\(\int (a+b x) (1+(c+a x+\frac {b x^2}{2})^4) \, dx\) [207]

Optimal result

Integrand size = 23, antiderivative size = 31 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=a x+\frac {b x^2}{2}+\frac {1}{5} \left (c+a x+\frac {b x^2}{2}\right )^5 \]

[Out]

a*x+1/2*b*x^2+1/5*(c+a*x+1/2*b*x^2)^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1605} \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{5} \left (a x+\frac {b x^2}{2}+c\right )^5+a x+\frac {b x^2}{2} \]

[In]

Int[(a + b*x)*(1 + (c + a*x + (b*x^2)/2)^4),x]

[Out]

a*x + (b*x^2)/2 + (c + a*x + (b*x^2)/2)^5/5

Rule 1605

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1+x^4\right ) \, dx,x,c+a x+\frac {b x^2}{2}\right ) \\ & = a x+\frac {b x^2}{2}+\frac {1}{5} \left (c+a x+\frac {b x^2}{2}\right )^5 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(31)=62\).

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.48 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} x (2 a+b x) \left (80+80 c^4+16 a^4 x^4+32 a^3 b x^5+24 a^2 b^2 x^6+8 a b^3 x^7+b^4 x^8+80 c^3 x (2 a+b x)+40 c^2 x^2 (2 a+b x)^2+10 c x^3 (2 a+b x)^3\right ) \]

[In]

Integrate[(a + b*x)*(1 + (c + a*x + (b*x^2)/2)^4),x]

[Out]

(x*(2*a + b*x)*(80 + 80*c^4 + 16*a^4*x^4 + 32*a^3*b*x^5 + 24*a^2*b^2*x^6 + 8*a*b^3*x^7 + b^4*x^8 + 80*c^3*x*(2
*a + b*x) + 40*c^2*x^2*(2*a + b*x)^2 + 10*c*x^3*(2*a + b*x)^3))/160

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {\left (c +a x +\frac {1}{2} b \,x^{2}\right )^{5}}{5}+c +a x +\frac {b \,x^{2}}{2}\)	\(27\)
norman	\(\left (\frac {1}{4} a^{2} b^{3}+\frac {1}{16} b^{4} c \right ) x^{8}+\left (\frac {1}{2} a^{3} b^{2}+\frac {1}{2} a \,b^{3} c \right ) x^{7}+\left (\frac {1}{5} a^{5}+2 a^{3} b c +\frac {3}{2} b^{2} a \,c^{2}\right ) x^{5}+\left (2 a^{2} c^{3}+\frac {1}{2} b \,c^{4}+\frac {1}{2} b \right ) x^{2}+\left (\frac {1}{2} b \,a^{4}+\frac {3}{2} a^{2} b^{2} c +\frac {1}{4} b^{3} c^{2}\right ) x^{6}+\left (c \,a^{4}+3 a^{2} b \,c^{2}+\frac {1}{2} b^{2} c^{3}\right ) x^{4}+\left (a \,c^{4}+a \right ) x +\left (2 a^{3} c^{2}+2 a b \,c^{3}\right ) x^{3}+\frac {x^{10} b^{5}}{160}+\frac {b^{4} a \,x^{9}}{16}\)	\(190\)
gosper	\(\frac {x \left (x^{9} b^{5}+10 a \,x^{8} b^{4}+40 a^{2} b^{3} x^{7}+10 b^{4} c \,x^{7}+80 a^{3} b^{2} x^{6}+80 a \,b^{3} c \,x^{6}+80 a^{4} b \,x^{5}+240 a^{2} b^{2} c \,x^{5}+40 b^{3} c^{2} x^{5}+32 a^{5} x^{4}+320 x^{4} a^{3} b c +240 x^{4} b^{2} a \,c^{2}+160 a^{4} c \,x^{3}+480 a^{2} b \,c^{2} x^{3}+80 b^{2} c^{3} x^{3}+320 a^{3} c^{2} x^{2}+320 a b \,c^{3} x^{2}+320 a^{2} c^{3} x +80 b \,c^{4} x +160 a \,c^{4}+80 b x +160 a \right )}{160}\)	\(206\)
risch	\(\frac {1}{160} x^{10} b^{5}+\frac {1}{16} b^{4} a \,x^{9}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{16} x^{8} b^{4} c +\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} x^{7} a \,b^{3} c +\frac {1}{2} a^{4} b \,x^{6}+\frac {3}{2} x^{6} a^{2} b^{2} c +\frac {1}{4} x^{6} b^{3} c^{2}+\frac {1}{5} a^{5} x^{5}+2 x^{5} a^{3} b c +\frac {3}{2} x^{5} b^{2} a \,c^{2}+x^{4} c \,a^{4}+3 x^{4} a^{2} b \,c^{2}+\frac {1}{2} b^{2} c^{3} x^{4}+2 a^{3} c^{2} x^{3}+2 a b \,c^{3} x^{3}+2 a^{2} c^{3} x^{2}+\frac {1}{2} x^{2} b \,c^{4}+\frac {1}{2} b \,x^{2}+a \,c^{4} x +a x\)	\(209\)
parallelrisch	\(\frac {1}{160} x^{10} b^{5}+\frac {1}{16} b^{4} a \,x^{9}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{16} x^{8} b^{4} c +\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} x^{7} a \,b^{3} c +\frac {1}{2} a^{4} b \,x^{6}+\frac {3}{2} x^{6} a^{2} b^{2} c +\frac {1}{4} x^{6} b^{3} c^{2}+\frac {1}{5} a^{5} x^{5}+2 x^{5} a^{3} b c +\frac {3}{2} x^{5} b^{2} a \,c^{2}+x^{4} c \,a^{4}+3 x^{4} a^{2} b \,c^{2}+\frac {1}{2} b^{2} c^{3} x^{4}+2 a^{3} c^{2} x^{3}+2 a b \,c^{3} x^{3}+2 a^{2} c^{3} x^{2}+\frac {1}{2} x^{2} b \,c^{4}+\frac {1}{2} b \,x^{2}+a \,c^{4} x +a x\)	\(209\)

