\(\int x (c+d x)^n (a+b x^2) \, dx\) [187]

Optimal result

Integrand size = 16, antiderivative size = 102 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=-\frac {c \left (b c^2+a d^2\right ) (c+d x)^{1+n}}{d^4 (1+n)}+\frac {\left (3 b c^2+a d^2\right ) (c+d x)^{2+n}}{d^4 (2+n)}-\frac {3 b c (c+d x)^{3+n}}{d^4 (3+n)}+\frac {b (c+d x)^{4+n}}{d^4 (4+n)} \] Output:

-c*(a*d^2+b*c^2)*(d*x+c)^(1+n)/d^4/(1+n)+(a*d^2+3*b*c^2)*(d*x+c)^(2+n)/d^4 
/(2+n)-3*b*c*(d*x+c)^(3+n)/d^4/(3+n)+b*(d*x+c)^(4+n)/d^4/(4+n)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.10 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {(c+d x)^{1+n} \left (a d^2 \left (12+7 n+n^2\right ) (-c+d (1+n) x)+b \left (-6 c^3+6 c^2 d (1+n) x-3 c d^2 \left (2+3 n+n^2\right ) x^2+d^3 \left (6+11 n+6 n^2+n^3\right ) x^3\right )\right )}{d^4 (1+n) (2+n) (3+n) (4+n)} \] Input:

Integrate[x*(c + d*x)^n*(a + b*x^2),x]
 

Output:

((c + d*x)^(1 + n)*(a*d^2*(12 + 7*n + n^2)*(-c + d*(1 + n)*x) + b*(-6*c^3 
+ 6*c^2*d*(1 + n)*x - 3*c*d^2*(2 + 3*n + n^2)*x^2 + d^3*(6 + 11*n + 6*n^2 
+ n^3)*x^3)))/(d^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {522, 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.

\(\displaystyle \int x \left (a+b x^2\right ) (c+d x)^n \, dx\)

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {c \left (-a d^2-b c^2\right ) (c+d x)^n}{d^3}+\frac {\left (a d^2+3 b c^2\right ) (c+d x)^{n+1}}{d^3}-\frac {3 b c (c+d x)^{n+2}}{d^3}+\frac {b (c+d x)^{n+3}}{d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c \left (a d^2+b c^2\right ) (c+d x)^{n+1}}{d^4 (n+1)}+\frac {\left (a d^2+3 b c^2\right ) (c+d x)^{n+2}}{d^4 (n+2)}-\frac {3 b c (c+d x)^{n+3}}{d^4 (n+3)}+\frac {b (c+d x)^{n+4}}{d^4 (n+4)}\)

Input:

Int[x*(c + d*x)^n*(a + b*x^2),x]
 

Output:

-((c*(b*c^2 + a*d^2)*(c + d*x)^(1 + n))/(d^4*(1 + n))) + ((3*b*c^2 + a*d^2 
)*(c + d*x)^(2 + n))/(d^4*(2 + n)) - (3*b*c*(c + d*x)^(3 + n))/(d^4*(3 + n 
)) + (b*(c + d*x)^(4 + n))/(d^4*(4 + n))
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.91

