\(\int (c+d x)^3 (a+b (c+d x)^3) \, dx\) [2853]

Optimal result

Integrand size = 19, antiderivative size = 31 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^7}{7 d} \]

[Out]

1/4*a*(d*x+c)^4/d+1/7*b*(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {379, 14} \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^7}{7 d} \]

[In]

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

[Out]

(a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^7)/(7*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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^3\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (a x^3+b x^6\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^7}{7 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(31)=62\).

Time = 0.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.16 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=c^3 \left (a+b c^3\right ) x+\frac {3}{2} c^2 \left (a+2 b c^3\right ) d x^2+c \left (a+5 b c^3\right ) d^2 x^3+\frac {1}{4} \left (a+20 b c^3\right ) d^3 x^4+3 b c^2 d^4 x^5+b c d^5 x^6+\frac {1}{7} b d^6 x^7 \]

[In]

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

[Out]

c^3*(a + b*c^3)*x + (3*c^2*(a + 2*b*c^3)*d*x^2)/2 + c*(a + 5*b*c^3)*d^2*x^3 + ((a + 20*b*c^3)*d^3*x^4)/4 + 3*b
*c^2*d^4*x^5 + b*c*d^5*x^6 + (b*d^6*x^7)/7

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(27)=54\).

Time = 3.86 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35

method	result	size
norman	\(\frac {d^{6} b \,x^{7}}{7}+c \,d^{5} b \,x^{6}+3 c^{2} d^{4} b \,x^{5}+\left (5 c^{3} b \,d^{3}+\frac {1}{4} a \,d^{3}\right ) x^{4}+\left (5 c^{4} b \,d^{2}+a c \,d^{2}\right ) x^{3}+\left (3 c^{5} b d +\frac {3}{2} a \,c^{2} d \right ) x^{2}+\left (b \,c^{6}+c^{3} a \right ) x\)	\(104\)
risch	\(\frac {1}{7} d^{6} b \,x^{7}+c \,d^{5} b \,x^{6}+3 c^{2} d^{4} b \,x^{5}+5 x^{4} c^{3} b \,d^{3}+\frac {1}{4} d^{3} a \,x^{4}+5 b \,c^{4} d^{2} x^{3}+d^{2} x^{3} c a +3 x^{2} c^{5} b d +\frac {3}{2} d a \,c^{2} x^{2}+b \,c^{6} x +a \,c^{3} x\)	\(106\)
parallelrisch	\(\frac {1}{7} d^{6} b \,x^{7}+c \,d^{5} b \,x^{6}+3 c^{2} d^{4} b \,x^{5}+5 x^{4} c^{3} b \,d^{3}+\frac {1}{4} d^{3} a \,x^{4}+5 b \,c^{4} d^{2} x^{3}+d^{2} x^{3} c a +3 x^{2} c^{5} b d +\frac {3}{2} d a \,c^{2} x^{2}+b \,c^{6} x +a \,c^{3} x\)	\(106\)
gosper	\(\frac {x \left (4 d^{6} b \,x^{6}+28 c \,d^{5} b \,x^{5}+84 c^{2} d^{4} b \,x^{4}+140 x^{3} c^{3} b \,d^{3}+140 b \,c^{4} d^{2} x^{2}+84 x \,c^{5} b d +7 x^{3} a \,d^{3}+28 b \,c^{6}+28 a c \,d^{2} x^{2}+42 x a \,c^{2} d +28 c^{3} a \right )}{28}\)	\(107\)
default	\(\frac {d^{6} b \,x^{7}}{7}+c \,d^{5} b \,x^{6}+3 c^{2} d^{4} b \,x^{5}+\frac {\left (19 c^{3} b \,d^{3}+d^{3} \left (c^{3} b +a \right )\right ) x^{4}}{4}+\frac {\left (12 c^{4} b \,d^{2}+3 c \,d^{2} \left (c^{3} b +a \right )\right ) x^{3}}{3}+\frac {\left (3 c^{5} b d +3 c^{2} d \left (c^{3} b +a \right )\right ) x^{2}}{2}+c^{3} \left (c^{3} b +a \right ) x\)	\(124\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=\frac {1}{7} \, b d^{6} x^{7} + b c d^{5} x^{6} + 3 \, b c^{2} d^{4} x^{5} + \frac {1}{4} \, {\left (20 \, b c^{3} + a\right )} d^{3} x^{4} + {\left (5 \, b c^{4} + a c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (2 \, b c^{5} + a c^{2}\right )} d x^{2} + {\left (b c^{6} + a c^{3}\right )} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (22) = 44\).

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.45 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=3 b c^{2} d^{4} x^{5} + b c d^{5} x^{6} + \frac {b d^{6} x^{7}}{7} + x^{4} \left (\frac {a d^{3}}{4} + 5 b c^{3} d^{3}\right ) + x^{3} \left (a c d^{2} + 5 b c^{4} d^{2}\right ) + x^{2} \cdot \left (\frac {3 a c^{2} d}{2} + 3 b c^{5} d\right ) + x \left (a c^{3} + b c^{6}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (27) = 54\).

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=\frac {1}{7} \, b d^{6} x^{7} + b c d^{5} x^{6} + 3 \, b c^{2} d^{4} x^{5} + \frac {1}{4} \, {\left (20 \, b c^{3} + a\right )} d^{3} x^{4} + {\left (5 \, b c^{4} + a c\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (2 \, b c^{5} + a c^{2}\right )} d x^{2} + {\left (b c^{6} + a c^{3}\right )} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (27) = 54\).

Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.39 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=\frac {1}{7} \, b d^{6} x^{7} + b c d^{5} x^{6} + 3 \, b c^{2} d^{4} x^{5} + 5 \, b c^{3} d^{3} x^{4} + 5 \, b c^{4} d^{2} x^{3} + 3 \, b c^{5} d x^{2} + b c^{6} x + \frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int (c+d x)^3 \left (a+b (c+d x)^3\right ) \, dx=x\,\left (b\,c^6+a\,c^3\right )+\frac {d^3\,x^4\,\left (20\,b\,c^3+a\right )}{4}+\frac {b\,d^6\,x^7}{7}+3\,b\,c^2\,d^4\,x^5+\frac {3\,c^2\,d\,x^2\,\left (2\,b\,c^3+a\right )}{2}+c\,d^2\,x^3\,\left (5\,b\,c^3+a\right )+b\,c\,d^5\,x^6 \]

[In]

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

[Out]

x*(a*c^3 + b*c^6) + (d^3*x^4*(a + 20*b*c^3))/4 + (b*d^6*x^7)/7 + 3*b*c^2*d^4*x^5 + (3*c^2*d*x^2*(a + 2*b*c^3))
/2 + c*d^2*x^3*(a + 5*b*c^3) + b*c*d^5*x^6