[In]

int((b*x+a)*(1+(c+a*x+1/2*b*x^2)^4),x,method=_RETURNVERBOSE)

[Out]

1/5*(c+a*x+1/2*b*x^2)^5+c+a*x+1/2*b*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 6.03 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{16} \, {\left (4 \, a^{2} b^{3} + b^{4} c\right )} x^{8} + \frac {1}{2} \, {\left (a^{3} b^{2} + a b^{3} c\right )} x^{7} + \frac {1}{4} \, {\left (2 \, a^{4} b + 6 \, a^{2} b^{2} c + b^{3} c^{2}\right )} x^{6} + \frac {1}{10} \, {\left (2 \, a^{5} + 20 \, a^{3} b c + 15 \, a b^{2} c^{2}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a^{4} c + 6 \, a^{2} b c^{2} + b^{2} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} c^{2} + a b c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{2} c^{3} + b c^{4} + b\right )} x^{2} + {\left (a c^{4} + a\right )} x \]

[In]

integrate((b*x+a)*(1+(c+a*x+1/2*b*x^2)^4),x, algorithm="fricas")

[Out]

1/160*b^5*x^10 + 1/16*a*b^4*x^9 + 1/16*(4*a^2*b^3 + b^4*c)*x^8 + 1/2*(a^3*b^2 + a*b^3*c)*x^7 + 1/4*(2*a^4*b +
6*a^2*b^2*c + b^3*c^2)*x^6 + 1/10*(2*a^5 + 20*a^3*b*c + 15*a*b^2*c^2)*x^5 + 1/2*(2*a^4*c + 6*a^2*b*c^2 + b^2*c
^3)*x^4 + 2*(a^3*c^2 + a*b*c^3)*x^3 + 1/2*(4*a^2*c^3 + b*c^4 + b)*x^2 + (a*c^4 + a)*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (24) = 48\).

Time = 0.05 (sec) , antiderivative size = 194, normalized size of antiderivative = 6.26 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {a b^{4} x^{9}}{16} + \frac {b^{5} x^{10}}{160} + x^{8} \left (\frac {a^{2} b^{3}}{4} + \frac {b^{4} c}{16}\right ) + x^{7} \left (\frac {a^{3} b^{2}}{2} + \frac {a b^{3} c}{2}\right ) + x^{6} \left (\frac {a^{4} b}{2} + \frac {3 a^{2} b^{2} c}{2} + \frac {b^{3} c^{2}}{4}\right ) + x^{5} \left (\frac {a^{5}}{5} + 2 a^{3} b c + \frac {3 a b^{2} c^{2}}{2}\right ) + x^{4} \left (a^{4} c + 3 a^{2} b c^{2} + \frac {b^{2} c^{3}}{2}\right ) + x^{3} \cdot \left (2 a^{3} c^{2} + 2 a b c^{3}\right ) + x^{2} \cdot \left (2 a^{2} c^{3} + \frac {b c^{4}}{2} + \frac {b}{2}\right ) + x \left (a c^{4} + a\right ) \]

[In]

integrate((b*x+a)*(1+(c+a*x+1/2*b*x**2)**4),x)

[Out]

a*b**4*x**9/16 + b**5*x**10/160 + x**8*(a**2*b**3/4 + b**4*c/16) + x**7*(a**3*b**2/2 + a*b**3*c/2) + x**6*(a**
4*b/2 + 3*a**2*b**2*c/2 + b**3*c**2/4) + x**5*(a**5/5 + 2*a**3*b*c + 3*a*b**2*c**2/2) + x**4*(a**4*c + 3*a**2*
b*c**2 + b**2*c**3/2) + x**3*(2*a**3*c**2 + 2*a*b*c**3) + x**2*(2*a**2*c**3 + b*c**4/2 + b/2) + x*(a*c**4 + a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (27) = 54\).

