\(\int (c+d x)^n (a+b x^2)^3 \, dx\) [205]

Optimal result

Integrand size = 17, antiderivative size = 223 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {\left (b c^2+a d^2\right )^3 (c+d x)^{1+n}}{d^7 (1+n)}-\frac {6 b c \left (b c^2+a d^2\right )^2 (c+d x)^{2+n}}{d^7 (2+n)}+\frac {3 b \left (b c^2+a d^2\right ) \left (5 b c^2+a d^2\right ) (c+d x)^{3+n}}{d^7 (3+n)}-\frac {4 b^2 c \left (5 b c^2+3 a d^2\right ) (c+d x)^{4+n}}{d^7 (4+n)}+\frac {3 b^2 \left (5 b c^2+a d^2\right ) (c+d x)^{5+n}}{d^7 (5+n)}-\frac {6 b^3 c (c+d x)^{6+n}}{d^7 (6+n)}+\frac {b^3 (c+d x)^{7+n}}{d^7 (7+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.70 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {(c+d x)^{1+n} \left (\left (a+b x^2\right )^3+\frac {6 \left (\left (b c^2+a d^2\right ) (6+n) \left (d^4 (1+n) (2+n) (3+n) (4+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (4+n) \left (a d^2 \left (6+5 n+n^2\right )+b \left (2 c^2-2 c d (1+n) x+d^2 \left (2+3 n+n^2\right ) x^2\right )\right )-4 b c (1+n) (c+d x) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )\right )-b c (1+n) (c+d x) \left (d^4 (2+n) (3+n) (4+n) (5+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (5+n) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )-4 b c (2+n) (c+d x) \left (a d^2 \left (20+9 n+n^2\right )+b \left (2 c^2-2 c d (3+n) x+d^2 \left (12+7 n+n^2\right ) x^2\right )\right )\right )\right )}{d^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n)}\right )}{d (7+n)} \] Input:

Integrate[(c + d*x)^n*(a + b*x^2)^3,x]
 

Output:

((c + d*x)^(1 + n)*((a + b*x^2)^3 + (6*((b*c^2 + a*d^2)*(6 + n)*(d^4*(1 + 
n)*(2 + n)*(3 + n)*(4 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d^2)*(4 + n)*(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)) 
 - 4*b*c*(1 + n)*(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))) - b*c*(1 + n)*(c + d*x)*(d^4*(2 + n)*(3 
 + n)*(4 + n)*(5 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d^2)*(5 + n)*(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)) - 4 
*b*c*(2 + n)*(c + d*x)*(a*d^2*(20 + 9*n + n^2) + b*(2*c^2 - 2*c*d*(3 + n)* 
x + d^2*(12 + 7*n + n^2)*x^2)))))/(d^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 
+ n)*(6 + n))))/(d*(7 + n))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 223, 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.

\(\displaystyle \int \left (a+b x^2\right )^3 (c+d x)^n \, dx\)

\(\Big \downarrow \) 476

\(\displaystyle \int \left (-\frac {4 b^2 c \left (3 a d^2+5 b c^2\right ) (c+d x)^{n+3}}{d^6}+\frac {3 b^2 \left (a d^2+5 b c^2\right ) (c+d x)^{n+4}}{d^6}+\frac {\left (a d^2+b c^2\right )^3 (c+d x)^n}{d^6}-\frac {6 b c \left (a d^2+b c^2\right )^2 (c+d x)^{n+1}}{d^6}+\frac {3 b \left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right ) (c+d x)^{n+2}}{d^6}-\frac {6 b^3 c (c+d x)^{n+5}}{d^6}+\frac {b^3 (c+d x)^{n+6}}{d^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 b^2 c \left (3 a d^2+5 b c^2\right ) (c+d x)^{n+4}}{d^7 (n+4)}+\frac {3 b^2 \left (a d^2+5 b c^2\right ) (c+d x)^{n+5}}{d^7 (n+5)}+\frac {\left (a d^2+b c^2\right )^3 (c+d x)^{n+1}}{d^7 (n+1)}-\frac {6 b c \left (a d^2+b c^2\right )^2 (c+d x)^{n+2}}{d^7 (n+2)}+\frac {3 b \left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right ) (c+d x)^{n+3}}{d^7 (n+3)}-\frac {6 b^3 c (c+d x)^{n+6}}{d^7 (n+6)}+\frac {b^3 (c+d x)^{n+7}}{d^7 (n+7)}\)

