\(\int (c+d x)^5 (a+b (c+d x)^2)^p \, dx\) [46]

Optimal result

Integrand size = 21, antiderivative size = 93 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {a^2 \left (a+b (c+d x)^2\right )^{1+p}}{2 b^3 d (1+p)}-\frac {a \left (a+b (c+d x)^2\right )^{2+p}}{b^3 d (2+p)}+\frac {\left (a+b (c+d x)^2\right )^{3+p}}{2 b^3 d (3+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {\left (a+b (c+d x)^2\right )^{1+p} \left (\frac {a^2}{1+p}-\frac {2 a \left (a+b (c+d x)^2\right )}{2+p}+\frac {\left (a+b (c+d x)^2\right )^2}{3+p}\right )}{2 b^3 d} \] Input:

Integrate[(c + d*x)^5*(a + b*(c + d*x)^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 243, 53, 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.

\(\displaystyle \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\int (c+d x)^5 \left (b (c+d x)^2+a\right )^pd(c+d x)}{d}\)

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

Int[(c + d*x)^5*(a + b*(c + d*x)^2)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 895

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff 
icient[v, x, 1]*v^m)   Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ 
{a, b, m, n, p}, x] && LinearPairQ[u, v, x]
 

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. \(288\) vs. \(2(89)=178\).

