 \, {\left (C a + A b\right )} m^{4} + 121 \, {\left (C a + A b\right )} m^{3} + 372 \, {\left (C a + A b\right )} m^{2} + 240 \, C a + 240 \, A b + 508 \, {\left (C a + A b\right )} m\right )} x^{3} + {\left (B a m^{5} + 19 \, B a m^{4} + 137 \, B a m^{3} + 461 \, B a m^{2} + 702 \, B a m + 360 \, B a\right )} x^{2} + {\left (A a m^{5} + 20 \, A a m^{4} + 155 \, A a m^{3} + 580 \, A a m^{2} + 1044 \, A a m + 720 \, A a\right )} x\right )} \left (c x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \] Input:

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

Output:

((D*b*m^5 + 15*D*b*m^4 + 85*D*b*m^3 + 225*D*b*m^2 + 274*D*b*m + 120*D*b)*x 
^6 + (C*b*m^5 + 16*C*b*m^4 + 95*C*b*m^3 + 260*C*b*m^2 + 324*C*b*m + 144*C* 
b)*x^5 + ((D*a + B*b)*m^5 + 17*(D*a + B*b)*m^4 + 107*(D*a + B*b)*m^3 + 307 
*(D*a + B*b)*m^2 + 180*D*a + 180*B*b + 396*(D*a + B*b)*m)*x^4 + ((C*a + A* 
b)*m^5 + 18*(C*a + A*b)*m^4 + 121*(C*a + A*b)*m^3 + 372*(C*a + A*b)*m^2 + 
240*C*a + 240*A*b + 508*(C*a + A*b)*m)*x^3 + (B*a*m^5 + 19*B*a*m^4 + 137*B 
*a*m^3 + 461*B*a*m^2 + 702*B*a*m + 360*B*a)*x^2 + (A*a*m^5 + 20*A*a*m^4 + 
155*A*a*m^3 + 580*A*a*m^2 + 1044*A*a*m + 720*A*a)*x)*(c*x)^m/(m^6 + 21*m^5 
 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2504 vs. \(2 (105) = 210\).

Time = 0.61 (sec) , antiderivative size = 2504, normalized size of antiderivative = 21.04 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((c*x)**m*(b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Piecewise(((-A*a/(5*x**5) - A*b/(3*x**3) - B*a/(4*x**4) - B*b/(2*x**2) - C 
*a/(3*x**3) - C*b/x - D*a/(2*x**2) + D*b*log(x))/c**6, Eq(m, -6)), ((-A*a/ 
(4*x**4) - A*b/(2*x**2) - B*a/(3*x**3) - B*b/x - C*a/(2*x**2) + C*b*log(x) 
 - D*a/x + D*b*x)/c**5, Eq(m, -5)), ((-A*a/(3*x**3) - A*b/x - B*a/(2*x**2) 
 + B*b*log(x) - C*a/x + C*b*x + D*a*log(x) + D*b*x**2/2)/c**4, Eq(m, -4)), 
 ((-A*a/(2*x**2) + A*b*log(x) - B*a/x + B*b*x + C*a*log(x) + C*b*x**2/2 + 
D*a*x + D*b*x**3/3)/c**3, Eq(m, -3)), ((-A*a/x + A*b*x + B*a*log(x) + B*b* 
x**2/2 + C*a*x + C*b*x**3/3 + D*a*x**2/2 + D*b*x**4/4)/c**2, Eq(m, -2)), ( 
(A*a*log(x) + A*b*x**2/2 + B*a*x + B*b*x**3/3 + C*a*x**2/2 + C*b*x**4/4 + 
D*a*x**3/3 + D*b*x**5/5)/c, Eq(m, -1)), (A*a*m**5*x*(c*x)**m/(m**6 + 21*m* 
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*A*a*m**4*x*(c*x) 
**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15 
5*A*a*m**3*x*(c*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 
1764*m + 720) + 580*A*a*m**2*x*(c*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m 
**3 + 1624*m**2 + 1764*m + 720) + 1044*A*a*m*x*(c*x)**m/(m**6 + 21*m**5 + 
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*A*a*x*(c*x)**m/(m**6 
 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*b*m**5*x* 
*3*(c*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 7 
20) + 18*A*b*m**4*x**3*(c*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 16 
24*m**2 + 1764*m + 720) + 121*A*b*m**3*x**3*(c*x)**m/(m**6 + 21*m**5 + ...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.16 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {D b c^{m} x^{6} x^{m}}{m + 6} + \frac {C b c^{m} x^{5} x^{m}}{m + 5} + \frac {D a c^{m} x^{4} x^{m}}{m + 4} + \frac {B b c^{m} x^{4} x^{m}}{m + 4} + \frac {C a c^{m} x^{3} x^{m}}{m + 3} + \frac {A b c^{m} x^{3} x^{m}}{m + 3} + \frac {B a c^{m} x^{2} x^{m}}{m + 2} + \frac {\left (c x\right )^{m + 1} A a}{c {\left (m + 1\right )}} \] Input:

