\(\int (c+d x)^n (a+b x^2)^2 \, dx\) [196]

Optimal result

Integrand size = 17, antiderivative size = 140 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {\left (b c^2+a d^2\right )^2 (c+d x)^{1+n}}{d^5 (1+n)}-\frac {4 b c \left (b c^2+a d^2\right ) (c+d x)^{2+n}}{d^5 (2+n)}+\frac {2 b \left (3 b c^2+a d^2\right ) (c+d x)^{3+n}}{d^5 (3+n)}-\frac {4 b^2 c (c+d x)^{4+n}}{d^5 (4+n)}+\frac {b^2 (c+d x)^{5+n}}{d^5 (5+n)} \] Output:

(a*d^2+b*c^2)^2*(d*x+c)^(1+n)/d^5/(1+n)-4*b*c*(a*d^2+b*c^2)*(d*x+c)^(2+n)/ 
d^5/(2+n)+2*b*(a*d^2+3*b*c^2)*(d*x+c)^(3+n)/d^5/(3+n)-4*b^2*c*(d*x+c)^(4+n 
)/d^5/(4+n)+b^2*(d*x+c)^(5+n)/d^5/(5+n)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {(c+d x)^{1+n} \left (\left (a+b x^2\right )^2+\frac {4 \left (b c^2+a d^2\right ) \left (a d^2 \left (6+5 n+n^2\right )+b \left (2 c^2-2 c d (1+n) x+d^2 \left (2+3 n+n^2\right ) x^2\right )\right )}{d^4 (1+n) (2+n) (3+n)}-\frac {4 b c (c+d x) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )}{d^4 (2+n) (3+n) (4+n)}\right )}{d (5+n)} \] Input:

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

Output:

((c + d*x)^(1 + n)*((a + b*x^2)^2 + (4*(b*c^2 + a*d^2)*(a*d^2*(6 + 5*n + n 
^2) + b*(2*c^2 - 2*c*d*(1 + n)*x + d^2*(2 + 3*n + n^2)*x^2)))/(d^4*(1 + n) 
*(2 + n)*(3 + n)) - (4*b*c*(c + d*x)*(a*d^2*(12 + 7*n + n^2) + b*(2*c^2 - 
2*c*d*(2 + n)*x + d^2*(6 + 5*n + n^2)*x^2)))/(d^4*(2 + n)*(3 + n)*(4 + n)) 
))/(d*(5 + n))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {476, 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.

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

\(\Big \downarrow \) 476

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (a d^2+b c^2\right )^2 (c+d x)^{n+1}}{d^5 (n+1)}-\frac {4 b c \left (a d^2+b c^2\right ) (c+d x)^{n+2}}{d^5 (n+2)}+\frac {2 b \left (a d^2+3 b c^2\right ) (c+d x)^{n+3}}{d^5 (n+3)}-\frac {4 b^2 c (c+d x)^{n+4}}{d^5 (n+4)}+\frac {b^2 (c+d x)^{n+5}}{d^5 (n+5)}\)

Input:

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

Output:

((b*c^2 + a*d^2)^2*(c + d*x)^(1 + n))/(d^5*(1 + n)) - (4*b*c*(b*c^2 + a*d^ 
2)*(c + d*x)^(2 + n))/(d^5*(2 + n)) + (2*b*(3*b*c^2 + a*d^2)*(c + d*x)^(3 
+ n))/(d^5*(3 + n)) - (4*b^2*c*(c + d*x)^(4 + n))/(d^5*(4 + n)) + (b^2*(c 
+ d*x)^(5 + n))/(d^5*(5 + n))
 

Defintions of rubi rules used

rule 476

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

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. \(419\) vs. \(2(140)=280\).