method	result	size
gosper	\(-\frac {\left (d x +c \right )^{1+n} \left (-b \,d^{3} n^{3} x^{3}-6 b \,d^{3} n^{2} x^{3}-a \,d^{3} n^{3} x +3 b c \,d^{2} n^{2} x^{2}-11 b \,d^{3} n \,x^{3}-8 a \,d^{3} n^{2} x +9 b c \,d^{2} n \,x^{2}-6 b \,d^{3} x^{3}+a c \,d^{2} n^{2}-19 a \,d^{3} n x -6 b \,c^{2} d n x +6 b c \,d^{2} x^{2}+7 a c \,d^{2} n -12 a x \,d^{3}-6 b \,c^{2} d x +12 a \,d^{2} c +6 b \,c^{3}\right )}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(195\)
orering	\(-\frac {\left (d x +c \right )^{n} \left (-b \,d^{3} n^{3} x^{3}-6 b \,d^{3} n^{2} x^{3}-a \,d^{3} n^{3} x +3 b c \,d^{2} n^{2} x^{2}-11 b \,d^{3} n \,x^{3}-8 a \,d^{3} n^{2} x +9 b c \,d^{2} n \,x^{2}-6 b \,d^{3} x^{3}+a c \,d^{2} n^{2}-19 a \,d^{3} n x -6 b \,c^{2} d n x +6 b c \,d^{2} x^{2}+7 a c \,d^{2} n -12 a x \,d^{3}-6 b \,c^{2} d x +12 a \,d^{2} c +6 b \,c^{3}\right ) \left (d x +c \right )}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(198\)
norman	\(\frac {b \,x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{4+n}+\frac {\left (a \,d^{2} n^{2}+7 a \,d^{2} n -3 b \,c^{2} n +12 a \,d^{2}\right ) x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}+\frac {n c \left (a \,d^{2} n^{2}+7 a \,d^{2} n +12 a \,d^{2}+6 b \,c^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {n b c \,x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+7 n +12\right )}-\frac {c^{2} \left (a \,d^{2} n^{2}+7 a \,d^{2} n +12 a \,d^{2}+6 b \,c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(239\)
risch	\(-\frac {\left (-b \,d^{4} n^{3} x^{4}-b c \,d^{3} n^{3} x^{3}-6 b \,d^{4} n^{2} x^{4}-a \,d^{4} n^{3} x^{2}-3 b c \,d^{3} n^{2} x^{3}-11 b \,d^{4} n \,x^{4}-a c \,d^{3} n^{3} x -8 a \,d^{4} n^{2} x^{2}+3 b \,c^{2} d^{2} n^{2} x^{2}-2 b c \,d^{3} n \,x^{3}-6 b \,x^{4} d^{4}-7 a c \,d^{3} n^{2} x -19 a \,d^{4} n \,x^{2}+3 b \,c^{2} d^{2} n \,x^{2}+a \,c^{2} d^{2} n^{2}-12 a c \,d^{3} n x -12 a \,d^{4} x^{2}-6 b \,c^{3} d n x +7 a \,c^{2} d^{2} n +12 a \,c^{2} d^{2}+6 b \,c^{4}\right ) \left (d x +c \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) d^{4}}\)	\(261\)
parallelrisch	\(\frac {11 x^{4} \left (d x +c \right )^{n} b c \,d^{4} n +3 x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n^{2}+x^{4} \left (d x +c \right )^{n} b c \,d^{4} n^{3}+6 x^{4} \left (d x +c \right )^{n} b c \,d^{4} n^{2}+x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n^{3}+6 x^{4} \left (d x +c \right )^{n} b c \,d^{4}+2 x^{3} \left (d x +c \right )^{n} b \,c^{2} d^{3} n +8 x^{2} \left (d x +c \right )^{n} a c \,d^{4} n^{2}-3 x^{2} \left (d x +c \right )^{n} b \,c^{3} d^{2} n^{2}+x \left (d x +c \right )^{n} a \,c^{2} d^{3} n^{3}+19 x^{2} \left (d x +c \right )^{n} a c \,d^{4} n -3 x^{2} \left (d x +c \right )^{n} b \,c^{3} d^{2} n +7 x \left (d x +c \right )^{n} a \,c^{2} d^{3} n^{2}+12 x \left (d x +c \right )^{n} a \,c^{2} d^{3} n +6 x \left (d x +c \right )^{n} b \,c^{4} d n -7 \left (d x +c \right )^{n} a \,c^{3} d^{2} n +x^{2} \left (d x +c \right )^{n} a c \,d^{4} n^{3}-\left (d x +c \right )^{n} a \,c^{3} d^{2} n^{2}+12 x^{2} \left (d x +c \right )^{n} a c \,d^{4}-12 \left (d x +c \right )^{n} a \,c^{3} d^{2}-6 \left (d x +c \right )^{n} b \,c^{5}}{\left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) c \,d^{4}}\)	\(420\)