Time = 1.96 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.11

method	result	size
gosper	\(\frac {\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p +1} \left (b^{2} d^{4} p^{2} x^{4}+4 b^{2} c \,d^{3} p^{2} x^{3}+3 b^{2} d^{4} p \,x^{4}+6 b^{2} c^{2} d^{2} p^{2} x^{2}+12 b^{2} c \,d^{3} p \,x^{3}+2 d^{4} x^{4} b^{2}+4 b^{2} c^{3} d \,p^{2} x +18 b^{2} c^{2} d^{2} p \,x^{2}+8 c \,x^{3} d^{3} b^{2}+b^{2} c^{4} p^{2}+12 b^{2} c^{3} d p x +12 b^{2} c^{2} d^{2} x^{2}-2 a b \,d^{2} p \,x^{2}+3 b^{2} c^{4} p +8 d \,b^{2} x \,c^{3}-4 a b c d p x -2 x^{2} a b \,d^{2}+2 b^{2} c^{4}-2 a b \,c^{2} p -4 a b c d x -2 b \,c^{2} a +2 a^{2}\right )}{2 b^{3} d \left (p^{3}+6 p^{2}+11 p +6\right )}\)	\(289\)
orering	\(\frac {\left (b^{2} d^{4} p^{2} x^{4}+4 b^{2} c \,d^{3} p^{2} x^{3}+3 b^{2} d^{4} p \,x^{4}+6 b^{2} c^{2} d^{2} p^{2} x^{2}+12 b^{2} c \,d^{3} p \,x^{3}+2 d^{4} x^{4} b^{2}+4 b^{2} c^{3} d \,p^{2} x +18 b^{2} c^{2} d^{2} p \,x^{2}+8 c \,x^{3} d^{3} b^{2}+b^{2} c^{4} p^{2}+12 b^{2} c^{3} d p x +12 b^{2} c^{2} d^{2} x^{2}-2 a b \,d^{2} p \,x^{2}+3 b^{2} c^{4} p +8 d \,b^{2} x \,c^{3}-4 a b c d p x -2 x^{2} a b \,d^{2}+2 b^{2} c^{4}-2 a b \,c^{2} p -4 a b c d x -2 b \,c^{2} a +2 a^{2}\right ) \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) \left (a +b \left (x d +c \right )^{2}\right )^{p}}{2 b^{3} d \left (p^{3}+6 p^{2}+11 p +6\right )}\)	\(298\)
norman	\(\frac {d^{5} x^{6} {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{6+2 p}+\frac {3 c \,d^{4} x^{5} {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{3+p}+\frac {\left (b^{3} c^{6} p^{2}+3 b^{3} c^{6} p +a \,b^{2} c^{4} p^{2}+2 b^{3} c^{6}+a \,b^{2} c^{4} p -2 a^{2} b \,c^{2} p +2 a^{3}\right ) {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{2 b^{3} \left (p^{3}+6 p^{2}+11 p +6\right ) d}-\frac {c \left (-3 b^{2} c^{4} p^{2}-9 b^{2} c^{4} p -2 a b \,c^{2} p^{2}-6 b^{2} c^{4}-2 a b \,c^{2} p +2 a^{2} p \right ) x \,{\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{b^{2} \left (p^{3}+6 p^{2}+11 p +6\right )}+\frac {d^{3} \left (15 b \,c^{2} p +30 b \,c^{2}+a p \right ) x^{4} {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{2 b \left (p^{2}+5 p +6\right )}-\frac {\left (-15 b^{2} c^{4} p^{2}-45 b^{2} c^{4} p -6 a b \,c^{2} p^{2}-30 b^{2} c^{4}-6 a b \,c^{2} p +2 a^{2} p \right ) d \,x^{2} {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{2 b^{2} \left (p^{3}+6 p^{2}+11 p +6\right )}+\frac {2 c \,d^{2} \left (5 b \,c^{2} p +10 b \,c^{2}+a p \right ) x^{3} {\mathrm e}^{p \ln \left (a +b \left (x d +c \right )^{2}\right )}}{b \left (p^{2}+5 p +6\right )}\)	\(449\)
risch	\(\frac {\left (b^{3} d^{6} p^{2} x^{6}+6 b^{3} c \,d^{5} p^{2} x^{5}+3 b^{3} d^{6} p \,x^{6}+15 b^{3} c^{2} d^{4} p^{2} x^{4}+18 b^{3} c \,d^{5} p \,x^{5}+2 d^{6} x^{6} b^{3}+20 b^{3} c^{3} d^{3} p^{2} x^{3}+45 b^{3} c^{2} d^{4} p \,x^{4}+12 b^{3} c \,d^{5} x^{5}+a \,b^{2} d^{4} p^{2} x^{4}+15 b^{3} c^{4} d^{2} p^{2} x^{2}+60 b^{3} c^{3} d^{3} p \,x^{3}+30 b^{3} c^{2} d^{4} x^{4}+4 a \,b^{2} c \,d^{3} p^{2} x^{3}+a \,b^{2} d^{4} p \,x^{4}+6 b^{3} c^{5} d \,p^{2} x +45 b^{3} c^{4} d^{2} p \,x^{2}+40 b^{3} c^{3} d^{3} x^{3}+6 a \,b^{2} c^{2} d^{2} p^{2} x^{2}+4 a \,b^{2} c \,d^{3} p \,x^{3}+b^{3} c^{6} p^{2}+18 b^{3} c^{5} d p x +30 b^{3} c^{4} d^{2} x^{2}+4 a \,b^{2} c^{3} d \,p^{2} x +6 a \,b^{2} c^{2} d^{2} p \,x^{2}+3 b^{3} c^{6} p +12 b^{3} c^{5} d x +a \,b^{2} c^{4} p^{2}+4 a \,b^{2} c^{3} d p x +2 b^{3} c^{6}-2 a^{2} b \,d^{2} p \,x^{2}+a \,b^{2} c^{4} p -4 a^{2} b c d p x -2 a^{2} b \,c^{2} p +2 a^{3}\right ) \left (a +b \left (x d +c \right )^{2}\right )^{p}}{2 \left (2+p \right ) \left (3+p \right ) \left (p +1\right ) b^{3} d}\)	\(487\)
parallelrisch	\(\frac {4 x \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{3} d^{2} p +4 x \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{3} d^{2} p^{2}-4 x \left (a +b \left (x d +c \right )^{2}\right )^{p} a^{2} b c \,d^{2} p +4 x^{3} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c \,d^{4} p^{2}+4 x^{3} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c \,d^{4} p +6 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{2} d^{3} p^{2}+6 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{2} d^{3} p +2 d^{7} \left (a +b \left (x d +c \right )^{2}\right )^{p} x^{6} b^{3}+2 \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{6} d +x^{4} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} d^{5} p +45 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{4} d^{3} p +6 x \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{5} d^{2} p^{2}+18 x \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{5} d^{2} p +\left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{4} d \,p^{2}-2 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} a^{2} b \,d^{3} p +\left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} c^{4} d p -2 \left (a +b \left (x d +c \right )^{2}\right )^{p} a^{2} b \,c^{2} d p +6 x^{5} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c \,d^{6} p^{2}+18 x^{5} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c \,d^{6} p +15 x^{4} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{2} d^{5} p^{2}+45 x^{4} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{2} d^{5} p +20 x^{3} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{3} d^{4} p^{2}+x^{4} \left (a +b \left (x d +c \right )^{2}\right )^{p} a \,b^{2} d^{5} p^{2}+60 x^{3} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{3} d^{4} p +15 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{4} d^{3} p^{2}+x^{6} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} d^{7} p^{2}+3 x^{6} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} d^{7} p +12 c \,d^{6} \left (a +b \left (x d +c \right )^{2}\right )^{p} x^{5} b^{3}+30 x^{4} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{2} d^{5}+40 x^{3} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{3} d^{4}+30 x^{2} \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{4} d^{3}+\left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{6} d \,p^{2}+12 x \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{5} d^{2}+3 \left (a +b \left (x d +c \right )^{2}\right )^{p} b^{3} c^{6} d p +2 \left (a +b \left (x d +c \right )^{2}\right )^{p} a^{3} d}{2 \left (3+p \right ) \left (2+p \right ) \left (p +1\right ) b^{3} d^{2}}\)	\(948\)