Input:

Int[(c + d*x)^n*(a + b*x^2)^3,x]
 

Output:

((b*c^2 + a*d^2)^3*(c + d*x)^(1 + n))/(d^7*(1 + n)) - (6*b*c*(b*c^2 + a*d^ 
2)^2*(c + d*x)^(2 + n))/(d^7*(2 + n)) + (3*b*(b*c^2 + a*d^2)*(5*b*c^2 + a* 
d^2)*(c + d*x)^(3 + n))/(d^7*(3 + n)) - (4*b^2*c*(5*b*c^2 + 3*a*d^2)*(c + 
d*x)^(4 + n))/(d^7*(4 + n)) + (3*b^2*(5*b*c^2 + a*d^2)*(c + d*x)^(5 + n))/ 
(d^7*(5 + n)) - (6*b^3*c*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*(c + d*x) 
^(7 + n))/(d^7*(7 + 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. \(953\) vs. \(2(223)=446\).

Time = 0.37 (sec) , antiderivative size = 954, normalized size of antiderivative = 4.28

method	result	size
norman	\(\frac {b^{3} x^{7} {\mathrm e}^{n \ln \left (d x +c \right )}}{7+n}+\frac {c \left (a^{3} d^{6} n^{6}+27 a^{3} d^{6} n^{5}+295 a^{3} d^{6} n^{4}+6 a^{2} b \,c^{2} d^{4} n^{4}+1665 a^{3} d^{6} n^{3}+132 a^{2} b \,c^{2} d^{4} n^{3}+5104 a^{3} d^{6} n^{2}+1074 a^{2} b \,c^{2} d^{4} n^{2}+72 a \,b^{2} c^{4} d^{2} n^{2}+8028 a^{3} d^{6} n +3828 a^{2} b \,c^{2} d^{4} n +936 a \,b^{2} c^{4} d^{2} n +5040 a^{3} d^{6}+5040 a^{2} b \,c^{2} d^{4}+3024 a \,b^{2} c^{4} d^{2}+720 b^{3} c^{6}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{7} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}+\frac {\left (a^{3} d^{6} n^{6}+27 a^{3} d^{6} n^{5}-6 a^{2} b \,c^{2} d^{4} n^{5}+295 a^{3} d^{6} n^{4}-132 a^{2} b \,c^{2} d^{4} n^{4}+1665 a^{3} d^{6} n^{3}-1074 a^{2} b \,c^{2} d^{4} n^{3}-72 a \,b^{2} c^{4} d^{2} n^{3}+5104 a^{3} d^{6} n^{2}-3828 a^{2} b \,c^{2} d^{4} n^{2}-936 a \,b^{2} c^{4} d^{2} n^{2}+8028 a^{3} d^{6} n -5040 a^{2} b \,c^{2} d^{4} n -3024 a \,b^{2} c^{4} d^{2} n -720 b^{3} c^{6} n +5040 a^{3} d^{6}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{6} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}+\frac {b^{3} c n \,x^{6} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+13 n +42\right )}+\frac {3 \left (a \,d^{2} n^{2}+13 a \,d^{2} n -2 b \,c^{2} n +42 a \,d^{2}\right ) b^{2} x^{5} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+18 n^{2}+107 n +210\right )}+\frac {3 \left (a^{2} d^{4} n^{4}+22 a^{2} d^{4} n^{3}-4 a b \,c^{2} d^{2} n^{3}+179 a^{2} d^{4} n^{2}-52 a b \,c^{2} d^{2} n^{2}+638 a^{2} d^{4} n -168 a b \,c^{2} d^{2} n -40 b^{2} c^{4} n +840 a^{2} d^{4}\right ) b \,x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{5}+25 n^{4}+245 n^{3}+1175 n^{2}+2754 n +2520\right )}+\frac {3 \left (a \,d^{2} n^{2}+13 a \,d^{2} n +42 a \,d^{2}+10 b \,c^{2}\right ) b^{2} c n \,x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+22 n^{3}+179 n^{2}+638 n +840\right )}+\frac {3 \left (a^{2} d^{4} n^{4}+22 a^{2} d^{4} n^{3}+179 a^{2} d^{4} n^{2}+12 a b \,c^{2} d^{2} n^{2}+638 a^{2} d^{4} n +156 a b \,c^{2} d^{2} n +840 a^{2} d^{4}+504 b \,c^{2} d^{2} a +120 b^{2} c^{4}\right ) b c n \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{5} \left (n^{6}+27 n^{5}+295 n^{4}+1665 n^{3}+5104 n^{2}+8028 n +5040\right )}\)	\(954\)
gosper	\(\text {Expression too large to display}\)	\(1140\)
orering	\(\text {Expression too large to display}\)	\(1143\)
risch	\(\text {Expression too large to display}\)	\(1412\)
parallelrisch	\(\text {Expression too large to display}\)	\(2063\)