Time = 0.19 (sec) , antiderivative size = 187, normalized size of antiderivative = 6.03 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{16} \, {\left (4 \, a^{2} b^{3} + b^{4} c\right )} x^{8} + \frac {1}{2} \, {\left (a^{3} b^{2} + a b^{3} c\right )} x^{7} + \frac {1}{4} \, {\left (2 \, a^{4} b + 6 \, a^{2} b^{2} c + b^{3} c^{2}\right )} x^{6} + \frac {1}{10} \, {\left (2 \, a^{5} + 20 \, a^{3} b c + 15 \, a b^{2} c^{2}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a^{4} c + 6 \, a^{2} b c^{2} + b^{2} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} c^{2} + a b c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{2} c^{3} + b c^{4} + b\right )} x^{2} + {\left (a c^{4} + a\right )} x \]

[In]

integrate((b*x+a)*(1+(c+a*x+1/2*b*x^2)^4),x, algorithm="maxima")

[Out]

1/160*b^5*x^10 + 1/16*a*b^4*x^9 + 1/16*(4*a^2*b^3 + b^4*c)*x^8 + 1/2*(a^3*b^2 + a*b^3*c)*x^7 + 1/4*(2*a^4*b +
6*a^2*b^2*c + b^3*c^2)*x^6 + 1/10*(2*a^5 + 20*a^3*b*c + 15*a*b^2*c^2)*x^5 + 1/2*(2*a^4*c + 6*a^2*b*c^2 + b^2*c
^3)*x^4 + 2*(a^3*c^2 + a*b*c^3)*x^3 + 1/2*(4*a^2*c^3 + b*c^4 + b)*x^2 + (a*c^4 + a)*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, {\left (b x^{2} + 2 \, a x\right )}^{5} + \frac {1}{16} \, {\left (b x^{2} + 2 \, a x\right )}^{4} c + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{3} c^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )}^{2} c^{3} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{4} + \frac {1}{2} \, b x^{2} + a x \]

[In]

integrate((b*x+a)*(1+(c+a*x+1/2*b*x^2)^4),x, algorithm="giac")

[Out]

1/160*(b*x^2 + 2*a*x)^5 + 1/16*(b*x^2 + 2*a*x)^4*c + 1/4*(b*x^2 + 2*a*x)^3*c^2 + 1/2*(b*x^2 + 2*a*x)^2*c^3 + 1
/2*(b*x^2 + 2*a*x)*c^4 + 1/2*b*x^2 + a*x

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.81 \[ \int (a+b x) \left (1+\left (c+a x+\frac {b x^2}{2}\right )^4\right ) \, dx=x^6\,\left (\frac {a^4\,b}{2}+\frac {3\,a^2\,b^2\,c}{2}+\frac {b^3\,c^2}{4}\right )+x^4\,\left (a^4\,c+3\,a^2\,b\,c^2+\frac {b^2\,c^3}{2}\right )+x^2\,\left (2\,a^2\,c^3+\frac {b\,c^4}{2}+\frac {b}{2}\right )+x^5\,\left (\frac {a^5}{5}+2\,a^3\,b\,c+\frac {3\,a\,b^2\,c^2}{2}\right )+\frac {b^5\,x^{10}}{160}+x^8\,\left (\frac {a^2\,b^3}{4}+\frac {c\,b^4}{16}\right )+\frac {a\,b^4\,x^9}{16}+a\,x\,\left (c^4+1\right )+\frac {a\,b^2\,x^7\,\left (a^2+b\,c\right )}{2}+2\,a\,c^2\,x^3\,\left (a^2+b\,c\right ) \]

[In]

int(((c + a*x + (b*x^2)/2)^4 + 1)*(a + b*x),x)

[Out]

x^6*((a^4*b)/2 + (b^3*c^2)/4 + (3*a^2*b^2*c)/2) + x^4*(a^4*c + (b^2*c^3)/2 + 3*a^2*b*c^2) + x^2*(b/2 + (b*c^4)
/2 + 2*a^2*c^3) + x^5*(a^5/5 + (3*a*b^2*c^2)/2 + 2*a^3*b*c) + (b^5*x^10)/160 + x^8*((b^4*c)/16 + (a^2*b^3)/4)
+ (a*b^4*x^9)/16 + a*x*(c^4 + 1) + (a*b^2*x^7*(b*c + a^2))/2 + 2*a*c^2*x^3*(b*c + a^2)