Time = 0.38 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.00

method	result	size
gosper	\(\frac {\left (d x +c \right )^{1+n} \left (b^{2} d^{4} n^{4} x^{4}+10 b^{2} d^{4} n^{3} x^{4}+2 a b \,d^{4} n^{4} x^{2}-4 b^{2} c \,d^{3} n^{3} x^{3}+35 b^{2} d^{4} n^{2} x^{4}+24 a b \,d^{4} n^{3} x^{2}-24 b^{2} c \,d^{3} n^{2} x^{3}+50 b^{2} d^{4} n \,x^{4}+a^{2} d^{4} n^{4}-4 a b c \,d^{3} n^{3} x +98 a b \,d^{4} n^{2} x^{2}+12 b^{2} c^{2} d^{2} n^{2} x^{2}-44 b^{2} c \,d^{3} n \,x^{3}+24 b^{2} d^{4} x^{4}+14 a^{2} d^{4} n^{3}-40 a b c \,d^{3} n^{2} x +156 a b \,d^{4} n \,x^{2}+36 b^{2} c^{2} d^{2} n \,x^{2}-24 b^{2} c \,d^{3} x^{3}+71 a^{2} d^{4} n^{2}+4 a b \,c^{2} d^{2} n^{2}-116 a b c \,d^{3} n x +80 a b \,d^{4} x^{2}-24 b^{2} c^{3} d n x +24 d^{2} c^{2} x^{2} b^{2}+154 a^{2} d^{4} n +36 a b \,c^{2} d^{2} n -80 a b c \,d^{3} x -24 b^{2} c^{3} d x +120 a^{2} d^{4}+80 b \,c^{2} d^{2} a +24 b^{2} c^{4}\right )}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\)	\(420\)
orering	\(\frac {\left (d x +c \right ) \left (b^{2} d^{4} n^{4} x^{4}+10 b^{2} d^{4} n^{3} x^{4}+2 a b \,d^{4} n^{4} x^{2}-4 b^{2} c \,d^{3} n^{3} x^{3}+35 b^{2} d^{4} n^{2} x^{4}+24 a b \,d^{4} n^{3} x^{2}-24 b^{2} c \,d^{3} n^{2} x^{3}+50 b^{2} d^{4} n \,x^{4}+a^{2} d^{4} n^{4}-4 a b c \,d^{3} n^{3} x +98 a b \,d^{4} n^{2} x^{2}+12 b^{2} c^{2} d^{2} n^{2} x^{2}-44 b^{2} c \,d^{3} n \,x^{3}+24 b^{2} d^{4} x^{4}+14 a^{2} d^{4} n^{3}-40 a b c \,d^{3} n^{2} x +156 a b \,d^{4} n \,x^{2}+36 b^{2} c^{2} d^{2} n \,x^{2}-24 b^{2} c \,d^{3} x^{3}+71 a^{2} d^{4} n^{2}+4 a b \,c^{2} d^{2} n^{2}-116 a b c \,d^{3} n x +80 a b \,d^{4} x^{2}-24 b^{2} c^{3} d n x +24 d^{2} c^{2} x^{2} b^{2}+154 a^{2} d^{4} n +36 a b \,c^{2} d^{2} n -80 a b c \,d^{3} x -24 b^{2} c^{3} d x +120 a^{2} d^{4}+80 b \,c^{2} d^{2} a +24 b^{2} c^{4}\right ) \left (d x +c \right )^{n}}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\)	\(423\)
norman	\(\frac {b^{2} x^{5} {\mathrm e}^{n \ln \left (d x +c \right )}}{5+n}+\frac {c \left (a^{2} d^{4} n^{4}+14 a^{2} d^{4} n^{3}+71 a^{2} d^{4} n^{2}+4 a b \,c^{2} d^{2} n^{2}+154 a^{2} d^{4} n +36 a b \,c^{2} d^{2} n +120 a^{2} d^{4}+80 b \,c^{2} d^{2} a +24 b^{2} c^{4}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {\left (a^{2} d^{4} n^{4}+14 a^{2} d^{4} n^{3}-4 a b \,c^{2} d^{2} n^{3}+71 a^{2} d^{4} n^{2}-36 a b \,c^{2} d^{2} n^{2}+154 a^{2} d^{4} n -80 a b \,c^{2} d^{2} n -24 b^{2} c^{4} n +120 a^{2} d^{4}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {b^{2} c n \,x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+9 n +20\right )}+\frac {2 \left (a \,d^{2} n^{2}+9 a \,d^{2} n -2 b \,c^{2} n +20 a \,d^{2}\right ) b \,x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}+\frac {2 \left (a \,d^{2} n^{2}+9 a \,d^{2} n +20 a \,d^{2}+6 b \,c^{2}\right ) b c n \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}\)	\(450\)