Input:

int((d*x+c)^n*(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

b^3/(7+n)*x^7*exp(n*ln(d*x+c))+c*(a^3*d^6*n^6+27*a^3*d^6*n^5+295*a^3*d^6*n 
^4+6*a^2*b*c^2*d^4*n^4+1665*a^3*d^6*n^3+132*a^2*b*c^2*d^4*n^3+5104*a^3*d^6 
*n^2+1074*a^2*b*c^2*d^4*n^2+72*a*b^2*c^4*d^2*n^2+8028*a^3*d^6*n+3828*a^2*b 
*c^2*d^4*n+936*a*b^2*c^4*d^2*n+5040*a^3*d^6+5040*a^2*b*c^2*d^4+3024*a*b^2* 
c^4*d^2+720*b^3*c^6)/d^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+1 
3068*n+5040)*exp(n*ln(d*x+c))+(a^3*d^6*n^6+27*a^3*d^6*n^5-6*a^2*b*c^2*d^4* 
n^5+295*a^3*d^6*n^4-132*a^2*b*c^2*d^4*n^4+1665*a^3*d^6*n^3-1074*a^2*b*c^2* 
d^4*n^3-72*a*b^2*c^4*d^2*n^3+5104*a^3*d^6*n^2-3828*a^2*b*c^2*d^4*n^2-936*a 
*b^2*c^4*d^2*n^2+8028*a^3*d^6*n-5040*a^2*b*c^2*d^4*n-3024*a*b^2*c^4*d^2*n- 
720*b^3*c^6*n+5040*a^3*d^6)/d^6/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+1313 
2*n^2+13068*n+5040)*x*exp(n*ln(d*x+c))+b^3*c*n/d/(n^2+13*n+42)*x^6*exp(n*l 
n(d*x+c))+3*(a*d^2*n^2+13*a*d^2*n-2*b*c^2*n+42*a*d^2)/d^2*b^2/(n^3+18*n^2+ 
107*n+210)*x^5*exp(n*ln(d*x+c))+3*(a^2*d^4*n^4+22*a^2*d^4*n^3-4*a*b*c^2*d^ 
2*n^3+179*a^2*d^4*n^2-52*a*b*c^2*d^2*n^2+638*a^2*d^4*n-168*a*b*c^2*d^2*n-4 
0*b^2*c^4*n+840*a^2*d^4)/d^4*b/(n^5+25*n^4+245*n^3+1175*n^2+2754*n+2520)*x 
^3*exp(n*ln(d*x+c))+3*(a*d^2*n^2+13*a*d^2*n+42*a*d^2+10*b*c^2)*b^2*c/d^3*n 
/(n^4+22*n^3+179*n^2+638*n+840)*x^4*exp(n*ln(d*x+c))+3*(a^2*d^4*n^4+22*a^2 
*d^4*n^3+179*a^2*d^4*n^2+12*a*b*c^2*d^2*n^2+638*a^2*d^4*n+156*a*b*c^2*d^2* 
n+840*a^2*d^4+504*a*b*c^2*d^2+120*b^2*c^4)*b*c/d^5*n/(n^6+27*n^5+295*n^4+1 
665*n^3+5104*n^2+8028*n+5040)*x^2*exp(n*ln(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1250 vs. \(2 (223) = 446\).

