\(\int (a+b x) (1+(a x+\frac {b x^2}{2})^4) \, dx\) [206]

Optimal result

Integrand size = 22, antiderivative size = 28 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=a x+\frac {b x^2}{2}+\frac {1}{160} x^5 (2 a+b x)^5 \]

[Out]

a*x+1/2*b*x^2+1/160*x^5*(b*x+2*a)^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1605} \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} x^5 (2 a+b x)^5+a x+\frac {b x^2}{2} \]

[In]

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

[Out]

a*x + (b*x^2)/2 + (x^5*(2*a + b*x)^5)/160

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,a x+\frac {b x^2}{2}\right ) \\ & = a x+\frac {b x^2}{2}+\frac {1}{160} x^5 (2 a+b x)^5 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(28)=56\).

Time = 0.00 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=a x+\frac {b x^2}{2}+\frac {a^5 x^5}{5}+\frac {1}{2} a^4 b x^6+\frac {1}{2} a^3 b^2 x^7+\frac {1}{4} a^2 b^3 x^8+\frac {1}{16} a b^4 x^9+\frac {b^5 x^{10}}{160} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {\left (a x +\frac {1}{2} b \,x^{2}\right )^{5}}{5}+a x +\frac {b \,x^{2}}{2}\)	\(25\)
gosper	\(\frac {x \left (x^{9} b^{5}+10 a \,x^{8} b^{4}+40 a^{2} b^{3} x^{7}+80 a^{3} b^{2} x^{6}+80 a^{4} b \,x^{5}+32 a^{5} x^{4}+80 b x +160 a \right )}{160}\)	\(67\)
norman	\(a x +\frac {1}{5} a^{5} x^{5}+\frac {1}{2} b \,x^{2}+\frac {1}{160} x^{10} b^{5}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} a^{4} b \,x^{6}+\frac {1}{16} b^{4} a \,x^{9}\)	\(67\)
risch	\(a x +\frac {1}{5} a^{5} x^{5}+\frac {1}{2} b \,x^{2}+\frac {1}{160} x^{10} b^{5}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} a^{4} b \,x^{6}+\frac {1}{16} b^{4} a \,x^{9}\)	\(67\)
parallelrisch	\(a x +\frac {1}{5} a^{5} x^{5}+\frac {1}{2} b \,x^{2}+\frac {1}{160} x^{10} b^{5}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} a^{4} b \,x^{6}+\frac {1}{16} b^{4} a \,x^{9}\)	\(67\)

[In]

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

[Out]

1/5*(a*x+1/2*b*x^2)^5+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. 66 vs. \(2 (24) = 48\).

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} x^{8} + \frac {1}{2} \, a^{3} b^{2} x^{7} + \frac {1}{2} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} + \frac {1}{2} \, b x^{2} + a x \]

[In]

integrate((b*x+a)*(1+(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 + 1/4*a^2*b^3*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a^4*b*x^6 + 1/5*a^5*x^5 + 1/2*b*x^2
+ a*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {a^{5} x^{5}}{5} + \frac {a^{4} b x^{6}}{2} + \frac {a^{3} b^{2} x^{7}}{2} + \frac {a^{2} b^{3} x^{8}}{4} + \frac {a b^{4} x^{9}}{16} + a x + \frac {b^{5} x^{10}}{160} + \frac {b x^{2}}{2} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} x^{8} + \frac {1}{2} \, a^{3} b^{2} x^{7} + \frac {1}{2} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} + \frac {1}{2} \, b x^{2} + a x \]

[In]

integrate((b*x+a)*(1+(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 + 1/4*a^2*b^3*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a^4*b*x^6 + 1/5*a^5*x^5 + 1/2*b*x^2
+ a*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {1}{160} \, {\left (b x^{2} + 2 \, a x\right )}^{5} + \frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

1/160*(b*x^2 + 2*a*x)^5 + 1/2*b*x^2 + a*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int (a+b x) \left (1+\left (a x+\frac {b x^2}{2}\right )^4\right ) \, dx=\frac {a^5\,x^5}{5}+\frac {a^4\,b\,x^6}{2}+\frac {a^3\,b^2\,x^7}{2}+\frac {a^2\,b^3\,x^8}{4}+\frac {a\,b^4\,x^9}{16}+a\,x+\frac {b^5\,x^{10}}{160}+\frac {b\,x^2}{2} \]

[In]

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

[Out]

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