Input:

int((d*x+c)^5*(a+b*(d*x+c)^2)^p,x,method=_RETURNVERBOSE)
 

Output:

1/2/b^3/d/(p^3+6*p^2+11*p+6)*(b*d^2*x^2+2*b*c*d*x+b*c^2+a)^(p+1)*(b^2*d^4* 
p^2*x^4+4*b^2*c*d^3*p^2*x^3+3*b^2*d^4*p*x^4+6*b^2*c^2*d^2*p^2*x^2+12*b^2*c 
*d^3*p*x^3+2*b^2*d^4*x^4+4*b^2*c^3*d*p^2*x+18*b^2*c^2*d^2*p*x^2+8*b^2*c*d^ 
3*x^3+b^2*c^4*p^2+12*b^2*c^3*d*p*x+12*b^2*c^2*d^2*x^2-2*a*b*d^2*p*x^2+3*b^ 
2*c^4*p+8*b^2*c^3*d*x-4*a*b*c*d*p*x-2*a*b*d^2*x^2+2*b^2*c^4-2*a*b*c^2*p-4* 
a*b*c*d*x-2*a*b*c^2+2*a^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (89) = 178\).

Time = 0.09 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.71 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {{\left (2 \, b^{3} c^{6} + {\left (b^{3} d^{6} p^{2} + 3 \, b^{3} d^{6} p + 2 \, b^{3} d^{6}\right )} x^{6} + 6 \, {\left (b^{3} c d^{5} p^{2} + 3 \, b^{3} c d^{5} p + 2 \, b^{3} c d^{5}\right )} x^{5} + {\left (30 \, b^{3} c^{2} d^{4} + {\left (15 \, b^{3} c^{2} + a b^{2}\right )} d^{4} p^{2} + {\left (45 \, b^{3} c^{2} + a b^{2}\right )} d^{4} p\right )} x^{4} + 4 \, {\left (10 \, b^{3} c^{3} d^{3} + {\left (5 \, b^{3} c^{3} + a b^{2} c\right )} d^{3} p^{2} + {\left (15 \, b^{3} c^{3} + a b^{2} c\right )} d^{3} p\right )} x^{3} + 2 \, a^{3} + {\left (b^{3} c^{6} + a b^{2} c^{4}\right )} p^{2} + {\left (30 \, b^{3} c^{4} d^{2} + 3 \, {\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c^{2}\right )} d^{2} p^{2} + {\left (45 \, b^{3} c^{4} + 6 \, a b^{2} c^{2} - 2 \, a^{2} b\right )} d^{2} p\right )} x^{2} + {\left (3 \, b^{3} c^{6} + a b^{2} c^{4} - 2 \, a^{2} b c^{2}\right )} p + 2 \, {\left (6 \, b^{3} c^{5} d + {\left (3 \, b^{3} c^{5} + 2 \, a b^{2} c^{3}\right )} d p^{2} + {\left (9 \, b^{3} c^{5} + 2 \, a b^{2} c^{3} - 2 \, a^{2} b c\right )} d p\right )} x\right )} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \, {\left (b^{3} d p^{3} + 6 \, b^{3} d p^{2} + 11 \, b^{3} d p + 6 \, b^{3} d\right )}} \] Input:

integrate((d*x+c)^5*(a+b*(d*x+c)^2)^p,x, algorithm="fricas")
 