Time = 0.10 (sec) , antiderivative size = 1250, normalized size of antiderivative = 5.61 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

(a^3*c*d^6*n^6 + 27*a^3*c*d^6*n^5 + 720*b^3*c^7 + 3024*a*b^2*c^5*d^2 + 504 
0*a^2*b*c^3*d^4 + 5040*a^3*c*d^6 + (b^3*d^7*n^6 + 21*b^3*d^7*n^5 + 175*b^3 
*d^7*n^4 + 735*b^3*d^7*n^3 + 1624*b^3*d^7*n^2 + 1764*b^3*d^7*n + 720*b^3*d 
^7)*x^7 + (b^3*c*d^6*n^6 + 15*b^3*c*d^6*n^5 + 85*b^3*c*d^6*n^4 + 225*b^3*c 
*d^6*n^3 + 274*b^3*c*d^6*n^2 + 120*b^3*c*d^6*n)*x^6 + 3*(a*b^2*d^7*n^6 + 1 
008*a*b^2*d^7 - (2*b^3*c^2*d^5 - 23*a*b^2*d^7)*n^5 - (20*b^3*c^2*d^5 - 207 
*a*b^2*d^7)*n^4 - 5*(14*b^3*c^2*d^5 - 185*a*b^2*d^7)*n^3 - 4*(25*b^3*c^2*d 
^5 - 536*a*b^2*d^7)*n^2 - 12*(4*b^3*c^2*d^5 - 201*a*b^2*d^7)*n)*x^5 + (6*a 
^2*b*c^3*d^4 + 295*a^3*c*d^6)*n^4 + 3*(a*b^2*c*d^6*n^6 + 19*a*b^2*c*d^6*n^ 
5 + (10*b^3*c^3*d^4 + 131*a*b^2*c*d^6)*n^4 + (60*b^3*c^3*d^4 + 401*a*b^2*c 
*d^6)*n^3 + 10*(11*b^3*c^3*d^4 + 54*a*b^2*c*d^6)*n^2 + 12*(5*b^3*c^3*d^4 + 
 21*a*b^2*c*d^6)*n)*x^4 + 3*(44*a^2*b*c^3*d^4 + 555*a^3*c*d^6)*n^3 + 3*(a^ 
2*b*d^7*n^6 + 1680*a^2*b*d^7 - (4*a*b^2*c^2*d^5 - 25*a^2*b*d^7)*n^5 - (64* 
a*b^2*c^2*d^5 - 247*a^2*b*d^7)*n^4 - (40*b^3*c^4*d^3 + 332*a*b^2*c^2*d^5 - 
 1219*a^2*b*d^7)*n^3 - 8*(15*b^3*c^4*d^3 + 76*a*b^2*c^2*d^5 - 389*a^2*b*d^ 
7)*n^2 - 4*(20*b^3*c^4*d^3 + 84*a*b^2*c^2*d^5 - 949*a^2*b*d^7)*n)*x^3 + 2* 
(36*a*b^2*c^5*d^2 + 537*a^2*b*c^3*d^4 + 2552*a^3*c*d^6)*n^2 + 3*(a^2*b*c*d 
^6*n^6 + 23*a^2*b*c*d^6*n^5 + 3*(4*a*b^2*c^3*d^4 + 67*a^2*b*c*d^6)*n^4 + ( 
168*a*b^2*c^3*d^4 + 817*a^2*b*c*d^6)*n^3 + 2*(60*b^3*c^5*d^2 + 330*a*b^2*c 
^3*d^4 + 739*a^2*b*c*d^6)*n^2 + 24*(5*b^3*c^5*d^2 + 21*a*b^2*c^3*d^4 + ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15990 vs. \(2 (207) = 414\).