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

Output:

D*b*c^m*x^6*x^m/(m + 6) + C*b*c^m*x^5*x^m/(m + 5) + D*a*c^m*x^4*x^m/(m + 4 
) + B*b*c^m*x^4*x^m/(m + 4) + C*a*c^m*x^3*x^m/(m + 3) + A*b*c^m*x^3*x^m/(m 
 + 3) + B*a*c^m*x^2*x^m/(m + 2) + (c*x)^(m + 1)*A*a/(c*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (119) = 238\).

Time = 0.13 (sec) , antiderivative size = 692, normalized size of antiderivative = 5.82 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

((c*x)^m*D*b*m^5*x^6 + (c*x)^m*C*b*m^5*x^5 + 15*(c*x)^m*D*b*m^4*x^6 + (c*x 
)^m*D*a*m^5*x^4 + (c*x)^m*B*b*m^5*x^4 + 16*(c*x)^m*C*b*m^4*x^5 + 85*(c*x)^ 
m*D*b*m^3*x^6 + (c*x)^m*C*a*m^5*x^3 + (c*x)^m*A*b*m^5*x^3 + 17*(c*x)^m*D*a 
*m^4*x^4 + 17*(c*x)^m*B*b*m^4*x^4 + 95*(c*x)^m*C*b*m^3*x^5 + 225*(c*x)^m*D 
*b*m^2*x^6 + (c*x)^m*B*a*m^5*x^2 + 18*(c*x)^m*C*a*m^4*x^3 + 18*(c*x)^m*A*b 
*m^4*x^3 + 107*(c*x)^m*D*a*m^3*x^4 + 107*(c*x)^m*B*b*m^3*x^4 + 260*(c*x)^m 
*C*b*m^2*x^5 + 274*(c*x)^m*D*b*m*x^6 + (c*x)^m*A*a*m^5*x + 19*(c*x)^m*B*a* 
m^4*x^2 + 121*(c*x)^m*C*a*m^3*x^3 + 121*(c*x)^m*A*b*m^3*x^3 + 307*(c*x)^m* 
D*a*m^2*x^4 + 307*(c*x)^m*B*b*m^2*x^4 + 324*(c*x)^m*C*b*m*x^5 + 120*(c*x)^ 
m*D*b*x^6 + 20*(c*x)^m*A*a*m^4*x + 137*(c*x)^m*B*a*m^3*x^2 + 372*(c*x)^m*C 
*a*m^2*x^3 + 372*(c*x)^m*A*b*m^2*x^3 + 396*(c*x)^m*D*a*m*x^4 + 396*(c*x)^m 
*B*b*m*x^4 + 144*(c*x)^m*C*b*x^5 + 155*(c*x)^m*A*a*m^3*x + 461*(c*x)^m*B*a 
*m^2*x^2 + 508*(c*x)^m*C*a*m*x^3 + 508*(c*x)^m*A*b*m*x^3 + 180*(c*x)^m*D*a 
*x^4 + 180*(c*x)^m*B*b*x^4 + 580*(c*x)^m*A*a*m^2*x + 702*(c*x)^m*B*a*m*x^2 
 + 240*(c*x)^m*C*a*x^3 + 240*(c*x)^m*A*b*x^3 + 1044*(c*x)^m*A*a*m*x + 360* 
(c*x)^m*B*a*x^2 + 720*(c*x)^m*A*a*x)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1 
624*m^2 + 1764*m + 720)
 