Input:

int(x*(d*x+c)^n*(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/d^4*(d*x+c)^(1+n)/(n^4+10*n^3+35*n^2+50*n+24)*(-b*d^3*n^3*x^3-6*b*d^3*n 
^2*x^3-a*d^3*n^3*x+3*b*c*d^2*n^2*x^2-11*b*d^3*n*x^3-8*a*d^3*n^2*x+9*b*c*d^ 
2*n*x^2-6*b*d^3*x^3+a*c*d^2*n^2-19*a*d^3*n*x-6*b*c^2*d*n*x+6*b*c*d^2*x^2+7 
*a*c*d^2*n-12*a*d^3*x-6*b*c^2*d*x+12*a*c*d^2+6*b*c^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (102) = 204\).

Time = 0.09 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.47 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=-\frac {{\left (a c^{2} d^{2} n^{2} + 7 \, a c^{2} d^{2} n + 6 \, b c^{4} + 12 \, a c^{2} d^{2} - {\left (b d^{4} n^{3} + 6 \, b d^{4} n^{2} + 11 \, b d^{4} n + 6 \, b d^{4}\right )} x^{4} - {\left (b c d^{3} n^{3} + 3 \, b c d^{3} n^{2} + 2 \, b c d^{3} n\right )} x^{3} - {\left (a d^{4} n^{3} + 12 \, a d^{4} - {\left (3 \, b c^{2} d^{2} - 8 \, a d^{4}\right )} n^{2} - {\left (3 \, b c^{2} d^{2} - 19 \, a d^{4}\right )} n\right )} x^{2} - {\left (a c d^{3} n^{3} + 7 \, a c d^{3} n^{2} + 6 \, {\left (b c^{3} d + 2 \, a c d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a),x, algorithm="fricas")
 

Output:

-(a*c^2*d^2*n^2 + 7*a*c^2*d^2*n + 6*b*c^4 + 12*a*c^2*d^2 - (b*d^4*n^3 + 6* 
b*d^4*n^2 + 11*b*d^4*n + 6*b*d^4)*x^4 - (b*c*d^3*n^3 + 3*b*c*d^3*n^2 + 2*b 
*c*d^3*n)*x^3 - (a*d^4*n^3 + 12*a*d^4 - (3*b*c^2*d^2 - 8*a*d^4)*n^2 - (3*b 
*c^2*d^2 - 19*a*d^4)*n)*x^2 - (a*c*d^3*n^3 + 7*a*c*d^3*n^2 + 6*(b*c^3*d + 
2*a*c*d^3)*n)*x)*(d*x + c)^n/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n 
 + 24*d^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2181 vs. \(2 (90) = 180\).

Time = 0.77 (sec) , antiderivative size = 2181, normalized size of antiderivative = 21.38 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=\text {Too large to display} \] Input:

integrate(x*(d*x+c)**n*(b*x**2+a),x)
 

Output:

Piecewise((c**n*(a*x**2/2 + b*x**4/4), Eq(d, 0)), (-a*c*d**2/(6*c**3*d**4 
+ 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 3*a*d**3*x/(6*c**3*d**4 
 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*b*c**3*log(c/d + x)/ 
(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 11*b*c**3/ 
(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*b*c**2* 
d*x*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x 
**3) + 27*b*c**2*d*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d* 
*7*x**3) + 18*b*c*d**2*x**2*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 1 
8*c*d**6*x**2 + 6*d**7*x**3) + 18*b*c*d**2*x**2/(6*c**3*d**4 + 18*c**2*d** 
5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*b*d**3*x**3*log(c/d + x)/(6*c**3*d 
**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3), Eq(n, -4)), (-a*c*d* 
*2/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 2*a*d**3*x/(2*c**2*d**4 + 4* 
c*d**5*x + 2*d**6*x**2) - 6*b*c**3*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x 
+ 2*d**6*x**2) - 9*b*c**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b* 
c**2*d*x*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b*c**2 
*d*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*b*c*d**2*x**2*log(c/d + 
x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*b*d**3*x**3/(2*c**2*d**4 + 
 4*c*d**5*x + 2*d**6*x**2), Eq(n, -3)), (2*a*c*d**2*log(c/d + x)/(2*c*d**4 
 + 2*d**5*x) + 2*a*c*d**2/(2*c*d**4 + 2*d**5*x) + 2*a*d**3*x*log(c/d + x)/ 
(2*c*d**4 + 2*d**5*x) + 6*b*c**3*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 6...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} b}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a),x, algorithm="maxima")
 