Time = 4.29 (sec) , antiderivative size = 15990, normalized size of antiderivative = 71.70 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)**n*(b*x**2+a)**3,x)
 

Output:

Piecewise((c**n*(a**3*x + a**2*b*x**3 + 3*a*b**2*x**5/5 + b**3*x**7/7), Eq 
(d, 0)), (-10*a**3*d**6/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x* 
*2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d* 
*13*x**6) - 3*a**2*b*c**2*d**4/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4* 
d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 
+ 60*d**13*x**6) - 18*a**2*b*c*d**5*x/(60*c**6*d**7 + 360*c**5*d**8*x + 90 
0*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**1 
2*x**5 + 60*d**13*x**6) - 45*a**2*b*d**6*x**2/(60*c**6*d**7 + 360*c**5*d** 
8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 36 
0*c*d**12*x**5 + 60*d**13*x**6) - 6*a*b**2*c**4*d**2/(60*c**6*d**7 + 360*c 
**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x* 
*4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 36*a*b**2*c**3*d**3*x/(60*c**6*d* 
*7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c** 
2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 90*a*b**2*c**2*d**4*x** 
2/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x 
**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 120*a*b**2 
*c*d**5*x**3/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c 
**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 
 90*a*b**2*d**6*x**4/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 
+ 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (223) = 446\).

Time = 0.05 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.12 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {{\left (d x + c\right )}^{n + 1} a^{3}}{d {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} a^{2} b}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {3 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c d^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{2} d^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} c^{3} d^{2} x^{2} - 24 \, c^{4} d n x + 24 \, c^{5}\right )} {\left (d x + c\right )}^{n} a b^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{5}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} d^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} c d^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} c^{2} d^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c^{3} d^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{4} d^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} c^{5} d^{2} x^{2} - 720 \, c^{6} d n x + 720 \, c^{7}\right )} {\left (d x + c\right )}^{n} b^{3}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} d^{7}} \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

(d*x + c)^(n + 1)*a^3/(d*(n + 1)) + 3*((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^2*b/((n^3 + 6*n^2 + 11*n + 
 6)*d^3) + 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*a*b^2/((n^5 + 15*n^4 + 
85*n^3 + 225*n^2 + 274*n + 120)*d^5) + ((n^6 + 21*n^5 + 175*n^4 + 735*n^3 
+ 1624*n^2 + 1764*n + 720)*d^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 27 
4*n^2 + 120*n)*c*d^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c^2*d 
^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^3*d^4*x^4 - 120*(n^3 + 3*n^2 + 
2*n)*c^4*d^3*x^3 + 360*(n^2 + n)*c^5*d^2*x^2 - 720*c^6*d*n*x + 720*c^7)*(d 
*x + c)^n*b^3/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 
 13068*n + 5040)*d^7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2085 vs. \(2 (223) = 446\).

Time = 0.15 (sec) , antiderivative size = 2085, normalized size of antiderivative = 9.35 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^n*(b*x^2+a)^3,x, algorithm="giac")
 

Output:

