\(\int (c+d x)^3 (a+b (c+d x)^4)^2 \, dx\) [2912]

Optimal result

Integrand size = 21, antiderivative size = 23 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {\left (a+b (c+d x)^4\right )^3}{12 b d} \]

[Out]

1/12*(a+b*(d*x+c)^4)^3/b/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {379, 267} \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {\left (a+b (c+d x)^4\right )^3}{12 b d} \]

[In]

Int[(c + d*x)^3*(a + b*(c + d*x)^4)^2,x]

[Out]

(a + b*(c + d*x)^4)^3/(12*b*d)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 379

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \left (a+b x^4\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b (c+d x)^4\right )^3}{12 b d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(23)=46\).

Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 7.48 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {1}{12} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \left (3 a^2+3 a b \left (2 c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )+b^2 \left (3 c^8+12 c^7 d x+34 c^6 d^2 x^2+60 c^5 d^3 x^3+71 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8\right )\right ) \]

[In]

Integrate[(c + d*x)^3*(a + b*(c + d*x)^4)^2,x]

[Out]

(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(3*a^2 + 3*a*b*(2*c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3
+ d^4*x^4) + b^2*(3*c^8 + 12*c^7*d*x + 34*c^6*d^2*x^2 + 60*c^5*d^3*x^3 + 71*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 28
*c^2*d^6*x^6 + 8*c*d^7*x^7 + d^8*x^8)))/12

Maple [A] (verified)

Time = 3.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\left (a +b \left (d x +c \right )^{4}\right )^{3}}{12 b d}\)	\(22\)
norman	\(\frac {d^{11} b^{2} x^{12}}{12}+c \,d^{10} b^{2} x^{11}+\frac {11 c^{2} d^{9} b^{2} x^{10}}{2}+\frac {55 c^{3} b^{2} d^{8} x^{9}}{3}+\left (\frac {165}{4} c^{4} b^{2} d^{7}+\frac {1}{4} a b \,d^{7}\right ) x^{8}+\left (66 c^{5} d^{6} b^{2}+2 a b c \,d^{6}\right ) x^{7}+\left (77 c^{6} b^{2} d^{5}+7 a b \,c^{2} d^{5}\right ) x^{6}+\left (66 b^{2} c^{7} d^{4}+14 a b \,c^{3} d^{4}\right ) x^{5}+\left (\frac {165}{4} b^{2} c^{8} d^{3}+\frac {35}{2} a b \,c^{4} d^{3}+\frac {1}{4} a^{2} d^{3}\right ) x^{4}+\left (\frac {55}{3} b^{2} c^{9} d^{2}+14 a b \,c^{5} d^{2}+c \,a^{2} d^{2}\right ) x^{3}+\left (\frac {11}{2} b^{2} c^{10} d +7 a b \,c^{6} d +\frac {3}{2} a^{2} c^{2} d \right ) x^{2}+\left (b^{2} c^{11}+2 a b \,c^{7}+a^{2} c^{3}\right ) x\)	\(274\)
gosper	\(\frac {x \left (d^{11} b^{2} x^{11}+12 c \,d^{10} b^{2} x^{10}+66 c^{2} d^{9} b^{2} x^{9}+220 c^{3} b^{2} d^{8} x^{8}+495 x^{7} c^{4} b^{2} d^{7}+792 b^{2} c^{5} d^{6} x^{6}+924 b^{2} c^{6} d^{5} x^{5}+792 b^{2} c^{7} d^{4} x^{4}+3 x^{7} a b \,d^{7}+495 x^{3} b^{2} c^{8} d^{3}+24 a b c \,d^{6} x^{6}+220 x^{2} b^{2} c^{9} d^{2}+84 a b \,c^{2} d^{5} x^{5}+66 x \,b^{2} c^{10} d +168 a b \,c^{3} d^{4} x^{4}+12 b^{2} c^{11}+210 x^{3} a b \,c^{4} d^{3}+168 x^{2} a b \,c^{5} d^{2}+84 x a b \,c^{6} d +24 a b \,c^{7}+3 a^{2} d^{3} x^{3}+12 a^{2} c \,d^{2} x^{2}+18 a^{2} c^{2} d x +12 a^{2} c^{3}\right )}{12}\)	\(287\)
parallelrisch	\(2 b \,d^{6} a c \,x^{7}+7 b \,d^{5} a \,c^{2} x^{6}+14 b \,d^{4} a \,c^{3} x^{5}+\frac {35}{2} b \,d^{3} a \,c^{4} x^{4}+14 b \,d^{2} a \,c^{5} x^{3}+a^{2} c \,d^{2} x^{3}+\frac {1}{12} d^{11} b^{2} x^{12}+a^{2} c^{3} x +\frac {1}{4} a^{2} d^{3} x^{4}+\frac {165}{4} b^{2} d^{7} c^{4} x^{8}+66 b^{2} d^{6} x^{7} c^{5}+77 b^{2} d^{5} x^{6} c^{6}+66 b^{2} d^{4} c^{7} x^{5}+\frac {1}{4} b \,d^{7} a \,x^{8}+\frac {3}{2} a^{2} c^{2} d \,x^{2}+c \,d^{10} b^{2} x^{11}+\frac {11}{2} c^{2} d^{9} b^{2} x^{10}+\frac {55}{3} c^{3} b^{2} d^{8} x^{9}+\frac {165}{4} b^{2} d^{3} c^{8} x^{4}+\frac {55}{3} b^{2} d^{2} c^{9} x^{3}+\frac {11}{2} b^{2} d \,c^{10} x^{2}+b^{2} x \,c^{11}+7 b d a \,c^{6} x^{2}+2 b a \,c^{7} x\)	\(290\)
risch	\(2 b \,d^{6} a c \,x^{7}+7 b \,d^{5} a \,c^{2} x^{6}+14 b \,d^{4} a \,c^{3} x^{5}+\frac {35 b \,d^{3} a \,c^{4} x^{4}}{2}+14 b \,d^{2} a \,c^{5} x^{3}+a^{2} c \,d^{2} x^{3}+\frac {d^{11} b^{2} x^{12}}{12}+a^{2} c^{3} x +\frac {a^{2} d^{3} x^{4}}{4}+\frac {a^{2} c^{4}}{4 d}+\frac {165 b^{2} d^{7} c^{4} x^{8}}{4}+66 b^{2} d^{6} x^{7} c^{5}+77 b^{2} d^{5} x^{6} c^{6}+66 b^{2} d^{4} c^{7} x^{5}+\frac {b \,d^{7} a \,x^{8}}{4}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+c \,d^{10} b^{2} x^{11}+\frac {11 c^{2} d^{9} b^{2} x^{10}}{2}+\frac {55 c^{3} b^{2} d^{8} x^{9}}{3}+\frac {b^{2} c^{12}}{12 d}+\frac {165 b^{2} d^{3} c^{8} x^{4}}{4}+\frac {55 b^{2} d^{2} c^{9} x^{3}}{3}+\frac {11 b^{2} d \,c^{10} x^{2}}{2}+b^{2} x \,c^{11}+\frac {b a \,c^{8}}{4 d}+7 b d a \,c^{6} x^{2}+2 b a \,c^{7} x +\frac {a^{3}}{12 b d}\)	\(333\)