Output:

1/2*(2*b^3*c^6 + (b^3*d^6*p^2 + 3*b^3*d^6*p + 2*b^3*d^6)*x^6 + 6*(b^3*c*d^ 
5*p^2 + 3*b^3*c*d^5*p + 2*b^3*c*d^5)*x^5 + (30*b^3*c^2*d^4 + (15*b^3*c^2 + 
 a*b^2)*d^4*p^2 + (45*b^3*c^2 + a*b^2)*d^4*p)*x^4 + 4*(10*b^3*c^3*d^3 + (5 
*b^3*c^3 + a*b^2*c)*d^3*p^2 + (15*b^3*c^3 + a*b^2*c)*d^3*p)*x^3 + 2*a^3 + 
(b^3*c^6 + a*b^2*c^4)*p^2 + (30*b^3*c^4*d^2 + 3*(5*b^3*c^4 + 2*a*b^2*c^2)* 
d^2*p^2 + (45*b^3*c^4 + 6*a*b^2*c^2 - 2*a^2*b)*d^2*p)*x^2 + (3*b^3*c^6 + a 
*b^2*c^4 - 2*a^2*b*c^2)*p + 2*(6*b^3*c^5*d + (3*b^3*c^5 + 2*a*b^2*c^3)*d*p 
^2 + (9*b^3*c^5 + 2*a*b^2*c^3 - 2*a^2*b*c)*d*p)*x)*(b*d^2*x^2 + 2*b*c*d*x 
+ b*c^2 + a)^p/(b^3*d*p^3 + 6*b^3*d*p^2 + 11*b^3*d*p + 6*b^3*d)
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**5*(a+b*(d*x+c)**2)**p,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (89) = 178\).

Time = 0.08 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.23 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} d^{6} x^{6} + 6 \, {\left (p^{2} + 3 \, p + 2\right )} b^{3} c d^{5} x^{5} + {\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{6} + {\left (p^{2} + p\right )} a b^{2} c^{4} - 2 \, a^{2} b c^{2} p + {\left (15 \, {\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{2} d^{4} + {\left (p^{2} + p\right )} a b^{2} d^{4}\right )} x^{4} + 4 \, {\left (5 \, {\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{3} d^{3} + {\left (p^{2} + p\right )} a b^{2} c d^{3}\right )} x^{3} + 2 \, a^{3} + {\left (15 \, {\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{4} d^{2} + 6 \, {\left (p^{2} + p\right )} a b^{2} c^{2} d^{2} - 2 \, a^{2} b d^{2} p\right )} x^{2} + 2 \, {\left (3 \, {\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{5} d + 2 \, {\left (p^{2} + p\right )} a b^{2} c^{3} d - 2 \, a^{2} b c d p\right )} x\right )} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \, {\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3} d} \] Input:

integrate((d*x+c)^5*(a+b*(d*x+c)^2)^p,x, algorithm="maxima")
 

Output:

1/2*((p^2 + 3*p + 2)*b^3*d^6*x^6 + 6*(p^2 + 3*p + 2)*b^3*c*d^5*x^5 + (p^2 
+ 3*p + 2)*b^3*c^6 + (p^2 + p)*a*b^2*c^4 - 2*a^2*b*c^2*p + (15*(p^2 + 3*p 
+ 2)*b^3*c^2*d^4 + (p^2 + p)*a*b^2*d^4)*x^4 + 4*(5*(p^2 + 3*p + 2)*b^3*c^3 
*d^3 + (p^2 + p)*a*b^2*c*d^3)*x^3 + 2*a^3 + (15*(p^2 + 3*p + 2)*b^3*c^4*d^ 
2 + 6*(p^2 + p)*a*b^2*c^2*d^2 - 2*a^2*b*d^2*p)*x^2 + 2*(3*(p^2 + 3*p + 2)* 
b^3*c^5*d + 2*(p^2 + p)*a*b^2*c^3*d - 2*a^2*b*c*d*p)*x)*(b*d^2*x^2 + 2*b*c 
*d*x + b*c^2 + a)^p/((p^3 + 6*p^2 + 11*p + 6)*b^3*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (89) = 178\).

Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.84 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{3} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} p - 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{2} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a p + 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{3} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} - 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{2} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a}{2 \, {\left (b^{3} d p^{2} + 5 \, b^{3} d p + 6 \, b^{3} d\right )}} + \frac {{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p + 1} a^{2}}{2 \, b^{3} d {\left (p + 1\right )}} \] Input:

integrate((d*x+c)^5*(a+b*(d*x+c)^2)^p,x, algorithm="giac")
 

Output:

1/2*((b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^3*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 
+ a)^p*p - 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^2*(b*d^2*x^2 + 2*b*c*d*x 
+ b*c^2 + a)^p*a*p + 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^3*(b*d^2*x^2 + 
2*b*c*d*x + b*c^2 + a)^p - 6*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^2*(b*d^2* 
x^2 + 2*b*c*d*x + b*c^2 + a)^p*a)/(b^3*d*p^2 + 5*b^3*d*p + 6*b^3*d) + 1/2* 
(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^(p + 1)*a^2/(b^3*d*(p + 1))
 

Mupad [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 401, normalized size of antiderivative = 4.31 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx={\left (a+b\,{\left (c+d\,x\right )}^2\right )}^p\,\left (\frac {d^5\,x^6\,\left (p^2+3\,p+2\right )}{2\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {\left (b\,c^2+a\right )\,\left (2\,a^2-2\,a\,b\,c^2\,p-2\,a\,b\,c^2+b^2\,c^4\,p^2+3\,b^2\,c^4\,p+2\,b^2\,c^4\right )}{2\,b^3\,d\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {c\,x\,\left (-2\,a^2\,p+2\,a\,b\,c^2\,p^2+2\,a\,b\,c^2\,p+3\,b^2\,c^4\,p^2+9\,b^2\,c^4\,p+6\,b^2\,c^4\right )}{b^2\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {d\,x^2\,\left (-2\,a^2\,p+6\,a\,b\,c^2\,p^2+6\,a\,b\,c^2\,p+15\,b^2\,c^4\,p^2+45\,b^2\,c^4\,p+30\,b^2\,c^4\right )}{2\,b^2\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {3\,c\,d^4\,x^5\,\left (p^2+3\,p+2\right )}{p^3+6\,p^2+11\,p+6}+\frac {d^3\,x^4\,\left (p+1\right )\,\left (a\,p+30\,b\,c^2+15\,b\,c^2\,p\right )}{2\,b\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {2\,c\,d^2\,x^3\,\left (p+1\right )\,\left (a\,p+10\,b\,c^2+5\,b\,c^2\,p\right )}{b\,\left (p^3+6\,p^2+11\,p+6\right )}\right ) \] Input:

int((a + b*(c + d*x)^2)^p*(c + d*x)^5,x)
 

Output:

