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

Optimal. Leaf size=31 \[ \frac{1}{5} \left (a x+\frac{b x^2}{2}+c\right )^5+a x+\frac{b x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.0311218, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1591} \[ \frac{1}{5} \left (a x+\frac{b x^2}{2}+c\right )^5+a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[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 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 (c+a x+\frac{b x^2}{2}\right )^4\right ) \, dx &=\operatorname{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]  time = 0.03429, size = 108, normalized size = 3.48 \[ \frac{1}{160} x (2 a+b x) \left (24 a^2 b^2 x^6+32 a^3 b x^5+16 a^4 x^4+8 a b^3 x^7+40 c^2 x^2 (2 a+b x)^2+80 c^3 x (2 a+b x)+10 c x^3 (2 a+b x)^3+b^4 x^8+80 c^4+80\right ) \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.002, size = 325, normalized size = 10.5 \begin{align*}{\frac{{b}^{5}{x}^{10}}{160}}+{\frac{a{b}^{4}{x}^{9}}{16}}+{\frac{{x}^{8}}{8} \left ({\frac{{a}^{2}{b}^{3}}{2}}+b \left ({\frac{ \left ({a}^{2}+bc \right ){b}^{2}}{2}}+{b}^{2}{a}^{2} \right ) \right ) }+{\frac{{x}^{7}}{7} \left ( a \left ({\frac{ \left ({a}^{2}+bc \right ){b}^{2}}{2}}+{b}^{2}{a}^{2} \right ) +b \left ( a{b}^{2}c+2\, \left ({a}^{2}+bc \right ) ab \right ) \right ) }+{\frac{{x}^{6}}{6} \left ( a \left ( a{b}^{2}c+2\, \left ({a}^{2}+bc \right ) ab \right ) +b \left ({\frac{{c}^{2}{b}^{2}}{2}}+4\,{a}^{2}bc+ \left ({a}^{2}+bc \right ) ^{2} \right ) \right ) }+{\frac{{x}^{5}}{5} \left ( a \left ({\frac{{c}^{2}{b}^{2}}{2}}+4\,{a}^{2}bc+ \left ({a}^{2}+bc \right ) ^{2} \right ) +b \left ( 2\,{c}^{2}ab+4\,ac \left ({a}^{2}+bc \right ) \right ) \right ) }+{\frac{ \left ( a \left ( 2\,{c}^{2}ab+4\,ac \left ({a}^{2}+bc \right ) \right ) +b \left ( 2\,{c}^{2} \left ({a}^{2}+bc \right ) +4\,{a}^{2}{c}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( a \left ( 2\,{c}^{2} \left ({a}^{2}+bc \right ) +4\,{a}^{2}{c}^{2} \right ) +4\,ab{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{2}{c}^{3}+b \left ({c}^{4}+1 \right ) \right ){x}^{2}}{2}}+a \left ({c}^{4}+1 \right ) x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.02107, size = 252, normalized size = 8.13 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.13951, size = 470, normalized size = 15.16 \begin{align*} \frac{1}{160} x^{10} b^{5} + \frac{1}{16} x^{9} b^{4} a + \frac{1}{16} x^{8} c b^{4} + \frac{1}{4} x^{8} b^{3} a^{2} + \frac{1}{2} x^{7} c b^{3} a + \frac{1}{2} x^{7} b^{2} a^{3} + \frac{1}{4} x^{6} c^{2} b^{3} + \frac{3}{2} x^{6} c b^{2} a^{2} + \frac{1}{2} x^{6} b a^{4} + \frac{3}{2} x^{5} c^{2} b^{2} a + 2 x^{5} c b a^{3} + \frac{1}{5} x^{5} a^{5} + \frac{1}{2} x^{4} c^{3} b^{2} + 3 x^{4} c^{2} b a^{2} + x^{4} c a^{4} + 2 x^{3} c^{3} b a + 2 x^{3} c^{2} a^{3} + \frac{1}{2} x^{2} c^{4} b + 2 x^{2} c^{3} a^{2} + x c^{4} a + \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+(c+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 + 1/16*x^8*c*b^4 + 1/4*x^8*b^3*a^2 + 1/2*x^7*c*b^3*a + 1/2*x^7*b^2*a^3 + 1/4*x
^6*c^2*b^3 + 3/2*x^6*c*b^2*a^2 + 1/2*x^6*b*a^4 + 3/2*x^5*c^2*b^2*a + 2*x^5*c*b*a^3 + 1/5*x^5*a^5 + 1/2*x^4*c^3
*b^2 + 3*x^4*c^2*b*a^2 + x^4*c*a^4 + 2*x^3*c^3*b*a + 2*x^3*c^2*a^3 + 1/2*x^2*c^4*b + 2*x^2*c^3*a^2 + x*c^4*a +
1/2*x^2*b + x*a

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Sympy [B]  time = 0.107408, size = 194, normalized size = 6.26 \begin{align*} \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} \left (2 a^{3} c^{2} + 2 a b c^{3}\right ) + x^{2} \left (2 a^{2} c^{3} + \frac{b c^{4}}{2} + \frac{b}{2}\right ) + x \left (a c^{4} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.17276, size = 281, normalized size = 9.06 \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}{16} \, b^{4} c x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a b^{3} c x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{3}{2} \, a^{2} b^{2} c x^{6} + \frac{1}{4} \, b^{3} c^{2} x^{6} + \frac{1}{5} \, a^{5} x^{5} + 2 \, a^{3} b c x^{5} + \frac{3}{2} \, a b^{2} c^{2} x^{5} + a^{4} c x^{4} + 3 \, a^{2} b c^{2} x^{4} + \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} \, b c^{4} x^{2} + a c^{4} x + \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+(c+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 + 1/16*b^4*c*x^8 + 1/2*a^3*b^2*x^7 + 1/2*a*b^3*c*x^7 + 1/2*a
^4*b*x^6 + 3/2*a^2*b^2*c*x^6 + 1/4*b^3*c^2*x^6 + 1/5*a^5*x^5 + 2*a^3*b*c*x^5 + 3/2*a*b^2*c^2*x^5 + a^4*c*x^4 +
3*a^2*b*c^2*x^4 + 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 + 1/2*b*c^4*x^2 + a*c^4*x +
1/2*b*x^2 + a*x