((d*x + c)^n*b^3*d^7*n^6*x^7 + (d*x + c)^n*b^3*c*d^6*n^6*x^6 + 21*(d*x + c 
)^n*b^3*d^7*n^5*x^7 + 3*(d*x + c)^n*a*b^2*d^7*n^6*x^5 + 15*(d*x + c)^n*b^3 
*c*d^6*n^5*x^6 + 175*(d*x + c)^n*b^3*d^7*n^4*x^7 + 3*(d*x + c)^n*a*b^2*c*d 
^6*n^6*x^4 - 6*(d*x + c)^n*b^3*c^2*d^5*n^5*x^5 + 69*(d*x + c)^n*a*b^2*d^7* 
n^5*x^5 + 85*(d*x + c)^n*b^3*c*d^6*n^4*x^6 + 735*(d*x + c)^n*b^3*d^7*n^3*x 
^7 + 3*(d*x + c)^n*a^2*b*d^7*n^6*x^3 + 57*(d*x + c)^n*a*b^2*c*d^6*n^5*x^4 
- 60*(d*x + c)^n*b^3*c^2*d^5*n^4*x^5 + 621*(d*x + c)^n*a*b^2*d^7*n^4*x^5 + 
 225*(d*x + c)^n*b^3*c*d^6*n^3*x^6 + 1624*(d*x + c)^n*b^3*d^7*n^2*x^7 + 3* 
(d*x + c)^n*a^2*b*c*d^6*n^6*x^2 - 12*(d*x + c)^n*a*b^2*c^2*d^5*n^5*x^3 + 7 
5*(d*x + c)^n*a^2*b*d^7*n^5*x^3 + 30*(d*x + c)^n*b^3*c^3*d^4*n^4*x^4 + 393 
*(d*x + c)^n*a*b^2*c*d^6*n^4*x^4 - 210*(d*x + c)^n*b^3*c^2*d^5*n^3*x^5 + 2 
775*(d*x + c)^n*a*b^2*d^7*n^3*x^5 + 274*(d*x + c)^n*b^3*c*d^6*n^2*x^6 + 17 
64*(d*x + c)^n*b^3*d^7*n*x^7 + (d*x + c)^n*a^3*d^7*n^6*x + 69*(d*x + c)^n* 
a^2*b*c*d^6*n^5*x^2 - 192*(d*x + c)^n*a*b^2*c^2*d^5*n^4*x^3 + 741*(d*x + c 
)^n*a^2*b*d^7*n^4*x^3 + 180*(d*x + c)^n*b^3*c^3*d^4*n^3*x^4 + 1203*(d*x + 
c)^n*a*b^2*c*d^6*n^3*x^4 - 300*(d*x + c)^n*b^3*c^2*d^5*n^2*x^5 + 6432*(d*x 
 + c)^n*a*b^2*d^7*n^2*x^5 + 120*(d*x + c)^n*b^3*c*d^6*n*x^6 + 720*(d*x + c 
)^n*b^3*d^7*x^7 + (d*x + c)^n*a^3*c*d^6*n^6 - 6*(d*x + c)^n*a^2*b*c^2*d^5* 
n^5*x + 27*(d*x + c)^n*a^3*d^7*n^5*x + 36*(d*x + c)^n*a*b^2*c^3*d^4*n^4*x^ 
2 + 603*(d*x + c)^n*a^2*b*c*d^6*n^4*x^2 - 120*(d*x + c)^n*b^3*c^4*d^3*n...
 

Mupad [B] (verification not implemented)

Time = 9.17 (sec) , antiderivative size = 1144, normalized size of antiderivative = 5.13 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((a + b*x^2)^3*(c + d*x)^n,x)
 

Output:

((c + d*x)^n*(720*b^3*c^7 + 5040*a^3*c*d^6 + 3024*a*b^2*c^5*d^2 + 5040*a^2 
*b*c^3*d^4 + 5104*a^3*c*d^6*n^2 + 1665*a^3*c*d^6*n^3 + 295*a^3*c*d^6*n^4 + 
 27*a^3*c*d^6*n^5 + a^3*c*d^6*n^6 + 8028*a^3*c*d^6*n + 936*a*b^2*c^5*d^2*n 
 + 3828*a^2*b*c^3*d^4*n + 72*a*b^2*c^5*d^2*n^2 + 1074*a^2*b*c^3*d^4*n^2 + 
132*a^2*b*c^3*d^4*n^3 + 6*a^2*b*c^3*d^4*n^4))/(d^7*(13068*n + 13132*n^2 + 
6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (x*(c + d*x)^n*(72 
0*b^3*c^6*d*n - 8028*a^3*d^7*n - 5104*a^3*d^7*n^2 - 1665*a^3*d^7*n^3 - 295 
*a^3*d^7*n^4 - 27*a^3*d^7*n^5 - a^3*d^7*n^6 - 5040*a^3*d^7 + 3024*a*b^2*c^ 
4*d^3*n + 5040*a^2*b*c^2*d^5*n + 936*a*b^2*c^4*d^3*n^2 + 3828*a^2*b*c^2*d^ 
5*n^2 + 72*a*b^2*c^4*d^3*n^3 + 1074*a^2*b*c^2*d^5*n^3 + 132*a^2*b*c^2*d^5* 
n^4 + 6*a^2*b*c^2*d^5*n^5))/(d^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^ 
4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (b^3*x^7*(c + d*x)^n*(1764*n + 1624* 
n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769 
*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (3*b^2*x^5*(c + d*x)^n* 
(42*a*d^2 + a*d^2*n^2 + 13*a*d^2*n - 2*b*c^2*n)*(50*n + 35*n^2 + 10*n^3 + 
n^4 + 24))/(d^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28* 
n^6 + n^7 + 5040)) + (3*b*x^3*(c + d*x)^n*(3*n + n^2 + 2)*(840*a^2*d^4 + 6 
38*a^2*d^4*n - 40*b^2*c^4*n + 179*a^2*d^4*n^2 + 22*a^2*d^4*n^3 + a^2*d^4*n 
^4 - 168*a*b*c^2*d^2*n - 52*a*b*c^2*d^2*n^2 - 4*a*b*c^2*d^2*n^3))/(d^4*(13 
068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1411, normalized size of antiderivative = 6.33 \[ \int (c+d x)^n \left (a+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((d*x+c)^n*(b*x^2+a)^3,x)
 