risch	\(\frac {\left (b^{2} d^{5} n^{4} x^{5}+b^{2} c \,d^{4} n^{4} x^{4}+10 b^{2} d^{5} n^{3} x^{5}+2 a b \,d^{5} n^{4} x^{3}+6 b^{2} c \,d^{4} n^{3} x^{4}+35 b^{2} d^{5} n^{2} x^{5}+2 a b c \,d^{4} n^{4} x^{2}+24 a b \,d^{5} n^{3} x^{3}-4 b^{2} c^{2} d^{3} n^{3} x^{3}+11 b^{2} c \,d^{4} n^{2} x^{4}+50 b^{2} d^{5} n \,x^{5}+a^{2} d^{5} n^{4} x +20 a b c \,d^{4} n^{3} x^{2}+98 a b \,d^{5} n^{2} x^{3}-12 b^{2} c^{2} d^{3} n^{2} x^{3}+6 b^{2} c \,d^{4} n \,x^{4}+24 b^{2} x^{5} d^{5}+a^{2} c \,d^{4} n^{4}+14 a^{2} d^{5} n^{3} x -4 a b \,c^{2} d^{3} n^{3} x +58 a b c \,d^{4} n^{2} x^{2}+156 a b \,d^{5} n \,x^{3}+12 b^{2} c^{3} d^{2} n^{2} x^{2}-8 b^{2} c^{2} d^{3} n \,x^{3}+14 a^{2} c \,d^{4} n^{3}+71 a^{2} d^{5} n^{2} x -36 a b \,c^{2} d^{3} n^{2} x +40 a b c \,d^{4} n \,x^{2}+80 a b \,d^{5} x^{3}+12 b^{2} c^{3} d^{2} n \,x^{2}+71 a^{2} c \,d^{4} n^{2}+154 a^{2} d^{5} n x +4 a b \,c^{3} d^{2} n^{2}-80 a b \,c^{2} d^{3} n x -24 b^{2} c^{4} d n x +154 a^{2} c \,d^{4} n +120 a^{2} x \,d^{5}+36 a b \,c^{3} d^{2} n +120 a^{2} c \,d^{4}+80 a \,c^{3} d^{2} b +24 c^{5} b^{2}\right ) \left (d x +c \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{5}}\)	\(555\)
parallelrisch	\(\frac {24 x^{3} \left (d x +c \right )^{n} a b c \,d^{5} n^{3}+x^{4} \left (d x +c \right )^{n} b^{2} c^{2} d^{4} n^{4}+35 x^{5} \left (d x +c \right )^{n} b^{2} c \,d^{5} n^{2}+156 x^{3} \left (d x +c \right )^{n} a b c \,d^{5} n +2 x^{3} \left (d x +c \right )^{n} a b c \,d^{5} n^{4}+12 x^{2} \left (d x +c \right )^{n} b^{2} c^{4} d^{2} n^{2}+14 x \left (d x +c \right )^{n} a^{2} c \,d^{5} n^{3}+12 x^{2} \left (d x +c \right )^{n} b^{2} c^{4} d^{2} n +71 x \left (d x +c \right )^{n} a^{2} c \,d^{5} n^{2}+6 x^{4} \left (d x +c \right )^{n} b^{2} c^{2} d^{4} n^{3}+50 x^{5} \left (d x +c \right )^{n} b^{2} c \,d^{5} n +11 x^{4} \left (d x +c \right )^{n} b^{2} c^{2} d^{4} n^{2}-4 x^{3} \left (d x +c \right )^{n} b^{2} c^{3} d^{3} n^{3}+6 x^{4} \left (d x +c \right )^{n} b^{2} c^{2} d^{4} n -12 x^{3} \left (d x +c \right )^{n} b^{2} c^{3} d^{3} n^{2}+x \left (d x +c \right )^{n} a^{2} c \,d^{5} n^{4}-8 x^{3} \left (d x +c \right )^{n} b^{2} c^{3} d^{3} n +80 \left (d x +c \right )^{n} a b \,c^{4} d^{2}+58 x^{2} \left (d x +c \right )^{n} a b \,c^{2} d^{4} n^{2}-4 x \left (d x +c \right )^{n} a b \,c^{3} d^{3} n^{3}+40 x^{2} \left (d x +c \right )^{n} a b \,c^{2} d^{4} n -36 x \left (d x +c \right )^{n} a b \,c^{3} d^{3} n^{2}-80 x \left (d x +c \right )^{n} a b \,c^{3} d^{3} n +80 x^{3} \left (d x +c \right )^{n} a b c \,d^{5}+24 x^{5} \left (d x +c \right )^{n} b^{2} c \,d^{5}+120 x \left (d x +c \right )^{n} a^{2} c \,d^{5}+2 x^{2} \left (d x +c \right )^{n} a b \,c^{2} d^{4} n^{4}+98 x^{3} \left (d x +c \right )^{n} a b c \,d^{5} n^{2}+20 x^{2} \left (d x +c \right )^{n} a b \,c^{2} d^{4} n^{3}+x^{5} \left (d x +c \right )^{n} b^{2} c \,d^{5} n^{4}+10 x^{5} \left (d x +c \right )^{n} b^{2} c \,d^{5} n^{3}+154 x \left (d x +c \right )^{n} a^{2} c \,d^{5} n -24 x \left (d x +c \right )^{n} b^{2} c^{5} d n +4 \left (d x +c \right )^{n} a b \,c^{4} d^{2} n^{2}+36 \left (d x +c \right )^{n} a b \,c^{4} d^{2} n +\left (d x +c \right )^{n} a^{2} c^{2} d^{4} n^{4}+14 \left (d x +c \right )^{n} a^{2} c^{2} d^{4} n^{3}+71 \left (d x +c \right )^{n} a^{2} c^{2} d^{4} n^{2}+154 \left (d x +c \right )^{n} a^{2} c^{2} d^{4} n +120 \left (d x +c \right )^{n} a^{2} c^{2} d^{4}+24 \left (d x +c \right )^{n} b^{2} c^{6}}{c \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{5}}\)	\(879\)