Output:

(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*a/((n^2 + 3*n + 2)*d^2) + (( 
n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + 
 n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*b/((n^4 + 10*n^3 + 35*n 
^2 + 50*n + 24)*d^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (102) = 204\).

Time = 0.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.02 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left (d x + c\right )}^{n} b d^{4} n^{3} x^{4} + {\left (d x + c\right )}^{n} b c d^{3} n^{3} x^{3} + 6 \, {\left (d x + c\right )}^{n} b d^{4} n^{2} x^{4} + {\left (d x + c\right )}^{n} a d^{4} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} b c d^{3} n^{2} x^{3} + 11 \, {\left (d x + c\right )}^{n} b d^{4} n x^{4} + {\left (d x + c\right )}^{n} a c d^{3} n^{3} x - 3 \, {\left (d x + c\right )}^{n} b c^{2} d^{2} n^{2} x^{2} + 8 \, {\left (d x + c\right )}^{n} a d^{4} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} b c d^{3} n x^{3} + 6 \, {\left (d x + c\right )}^{n} b d^{4} x^{4} + 7 \, {\left (d x + c\right )}^{n} a c d^{3} n^{2} x - 3 \, {\left (d x + c\right )}^{n} b c^{2} d^{2} n x^{2} + 19 \, {\left (d x + c\right )}^{n} a d^{4} n x^{2} - {\left (d x + c\right )}^{n} a c^{2} d^{2} n^{2} + 6 \, {\left (d x + c\right )}^{n} b c^{3} d n x + 12 \, {\left (d x + c\right )}^{n} a c d^{3} n x + 12 \, {\left (d x + c\right )}^{n} a d^{4} x^{2} - 7 \, {\left (d x + c\right )}^{n} a c^{2} d^{2} n - 6 \, {\left (d x + c\right )}^{n} b c^{4} - 12 \, {\left (d x + c\right )}^{n} a c^{2} d^{2}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a),x, algorithm="giac")
 

Output:

((d*x + c)^n*b*d^4*n^3*x^4 + (d*x + c)^n*b*c*d^3*n^3*x^3 + 6*(d*x + c)^n*b 
*d^4*n^2*x^4 + (d*x + c)^n*a*d^4*n^3*x^2 + 3*(d*x + c)^n*b*c*d^3*n^2*x^3 + 
 11*(d*x + c)^n*b*d^4*n*x^4 + (d*x + c)^n*a*c*d^3*n^3*x - 3*(d*x + c)^n*b* 
c^2*d^2*n^2*x^2 + 8*(d*x + c)^n*a*d^4*n^2*x^2 + 2*(d*x + c)^n*b*c*d^3*n*x^ 
3 + 6*(d*x + c)^n*b*d^4*x^4 + 7*(d*x + c)^n*a*c*d^3*n^2*x - 3*(d*x + c)^n* 
b*c^2*d^2*n*x^2 + 19*(d*x + c)^n*a*d^4*n*x^2 - (d*x + c)^n*a*c^2*d^2*n^2 + 
 6*(d*x + c)^n*b*c^3*d*n*x + 12*(d*x + c)^n*a*c*d^3*n*x + 12*(d*x + c)^n*a 
*d^4*x^2 - 7*(d*x + c)^n*a*c^2*d^2*n - 6*(d*x + c)^n*b*c^4 - 12*(d*x + c)^ 
n*a*c^2*d^2)/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)
 