Output:

((c + d*x)**n*(a**3*c*d**6*n**6 + 27*a**3*c*d**6*n**5 + 295*a**3*c*d**6*n* 
*4 + 1665*a**3*c*d**6*n**3 + 5104*a**3*c*d**6*n**2 + 8028*a**3*c*d**6*n + 
5040*a**3*c*d**6 + a**3*d**7*n**6*x + 27*a**3*d**7*n**5*x + 295*a**3*d**7* 
n**4*x + 1665*a**3*d**7*n**3*x + 5104*a**3*d**7*n**2*x + 8028*a**3*d**7*n* 
x + 5040*a**3*d**7*x + 6*a**2*b*c**3*d**4*n**4 + 132*a**2*b*c**3*d**4*n**3 
 + 1074*a**2*b*c**3*d**4*n**2 + 3828*a**2*b*c**3*d**4*n + 5040*a**2*b*c**3 
*d**4 - 6*a**2*b*c**2*d**5*n**5*x - 132*a**2*b*c**2*d**5*n**4*x - 1074*a** 
2*b*c**2*d**5*n**3*x - 3828*a**2*b*c**2*d**5*n**2*x - 5040*a**2*b*c**2*d** 
5*n*x + 3*a**2*b*c*d**6*n**6*x**2 + 69*a**2*b*c*d**6*n**5*x**2 + 603*a**2* 
b*c*d**6*n**4*x**2 + 2451*a**2*b*c*d**6*n**3*x**2 + 4434*a**2*b*c*d**6*n** 
2*x**2 + 2520*a**2*b*c*d**6*n*x**2 + 3*a**2*b*d**7*n**6*x**3 + 75*a**2*b*d 
**7*n**5*x**3 + 741*a**2*b*d**7*n**4*x**3 + 3657*a**2*b*d**7*n**3*x**3 + 9 
336*a**2*b*d**7*n**2*x**3 + 11388*a**2*b*d**7*n*x**3 + 5040*a**2*b*d**7*x* 
*3 + 72*a*b**2*c**5*d**2*n**2 + 936*a*b**2*c**5*d**2*n + 3024*a*b**2*c**5* 
d**2 - 72*a*b**2*c**4*d**3*n**3*x - 936*a*b**2*c**4*d**3*n**2*x - 3024*a*b 
**2*c**4*d**3*n*x + 36*a*b**2*c**3*d**4*n**4*x**2 + 504*a*b**2*c**3*d**4*n 
**3*x**2 + 1980*a*b**2*c**3*d**4*n**2*x**2 + 1512*a*b**2*c**3*d**4*n*x**2 
- 12*a*b**2*c**2*d**5*n**5*x**3 - 192*a*b**2*c**2*d**5*n**4*x**3 - 996*a*b 
**2*c**2*d**5*n**3*x**3 - 1824*a*b**2*c**2*d**5*n**2*x**3 - 1008*a*b**2*c* 
*2*d**5*n*x**3 + 3*a*b**2*c*d**6*n**6*x**4 + 57*a*b**2*c*d**6*n**5*x**4...