Input:

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

Output:

1/d^5*(d*x+c)^(1+n)/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)*(b^2*d^4*n^4*x^4 
+10*b^2*d^4*n^3*x^4+2*a*b*d^4*n^4*x^2-4*b^2*c*d^3*n^3*x^3+35*b^2*d^4*n^2*x 
^4+24*a*b*d^4*n^3*x^2-24*b^2*c*d^3*n^2*x^3+50*b^2*d^4*n*x^4+a^2*d^4*n^4-4* 
a*b*c*d^3*n^3*x+98*a*b*d^4*n^2*x^2+12*b^2*c^2*d^2*n^2*x^2-44*b^2*c*d^3*n*x 
^3+24*b^2*d^4*x^4+14*a^2*d^4*n^3-40*a*b*c*d^3*n^2*x+156*a*b*d^4*n*x^2+36*b 
^2*c^2*d^2*n*x^2-24*b^2*c*d^3*x^3+71*a^2*d^4*n^2+4*a*b*c^2*d^2*n^2-116*a*b 
*c*d^3*n*x+80*a*b*d^4*x^2-24*b^2*c^3*d*n*x+24*b^2*c^2*d^2*x^2+154*a^2*d^4* 
n+36*a*b*c^2*d^2*n-80*a*b*c*d^3*x-24*b^2*c^3*d*x+120*a^2*d^4+80*a*b*c^2*d^ 
2+24*b^2*c^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (140) = 280\).