Mupad [B] (verification not implemented)

Time = 8.50 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.50 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx={\left (c+d\,x\right )}^n\,\left (\frac {b\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {c^2\,\left (6\,b\,c^2+a\,d^2\,n^2+7\,a\,d^2\,n+12\,a\,d^2\right )}{d^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,b\,c^2\,n+a\,d^2\,n^2+7\,a\,d^2\,n+12\,a\,d^2\right )}{d^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {c\,n\,x\,\left (6\,b\,c^2+a\,d^2\,n^2+7\,a\,d^2\,n+12\,a\,d^2\right )}{d^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,c\,n\,x^3\,\left (n^2+3\,n+2\right )}{d\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \] Input:

int(x*(a + b*x^2)*(c + d*x)^n,x)
 

Output:

(c + d*x)^n*((b*x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^ 
4 + 24) - (c^2*(12*a*d^2 + 6*b*c^2 + a*d^2*n^2 + 7*a*d^2*n))/(d^4*(50*n + 
35*n^2 + 10*n^3 + n^4 + 24)) + (x^2*(n + 1)*(12*a*d^2 + a*d^2*n^2 + 7*a*d^ 
2*n - 3*b*c^2*n))/(d^2*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (c*n*x*(12*a 
*d^2 + 6*b*c^2 + a*d^2*n^2 + 7*a*d^2*n))/(d^3*(50*n + 35*n^2 + 10*n^3 + n^ 
4 + 24)) + (b*c*n*x^3*(3*n + n^2 + 2))/(d*(50*n + 35*n^2 + 10*n^3 + n^4 + 
24)))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.51 \[ \int x (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {\left (d x +c \right )^{n} \left (b \,d^{4} n^{3} x^{4}+b c \,d^{3} n^{3} x^{3}+6 b \,d^{4} n^{2} x^{4}+a \,d^{4} n^{3} x^{2}+3 b c \,d^{3} n^{2} x^{3}+11 b \,d^{4} n \,x^{4}+a c \,d^{3} n^{3} x +8 a \,d^{4} n^{2} x^{2}-3 b \,c^{2} d^{2} n^{2} x^{2}+2 b c \,d^{3} n \,x^{3}+6 b \,d^{4} x^{4}+7 a c \,d^{3} n^{2} x +19 a \,d^{4} n \,x^{2}-3 b \,c^{2} d^{2} n \,x^{2}-a \,c^{2} d^{2} n^{2}+12 a c \,d^{3} n x +12 a \,d^{4} x^{2}+6 b \,c^{3} d n x -7 a \,c^{2} d^{2} n -12 a \,c^{2} d^{2}-6 b \,c^{4}\right )}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )} \] Input:

int(x*(d*x+c)^n*(b*x^2+a),x)
 

Output:

((c + d*x)**n*( - a*c**2*d**2*n**2 - 7*a*c**2*d**2*n - 12*a*c**2*d**2 + a* 
c*d**3*n**3*x + 7*a*c*d**3*n**2*x + 12*a*c*d**3*n*x + a*d**4*n**3*x**2 + 8 
*a*d**4*n**2*x**2 + 19*a*d**4*n*x**2 + 12*a*d**4*x**2 - 6*b*c**4 + 6*b*c** 
3*d*n*x - 3*b*c**2*d**2*n**2*x**2 - 3*b*c**2*d**2*n*x**2 + b*c*d**3*n**3*x 
**3 + 3*b*c*d**3*n**2*x**3 + 2*b*c*d**3*n*x**3 + b*d**4*n**3*x**4 + 6*b*d* 
*4*n**2*x**4 + 11*b*d**4*n*x**4 + 6*b*d**4*x**4))/(d**4*(n**4 + 10*n**3 + 
35*n**2 + 50*n + 24))