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

Optimal. Leaf size=28 \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0270042, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1591} \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[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 1591

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*} \int (a+b x) \left (1+\left (a x+\frac{b x^2}{2}\right )^4\right ) \, dx &=\operatorname{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]  time = 0.0051486, size = 80, normalized size = 2.86 \[ \frac{1}{4} a^2 b^3 x^8+\frac{1}{2} a^3 b^2 x^7+\frac{1}{2} a^4 b x^6+\frac{a^5 x^5}{5}+\frac{1}{16} a b^4 x^9+a x+\frac{b^5 x^{10}}{160}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.002, size = 67, normalized size = 2.4 \begin{align*}{\frac{{b}^{5}{x}^{10}}{160}}+{\frac{a{b}^{4}{x}^{9}}{16}}+{\frac{{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{{a}^{3}{b}^{2}{x}^{7}}{2}}+{\frac{{a}^{4}b{x}^{6}}{2}}+{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{b{x}^{2}}{2}}+ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [B]  time = 1.00854, size = 89, normalized size = 3.18 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.11006, size = 158, normalized size = 5.64 \begin{align*} \frac{1}{160} x^{10} b^{5} + \frac{1}{16} x^{9} b^{4} a + \frac{1}{4} x^{8} b^{3} a^{2} + \frac{1}{2} x^{7} b^{2} a^{3} + \frac{1}{2} x^{6} b a^{4} + \frac{1}{5} x^{5} a^{5} + \frac{1}{2} x^{2} b + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.080792, size = 70, normalized size = 2.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.23564, size = 89, normalized size = 3.18 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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