[In]

int((d*x+c)^3*(a+b*(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/12*(a+b*(d*x+c)^4)^3/b/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 250, normalized size of antiderivative = 10.87 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {1}{12} \, b^{2} d^{11} x^{12} + b^{2} c d^{10} x^{11} + \frac {11}{2} \, b^{2} c^{2} d^{9} x^{10} + \frac {55}{3} \, b^{2} c^{3} d^{8} x^{9} + \frac {1}{4} \, {\left (165 \, b^{2} c^{4} + a b\right )} d^{7} x^{8} + 2 \, {\left (33 \, b^{2} c^{5} + a b c\right )} d^{6} x^{7} + 7 \, {\left (11 \, b^{2} c^{6} + a b c^{2}\right )} d^{5} x^{6} + 2 \, {\left (33 \, b^{2} c^{7} + 7 \, a b c^{3}\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (165 \, b^{2} c^{8} + 70 \, a b c^{4} + a^{2}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (55 \, b^{2} c^{9} + 42 \, a b c^{5} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (11 \, b^{2} c^{10} + 14 \, a b c^{6} + 3 \, a^{2} c^{2}\right )} d x^{2} + {\left (b^{2} c^{11} + 2 \, a b c^{7} + a^{2} c^{3}\right )} x \]

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/12*b^2*d^11*x^12 + b^2*c*d^10*x^11 + 11/2*b^2*c^2*d^9*x^10 + 55/3*b^2*c^3*d^8*x^9 + 1/4*(165*b^2*c^4 + a*b)*
d^7*x^8 + 2*(33*b^2*c^5 + a*b*c)*d^6*x^7 + 7*(11*b^2*c^6 + a*b*c^2)*d^5*x^6 + 2*(33*b^2*c^7 + 7*a*b*c^3)*d^4*x
^5 + 1/4*(165*b^2*c^8 + 70*a*b*c^4 + a^2)*d^3*x^4 + 1/3*(55*b^2*c^9 + 42*a*b*c^5 + 3*a^2*c)*d^2*x^3 + 1/2*(11*
b^2*c^10 + 14*a*b*c^6 + 3*a^2*c^2)*d*x^2 + (b^2*c^11 + 2*a*b*c^7 + a^2*c^3)*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (15) = 30\).