Time = 0.08 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.71 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {{\left (a^{2} c d^{4} n^{4} + 14 \, a^{2} c d^{4} n^{3} + 24 \, b^{2} c^{5} + 80 \, a b c^{3} d^{2} + 120 \, a^{2} c d^{4} + {\left (b^{2} d^{5} n^{4} + 10 \, b^{2} d^{5} n^{3} + 35 \, b^{2} d^{5} n^{2} + 50 \, b^{2} d^{5} n + 24 \, b^{2} d^{5}\right )} x^{5} + {\left (b^{2} c d^{4} n^{4} + 6 \, b^{2} c d^{4} n^{3} + 11 \, b^{2} c d^{4} n^{2} + 6 \, b^{2} c d^{4} n\right )} x^{4} + 2 \, {\left (a b d^{5} n^{4} + 40 \, a b d^{5} - 2 \, {\left (b^{2} c^{2} d^{3} - 6 \, a b d^{5}\right )} n^{3} - {\left (6 \, b^{2} c^{2} d^{3} - 49 \, a b d^{5}\right )} n^{2} - 2 \, {\left (2 \, b^{2} c^{2} d^{3} - 39 \, a b d^{5}\right )} n\right )} x^{3} + {\left (4 \, a b c^{3} d^{2} + 71 \, a^{2} c d^{4}\right )} n^{2} + 2 \, {\left (a b c d^{4} n^{4} + 10 \, a b c d^{4} n^{3} + {\left (6 \, b^{2} c^{3} d^{2} + 29 \, a b c d^{4}\right )} n^{2} + 2 \, {\left (3 \, b^{2} c^{3} d^{2} + 10 \, a b c d^{4}\right )} n\right )} x^{2} + 2 \, {\left (18 \, a b c^{3} d^{2} + 77 \, a^{2} c d^{4}\right )} n + {\left (a^{2} d^{5} n^{4} + 120 \, a^{2} d^{5} - 2 \, {\left (2 \, a b c^{2} d^{3} - 7 \, a^{2} d^{5}\right )} n^{3} - {\left (36 \, a b c^{2} d^{3} - 71 \, a^{2} d^{5}\right )} n^{2} - 2 \, {\left (12 \, b^{2} c^{4} d + 40 \, a b c^{2} d^{3} - 77 \, a^{2} d^{5}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{5} n^{5} + 15 \, d^{5} n^{4} + 85 \, d^{5} n^{3} + 225 \, d^{5} n^{2} + 274 \, d^{5} n + 120 \, d^{5}} \] Input:

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

Output:

(a^2*c*d^4*n^4 + 14*a^2*c*d^4*n^3 + 24*b^2*c^5 + 80*a*b*c^3*d^2 + 120*a^2* 
c*d^4 + (b^2*d^5*n^4 + 10*b^2*d^5*n^3 + 35*b^2*d^5*n^2 + 50*b^2*d^5*n + 24 
*b^2*d^5)*x^5 + (b^2*c*d^4*n^4 + 6*b^2*c*d^4*n^3 + 11*b^2*c*d^4*n^2 + 6*b^ 
2*c*d^4*n)*x^4 + 2*(a*b*d^5*n^4 + 40*a*b*d^5 - 2*(b^2*c^2*d^3 - 6*a*b*d^5) 
*n^3 - (6*b^2*c^2*d^3 - 49*a*b*d^5)*n^2 - 2*(2*b^2*c^2*d^3 - 39*a*b*d^5)*n 
)*x^3 + (4*a*b*c^3*d^2 + 71*a^2*c*d^4)*n^2 + 2*(a*b*c*d^4*n^4 + 10*a*b*c*d 
^4*n^3 + (6*b^2*c^3*d^2 + 29*a*b*c*d^4)*n^2 + 2*(3*b^2*c^3*d^2 + 10*a*b*c* 
d^4)*n)*x^2 + 2*(18*a*b*c^3*d^2 + 77*a^2*c*d^4)*n + (a^2*d^5*n^4 + 120*a^2 
*d^5 - 2*(2*a*b*c^2*d^3 - 7*a^2*d^5)*n^3 - (36*a*b*c^2*d^3 - 71*a^2*d^5)*n 
^2 - 2*(12*b^2*c^4*d + 40*a*b*c^2*d^3 - 77*a^2*d^5)*n)*x)*(d*x + c)^n/(d^5 
*n^5 + 15*d^5*n^4 + 85*d^5*n^3 + 225*d^5*n^2 + 274*d^5*n + 120*d^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5097 vs. \(2 (128) = 256\).