Mupad [F(-1)]

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

int((c*x)^m*(a + b*x^2)*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((c*x)^m*(a + b*x^2)*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 453, normalized size of antiderivative = 3.81 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x^{m} c^{m} x \left (b d \,m^{5} x^{5}+b c \,m^{5} x^{4}+15 b d \,m^{4} x^{5}+a d \,m^{5} x^{3}+b^{2} m^{5} x^{3}+16 b c \,m^{4} x^{4}+85 b d \,m^{3} x^{5}+a b \,m^{5} x^{2}+a c \,m^{5} x^{2}+17 a d \,m^{4} x^{3}+17 b^{2} m^{4} x^{3}+95 b c \,m^{3} x^{4}+225 b d \,m^{2} x^{5}+a b \,m^{5} x +18 a b \,m^{4} x^{2}+18 a c \,m^{4} x^{2}+107 a d \,m^{3} x^{3}+107 b^{2} m^{3} x^{3}+260 b c \,m^{2} x^{4}+274 b d m \,x^{5}+a^{2} m^{5}+19 a b \,m^{4} x +121 a b \,m^{3} x^{2}+121 a c \,m^{3} x^{2}+307 a d \,m^{2} x^{3}+307 b^{2} m^{2} x^{3}+324 b c m \,x^{4}+120 b d \,x^{5}+20 a^{2} m^{4}+137 a b \,m^{3} x +372 a b \,m^{2} x^{2}+372 a c \,m^{2} x^{2}+396 a d m \,x^{3}+396 b^{2} m \,x^{3}+144 b c \,x^{4}+155 a^{2} m^{3}+461 a b \,m^{2} x +508 a b m \,x^{2}+508 a c m \,x^{2}+180 a d \,x^{3}+180 b^{2} x^{3}+580 a^{2} m^{2}+702 a b m x +240 a b \,x^{2}+240 a c \,x^{2}+1044 a^{2} m +360 a b x +720 a^{2}\right )}{m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720} \] Input:

int((c*x)^m*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(x**m*c**m*x*(a**2*m**5 + 20*a**2*m**4 + 155*a**2*m**3 + 580*a**2*m**2 + 1 
044*a**2*m + 720*a**2 + a*b*m**5*x**2 + a*b*m**5*x + 18*a*b*m**4*x**2 + 19 
*a*b*m**4*x + 121*a*b*m**3*x**2 + 137*a*b*m**3*x + 372*a*b*m**2*x**2 + 461 
*a*b*m**2*x + 508*a*b*m*x**2 + 702*a*b*m*x + 240*a*b*x**2 + 360*a*b*x + a* 
c*m**5*x**2 + 18*a*c*m**4*x**2 + 121*a*c*m**3*x**2 + 372*a*c*m**2*x**2 + 5 
08*a*c*m*x**2 + 240*a*c*x**2 + a*d*m**5*x**3 + 17*a*d*m**4*x**3 + 107*a*d* 
m**3*x**3 + 307*a*d*m**2*x**3 + 396*a*d*m*x**3 + 180*a*d*x**3 + b**2*m**5* 
x**3 + 17*b**2*m**4*x**3 + 107*b**2*m**3*x**3 + 307*b**2*m**2*x**3 + 396*b 
**2*m*x**3 + 180*b**2*x**3 + b*c*m**5*x**4 + 16*b*c*m**4*x**4 + 95*b*c*m** 
3*x**4 + 260*b*c*m**2*x**4 + 324*b*c*m*x**4 + 144*b*c*x**4 + b*d*m**5*x**5 
 + 15*b*d*m**4*x**5 + 85*b*d*m**3*x**5 + 225*b*d*m**2*x**5 + 274*b*d*m*x** 
5 + 120*b*d*x**5))/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 176 
4*m + 720)