Time = 0.05 (sec) , antiderivative size = 299, normalized size of antiderivative = 13.00 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {55 b^{2} c^{3} d^{8} x^{9}}{3} + \frac {11 b^{2} c^{2} d^{9} x^{10}}{2} + b^{2} c d^{10} x^{11} + \frac {b^{2} d^{11} x^{12}}{12} + x^{8} \left (\frac {a b d^{7}}{4} + \frac {165 b^{2} c^{4} d^{7}}{4}\right ) + x^{7} \cdot \left (2 a b c d^{6} + 66 b^{2} c^{5} d^{6}\right ) + x^{6} \cdot \left (7 a b c^{2} d^{5} + 77 b^{2} c^{6} d^{5}\right ) + x^{5} \cdot \left (14 a b c^{3} d^{4} + 66 b^{2} c^{7} d^{4}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {35 a b c^{4} d^{3}}{2} + \frac {165 b^{2} c^{8} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 14 a b c^{5} d^{2} + \frac {55 b^{2} c^{9} d^{2}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d}{2} + 7 a b c^{6} d + \frac {11 b^{2} c^{10} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{7} + b^{2} c^{11}\right ) \]

[In]

integrate((d*x+c)**3*(a+b*(d*x+c)**4)**2,x)

[Out]

55*b**2*c**3*d**8*x**9/3 + 11*b**2*c**2*d**9*x**10/2 + b**2*c*d**10*x**11 + b**2*d**11*x**12/12 + x**8*(a*b*d*
*7/4 + 165*b**2*c**4*d**7/4) + x**7*(2*a*b*c*d**6 + 66*b**2*c**5*d**6) + x**6*(7*a*b*c**2*d**5 + 77*b**2*c**6*
d**5) + x**5*(14*a*b*c**3*d**4 + 66*b**2*c**7*d**4) + x**4*(a**2*d**3/4 + 35*a*b*c**4*d**3/2 + 165*b**2*c**8*d
**3/4) + x**3*(a**2*c*d**2 + 14*a*b*c**5*d**2 + 55*b**2*c**9*d**2/3) + x**2*(3*a**2*c**2*d/2 + 7*a*b*c**6*d +
11*b**2*c**10*d/2) + x*(a**2*c**3 + 2*a*b*c**7 + b**2*c**11)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \]

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/12*((d*x + c)^4*b + a)^3/(b*d)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=\frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \]

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/12*((d*x + c)^4*b + a)^3/(b*d)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 229, normalized size of antiderivative = 9.96 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx=c^3\,x\,{\left (b\,c^4+a\right )}^2+\frac {b^2\,d^{11}\,x^{12}}{12}+\frac {d^3\,x^4\,\left (a^2+70\,a\,b\,c^4+165\,b^2\,c^8\right )}{4}+b^2\,c\,d^{10}\,x^{11}+\frac {55\,b^2\,c^3\,d^8\,x^9}{3}+\frac {11\,b^2\,c^2\,d^9\,x^{10}}{2}+\frac {b\,d^7\,x^8\,\left (165\,b\,c^4+a\right )}{4}+\frac {c^2\,d\,x^2\,\left (3\,a^2+14\,a\,b\,c^4+11\,b^2\,c^8\right )}{2}+\frac {c\,d^2\,x^3\,\left (3\,a^2+42\,a\,b\,c^4+55\,b^2\,c^8\right )}{3}+2\,b\,c\,d^6\,x^7\,\left (33\,b\,c^4+a\right )+7\,b\,c^2\,d^5\,x^6\,\left (11\,b\,c^4+a\right )+2\,b\,c^3\,d^4\,x^5\,\left (33\,b\,c^4+7\,a\right ) \]

[In]

int((a + b*(c + d*x)^4)^2*(c + d*x)^3,x)

[Out]

c^3*x*(a + b*c^4)^2 + (b^2*d^11*x^12)/12 + (d^3*x^4*(a^2 + 165*b^2*c^8 + 70*a*b*c^4))/4 + b^2*c*d^10*x^11 + (5
5*b^2*c^3*d^8*x^9)/3 + (11*b^2*c^2*d^9*x^10)/2 + (b*d^7*x^8*(a + 165*b*c^4))/4 + (c^2*d*x^2*(3*a^2 + 11*b^2*c^
8 + 14*a*b*c^4))/2 + (c*d^2*x^3*(3*a^2 + 55*b^2*c^8 + 42*a*b*c^4))/3 + 2*b*c*d^6*x^7*(a + 33*b*c^4) + 7*b*c^2*
d^5*x^6*(a + 11*b*c^4) + 2*b*c^3*d^4*x^5*(7*a + 33*b*c^4)