Time = 1.38 (sec) , antiderivative size = 5097, normalized size of antiderivative = 36.41 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((c**n*(a**2*x + 2*a*b*x**3/3 + b**2*x**5/5), Eq(d, 0)), (-3*a**2 
*d**4/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 
+ 12*d**9*x**4) - 2*a*b*c**2*d**2/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2 
*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) - 8*a*b*c*d**3*x/(12*c**4*d**5 
 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) - 1 
2*a*b*d**4*x**2/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c* 
d**8*x**3 + 12*d**9*x**4) + 12*b**2*c**4*log(c/d + x)/(12*c**4*d**5 + 48*c 
**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 25*b**2* 
c**4/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 
 12*d**9*x**4) + 48*b**2*c**3*d*x*log(c/d + x)/(12*c**4*d**5 + 48*c**3*d** 
6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 88*b**2*c**3*d* 
x/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12 
*d**9*x**4) + 72*b**2*c**2*d**2*x**2*log(c/d + x)/(12*c**4*d**5 + 48*c**3* 
d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 108*b**2*c** 
2*d**2*x**2/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8 
*x**3 + 12*d**9*x**4) + 48*b**2*c*d**3*x**3*log(c/d + x)/(12*c**4*d**5 + 4 
8*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4) + 48*b* 
*2*c*d**3*x**3/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d 
**8*x**3 + 12*d**9*x**4) + 12*b**2*d**4*x**4*log(c/d + x)/(12*c**4*d**5 + 
48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d**8*x**3 + 12*d**9*x**4), Eq...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.68 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {{\left (d x + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {2 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} a b}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c d^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{2} d^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} c^{3} d^{2} x^{2} - 24 \, c^{4} d n x + 24 \, c^{5}\right )} {\left (d x + c\right )}^{n} b^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{5}} \] Input:

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

Output:

(d*x + c)^(n + 1)*a^2/(d*(n + 1)) + 2*((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n) 
*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*a*b/((n^3 + 6*n^2 + 11*n + 6 
)*d^3) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11* 
n^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^ 
3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*b^2/((n^5 + 15*n^4 + 85*n^3 
 + 225*n^2 + 274*n + 120)*d^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (140) = 280\).

Time = 0.13 (sec) , antiderivative size = 851, normalized size of antiderivative = 6.08 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

((d*x + c)^n*b^2*d^5*n^4*x^5 + (d*x + c)^n*b^2*c*d^4*n^4*x^4 + 10*(d*x + c 
)^n*b^2*d^5*n^3*x^5 + 2*(d*x + c)^n*a*b*d^5*n^4*x^3 + 6*(d*x + c)^n*b^2*c* 
d^4*n^3*x^4 + 35*(d*x + c)^n*b^2*d^5*n^2*x^5 + 2*(d*x + c)^n*a*b*c*d^4*n^4 
*x^2 - 4*(d*x + c)^n*b^2*c^2*d^3*n^3*x^3 + 24*(d*x + c)^n*a*b*d^5*n^3*x^3 
+ 11*(d*x + c)^n*b^2*c*d^4*n^2*x^4 + 50*(d*x + c)^n*b^2*d^5*n*x^5 + (d*x + 
 c)^n*a^2*d^5*n^4*x + 20*(d*x + c)^n*a*b*c*d^4*n^3*x^2 - 12*(d*x + c)^n*b^ 
2*c^2*d^3*n^2*x^3 + 98*(d*x + c)^n*a*b*d^5*n^2*x^3 + 6*(d*x + c)^n*b^2*c*d 
^4*n*x^4 + 24*(d*x + c)^n*b^2*d^5*x^5 + (d*x + c)^n*a^2*c*d^4*n^4 - 4*(d*x 
 + c)^n*a*b*c^2*d^3*n^3*x + 14*(d*x + c)^n*a^2*d^5*n^3*x + 12*(d*x + c)^n* 
b^2*c^3*d^2*n^2*x^2 + 58*(d*x + c)^n*a*b*c*d^4*n^2*x^2 - 8*(d*x + c)^n*b^2 
*c^2*d^3*n*x^3 + 156*(d*x + c)^n*a*b*d^5*n*x^3 + 14*(d*x + c)^n*a^2*c*d^4* 
n^3 - 36*(d*x + c)^n*a*b*c^2*d^3*n^2*x + 71*(d*x + c)^n*a^2*d^5*n^2*x + 12 
*(d*x + c)^n*b^2*c^3*d^2*n*x^2 + 40*(d*x + c)^n*a*b*c*d^4*n*x^2 + 80*(d*x 
+ c)^n*a*b*d^5*x^3 + 4*(d*x + c)^n*a*b*c^3*d^2*n^2 + 71*(d*x + c)^n*a^2*c* 
d^4*n^2 - 24*(d*x + c)^n*b^2*c^4*d*n*x - 80*(d*x + c)^n*a*b*c^2*d^3*n*x + 
154*(d*x + c)^n*a^2*d^5*n*x + 36*(d*x + c)^n*a*b*c^3*d^2*n + 154*(d*x + c) 
^n*a^2*c*d^4*n + 120*(d*x + c)^n*a^2*d^5*x + 24*(d*x + c)^n*b^2*c^5 + 80*( 
d*x + c)^n*a*b*c^3*d^2 + 120*(d*x + c)^n*a^2*c*d^4)/(d^5*n^5 + 15*d^5*n^4 
+ 85*d^5*n^3 + 225*d^5*n^2 + 274*d^5*n + 120*d^5)
 