(a + b*(c + d*x)^2)^p*((d^5*x^6*(3*p + p^2 + 2))/(2*(11*p + 6*p^2 + p^3 + 
6)) + ((a + b*c^2)*(2*a^2 + 2*b^2*c^4 + 3*b^2*c^4*p + b^2*c^4*p^2 - 2*a*b* 
c^2 - 2*a*b*c^2*p))/(2*b^3*d*(11*p + 6*p^2 + p^3 + 6)) + (c*x*(6*b^2*c^4 - 
 2*a^2*p + 9*b^2*c^4*p + 3*b^2*c^4*p^2 + 2*a*b*c^2*p + 2*a*b*c^2*p^2))/(b^ 
2*(11*p + 6*p^2 + p^3 + 6)) + (d*x^2*(30*b^2*c^4 - 2*a^2*p + 45*b^2*c^4*p 
+ 15*b^2*c^4*p^2 + 6*a*b*c^2*p + 6*a*b*c^2*p^2))/(2*b^2*(11*p + 6*p^2 + p^ 
3 + 6)) + (3*c*d^4*x^5*(3*p + p^2 + 2))/(11*p + 6*p^2 + p^3 + 6) + (d^3*x^ 
4*(p + 1)*(a*p + 30*b*c^2 + 15*b*c^2*p))/(2*b*(11*p + 6*p^2 + p^3 + 6)) + 
(2*c*d^2*x^3*(p + 1)*(a*p + 10*b*c^2 + 5*b*c^2*p))/(b*(11*p + 6*p^2 + p^3 
+ 6)))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 496, normalized size of antiderivative = 5.33 \[ \int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx=\frac {\left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{p} \left (b^{3} d^{6} p^{2} x^{6}+6 b^{3} c \,d^{5} p^{2} x^{5}+3 b^{3} d^{6} p \,x^{6}+15 b^{3} c^{2} d^{4} p^{2} x^{4}+18 b^{3} c \,d^{5} p \,x^{5}+2 b^{3} d^{6} x^{6}+20 b^{3} c^{3} d^{3} p^{2} x^{3}+45 b^{3} c^{2} d^{4} p \,x^{4}+12 b^{3} c \,d^{5} x^{5}+a \,b^{2} d^{4} p^{2} x^{4}+15 b^{3} c^{4} d^{2} p^{2} x^{2}+60 b^{3} c^{3} d^{3} p \,x^{3}+30 b^{3} c^{2} d^{4} x^{4}+4 a \,b^{2} c \,d^{3} p^{2} x^{3}+a \,b^{2} d^{4} p \,x^{4}+6 b^{3} c^{5} d \,p^{2} x +45 b^{3} c^{4} d^{2} p \,x^{2}+40 b^{3} c^{3} d^{3} x^{3}+6 a \,b^{2} c^{2} d^{2} p^{2} x^{2}+4 a \,b^{2} c \,d^{3} p \,x^{3}+b^{3} c^{6} p^{2}+18 b^{3} c^{5} d p x +30 b^{3} c^{4} d^{2} x^{2}+4 a \,b^{2} c^{3} d \,p^{2} x +6 a \,b^{2} c^{2} d^{2} p \,x^{2}+3 b^{3} c^{6} p +12 b^{3} c^{5} d x +a \,b^{2} c^{4} p^{2}+4 a \,b^{2} c^{3} d p x +2 b^{3} c^{6}-2 a^{2} b \,d^{2} p \,x^{2}+a \,b^{2} c^{4} p -4 a^{2} b c d p x -2 a^{2} b \,c^{2} p +2 a^{3}\right )}{2 b^{3} d \left (p^{3}+6 p^{2}+11 p +6\right )} \] Input:

int((d*x+c)^5*(a+b*(d*x+c)^2)^p,x)
 

Output:

((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)**p*(2*a**3 - 2*a**2*b*c**2*p - 4*a 
**2*b*c*d*p*x - 2*a**2*b*d**2*p*x**2 + a*b**2*c**4*p**2 + a*b**2*c**4*p + 
4*a*b**2*c**3*d*p**2*x + 4*a*b**2*c**3*d*p*x + 6*a*b**2*c**2*d**2*p**2*x** 
2 + 6*a*b**2*c**2*d**2*p*x**2 + 4*a*b**2*c*d**3*p**2*x**3 + 4*a*b**2*c*d** 
3*p*x**3 + a*b**2*d**4*p**2*x**4 + a*b**2*d**4*p*x**4 + b**3*c**6*p**2 + 3 
*b**3*c**6*p + 2*b**3*c**6 + 6*b**3*c**5*d*p**2*x + 18*b**3*c**5*d*p*x + 1 
2*b**3*c**5*d*x + 15*b**3*c**4*d**2*p**2*x**2 + 45*b**3*c**4*d**2*p*x**2 + 
 30*b**3*c**4*d**2*x**2 + 20*b**3*c**3*d**3*p**2*x**3 + 60*b**3*c**3*d**3* 
p*x**3 + 40*b**3*c**3*d**3*x**3 + 15*b**3*c**2*d**4*p**2*x**4 + 45*b**3*c* 
*2*d**4*p*x**4 + 30*b**3*c**2*d**4*x**4 + 6*b**3*c*d**5*p**2*x**5 + 18*b** 
3*c*d**5*p*x**5 + 12*b**3*c*d**5*x**5 + b**3*d**6*p**2*x**6 + 3*b**3*d**6* 
p*x**6 + 2*b**3*d**6*x**6))/(2*b**3*d*(p**3 + 6*p**2 + 11*p + 6))