Mupad [B] (verification not implemented)

Time = 8.81 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.54 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx={\left (c+d\,x\right )}^n\,\left (\frac {c\,\left (a^2\,d^4\,n^4+14\,a^2\,d^4\,n^3+71\,a^2\,d^4\,n^2+154\,a^2\,d^4\,n+120\,a^2\,d^4+4\,a\,b\,c^2\,d^2\,n^2+36\,a\,b\,c^2\,d^2\,n+80\,a\,b\,c^2\,d^2+24\,b^2\,c^4\right )}{d^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {b^2\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x\,\left (a^2\,d^5\,n^4+14\,a^2\,d^5\,n^3+71\,a^2\,d^5\,n^2+154\,a^2\,d^5\,n+120\,a^2\,d^5-4\,a\,b\,c^2\,d^3\,n^3-36\,a\,b\,c^2\,d^3\,n^2-80\,a\,b\,c^2\,d^3\,n-24\,b^2\,c^4\,d\,n\right )}{d^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {2\,b\,x^3\,\left (n^2+3\,n+2\right )\,\left (-2\,b\,c^2\,n+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {b^2\,c\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{d\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {2\,b\,c\,n\,x^2\,\left (n+1\right )\,\left (6\,b\,c^2+a\,d^2\,n^2+9\,a\,d^2\,n+20\,a\,d^2\right )}{d^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \] Input:

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

Output:

(c + d*x)^n*((c*(120*a^2*d^4 + 24*b^2*c^4 + 154*a^2*d^4*n + 71*a^2*d^4*n^2 
 + 14*a^2*d^4*n^3 + a^2*d^4*n^4 + 80*a*b*c^2*d^2 + 36*a*b*c^2*d^2*n + 4*a* 
b*c^2*d^2*n^2))/(d^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (b 
^2*x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^3 + 15 
*n^4 + n^5 + 120) + (x*(120*a^2*d^5 + 154*a^2*d^5*n + 71*a^2*d^5*n^2 + 14* 
a^2*d^5*n^3 + a^2*d^5*n^4 - 24*b^2*c^4*d*n - 80*a*b*c^2*d^3*n - 36*a*b*c^2 
*d^3*n^2 - 4*a*b*c^2*d^3*n^3))/(d^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n 
^5 + 120)) + (2*b*x^3*(3*n + n^2 + 2)*(20*a*d^2 + a*d^2*n^2 + 9*a*d^2*n - 
2*b*c^2*n))/(d^2*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (b^2*c 
*n*x^4*(11*n + 6*n^2 + n^3 + 6))/(d*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n 
^5 + 120)) + (2*b*c*n*x^2*(n + 1)*(20*a*d^2 + 6*b*c^2 + a*d^2*n^2 + 9*a*d^ 
2*n))/(d^3*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.96 \[ \int (c+d x)^n \left (a+b x^2\right )^2 \, dx=\frac {\left (d x +c \right )^{n} \left (b^{2} d^{5} n^{4} x^{5}+b^{2} c \,d^{4} n^{4} x^{4}+10 b^{2} d^{5} n^{3} x^{5}+2 a b \,d^{5} n^{4} x^{3}+6 b^{2} c \,d^{4} n^{3} x^{4}+35 b^{2} d^{5} n^{2} x^{5}+2 a b c \,d^{4} n^{4} x^{2}+24 a b \,d^{5} n^{3} x^{3}-4 b^{2} c^{2} d^{3} n^{3} x^{3}+11 b^{2} c \,d^{4} n^{2} x^{4}+50 b^{2} d^{5} n \,x^{5}+a^{2} d^{5} n^{4} x +20 a b c \,d^{4} n^{3} x^{2}+98 a b \,d^{5} n^{2} x^{3}-12 b^{2} c^{2} d^{3} n^{2} x^{3}+6 b^{2} c \,d^{4} n \,x^{4}+24 b^{2} d^{5} x^{5}+a^{2} c \,d^{4} n^{4}+14 a^{2} d^{5} n^{3} x -4 a b \,c^{2} d^{3} n^{3} x +58 a b c \,d^{4} n^{2} x^{2}+156 a b \,d^{5} n \,x^{3}+12 b^{2} c^{3} d^{2} n^{2} x^{2}-8 b^{2} c^{2} d^{3} n \,x^{3}+14 a^{2} c \,d^{4} n^{3}+71 a^{2} d^{5} n^{2} x -36 a b \,c^{2} d^{3} n^{2} x +40 a b c \,d^{4} n \,x^{2}+80 a b \,d^{5} x^{3}+12 b^{2} c^{3} d^{2} n \,x^{2}+71 a^{2} c \,d^{4} n^{2}+154 a^{2} d^{5} n x +4 a b \,c^{3} d^{2} n^{2}-80 a b \,c^{2} d^{3} n x -24 b^{2} c^{4} d n x +154 a^{2} c \,d^{4} n +120 a^{2} d^{5} x +36 a b \,c^{3} d^{2} n +120 a^{2} c \,d^{4}+80 a b \,c^{3} d^{2}+24 b^{2} c^{5}\right )}{d^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )} \] Input:

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

Output:

((c + d*x)**n*(a**2*c*d**4*n**4 + 14*a**2*c*d**4*n**3 + 71*a**2*c*d**4*n** 
2 + 154*a**2*c*d**4*n + 120*a**2*c*d**4 + a**2*d**5*n**4*x + 14*a**2*d**5* 
n**3*x + 71*a**2*d**5*n**2*x + 154*a**2*d**5*n*x + 120*a**2*d**5*x + 4*a*b 
*c**3*d**2*n**2 + 36*a*b*c**3*d**2*n + 80*a*b*c**3*d**2 - 4*a*b*c**2*d**3* 
n**3*x - 36*a*b*c**2*d**3*n**2*x - 80*a*b*c**2*d**3*n*x + 2*a*b*c*d**4*n** 
4*x**2 + 20*a*b*c*d**4*n**3*x**2 + 58*a*b*c*d**4*n**2*x**2 + 40*a*b*c*d**4 
*n*x**2 + 2*a*b*d**5*n**4*x**3 + 24*a*b*d**5*n**3*x**3 + 98*a*b*d**5*n**2* 
x**3 + 156*a*b*d**5*n*x**3 + 80*a*b*d**5*x**3 + 24*b**2*c**5 - 24*b**2*c** 
4*d*n*x + 12*b**2*c**3*d**2*n**2*x**2 + 12*b**2*c**3*d**2*n*x**2 - 4*b**2* 
c**2*d**3*n**3*x**3 - 12*b**2*c**2*d**3*n**2*x**3 - 8*b**2*c**2*d**3*n*x** 
3 + b**2*c*d**4*n**4*x**4 + 6*b**2*c*d**4*n**3*x**4 + 11*b**2*c*d**4*n**2* 
x**4 + 6*b**2*c*d**4*n*x**4 + b**2*d**5*n**4*x**5 + 10*b**2*d**5*n**3*x**5 
 + 35*b**2*d**5*n**2*x**5 + 50*b**2*d**5*n*x**5 + 24*b**2*d**5*x**5))/(d** 
5*(n**5 + 15*n**4 + 85*n**3 + 225*n**2 + 274*n + 120))