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

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.04, size = 108, normalized size = 3.48 \[ \frac {1}{160} x (2 a+b x) \left (16 a^4 x^4+32 a^3 b x^5+24 a^2 b^2 x^6+8 a b^3 x^7+80 c^3 x (2 a+b x)+40 c^2 x^2 (2 a+b x)^2+10 c x^3 (2 a+b x)^3+b^4 x^8+80 c^4+80\right ) \]

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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fricas [B]  time = 0.55, size = 208, normalized size = 6.71 \[ \frac {1}{160} x^{10} b^{5} + \frac {1}{16} x^{9} b^{4} a + \frac {1}{16} x^{8} c b^{4} + \frac {1}{4} x^{8} b^{3} a^{2} + \frac {1}{2} x^{7} c b^{3} a + \frac {1}{2} x^{7} b^{2} a^{3} + \frac {1}{4} x^{6} c^{2} b^{3} + \frac {3}{2} x^{6} c b^{2} a^{2} + \frac {1}{2} x^{6} b a^{4} + \frac {3}{2} x^{5} c^{2} b^{2} a + 2 x^{5} c b a^{3} + \frac {1}{5} x^{5} a^{5} + \frac {1}{2} x^{4} c^{3} b^{2} + 3 x^{4} c^{2} b a^{2} + x^{4} c a^{4} + 2 x^{3} c^{3} b a + 2 x^{3} c^{2} a^{3} + \frac {1}{2} x^{2} c^{4} b + 2 x^{2} c^{3} a^{2} + x c^{4} a + \frac {1}{2} x^{2} b + x a \]

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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giac [B]  time = 0.37, size = 88, normalized size = 2.84 \[ \frac {1}{160} \, {\left (b x^{2} + 2 \, a x\right )}^{5} + \frac {1}{16} \, {\left (b x^{2} + 2 \, a x\right )}^{4} c + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{3} c^{2} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )}^{2} c^{3} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{4} + \frac {1}{2} \, b x^{2} + a x \]

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*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

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maple [B]  time = 0.00, size = 325, normalized size = 10.48 \[ \frac {b^{5} x^{10}}{160}+\frac {a \,b^{4} x^{9}}{16}+\frac {\left (\frac {a^{2} b^{3}}{2}+\left (a^{2} b^{2}+\frac {\left (a^{2}+b c \right ) b^{2}}{2}\right ) b \right ) x^{8}}{8}+\frac {\left (\left (a^{2} b^{2}+\frac {\left (a^{2}+b c \right ) b^{2}}{2}\right ) a +\left (a \,b^{2} c +2 \left (a^{2}+b c \right ) a b \right ) b \right ) x^{7}}{7}+\frac {\left (\left (a \,b^{2} c +2 \left (a^{2}+b c \right ) a b \right ) a +\left (4 a^{2} b c +\frac {b^{2} c^{2}}{2}+\left (a^{2}+b c \right )^{2}\right ) b \right ) x^{6}}{6}+\frac {\left (\left (4 a^{2} b c +\frac {b^{2} c^{2}}{2}+\left (a^{2}+b c \right )^{2}\right ) a +\left (2 a b \,c^{2}+4 \left (a^{2}+b c \right ) a c \right ) b \right ) x^{5}}{5}+\frac {\left (\left (2 a b \,c^{2}+4 \left (a^{2}+b c \right ) a c \right ) a +\left (4 a^{2} c^{2}+2 \left (a^{2}+b c \right ) c^{2}\right ) b \right ) x^{4}}{4}+\frac {\left (4 a b \,c^{3}+\left (4 a^{2} c^{2}+2 \left (a^{2}+b c \right ) c^{2}\right ) a \right ) x^{3}}{3}+\left (c^{4}+1\right ) a x +\frac {\left (4 a^{2} c^{3}+\left (c^{4}+1\right ) b \right ) x^{2}}{2} \]

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+a^2*b^2))*x^8+1/7*(a*(1/2*(a^2+b*c)*b^2+a^
2*b^2)+b*(a*c*b^2+2*(a^2+b*c)*a*b))*x^7+1/6*(a*(a*c*b^2+2*(a^2+b*c)*a*b)+b*(1/2*c^2*b^2+4*a^2*c*b+(a^2+b*c)^2)
)*x^6+1/5*(a*(1/2*c^2*b^2+4*a^2*c*b+(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 = 0.44, size = 187, normalized size = 6.03 \[ \frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{16} \, {\left (4 \, a^{2} b^{3} + b^{4} c\right )} x^{8} + \frac {1}{2} \, {\left (a^{3} b^{2} + a b^{3} c\right )} x^{7} + \frac {1}{4} \, {\left (2 \, a^{4} b + 6 \, a^{2} b^{2} c + b^{3} c^{2}\right )} x^{6} + \frac {1}{10} \, {\left (2 \, a^{5} + 20 \, a^{3} b c + 15 \, a b^{2} c^{2}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a^{4} c + 6 \, a^{2} b c^{2} + b^{2} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} c^{2} + a b c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{2} c^{3} + b c^{4} + b\right )} x^{2} + {\left (a c^{4} + a\right )} x \]

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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mupad [B]  time = 0.10, size = 180, normalized size = 5.81 \[ x^6\,\left (\frac {a^4\,b}{2}+\frac {3\,a^2\,b^2\,c}{2}+\frac {b^3\,c^2}{4}\right )+x^4\,\left (a^4\,c+3\,a^2\,b\,c^2+\frac {b^2\,c^3}{2}\right )+x^2\,\left (2\,a^2\,c^3+\frac {b\,c^4}{2}+\frac {b}{2}\right )+x^5\,\left (\frac {a^5}{5}+2\,a^3\,b\,c+\frac {3\,a\,b^2\,c^2}{2}\right )+\frac {b^5\,x^{10}}{160}+x^8\,\left (\frac {a^2\,b^3}{4}+\frac {c\,b^4}{16}\right )+\frac {a\,b^4\,x^9}{16}+a\,x\,\left (c^4+1\right )+\frac {a\,b^2\,x^7\,\left (a^2+b\,c\right )}{2}+2\,a\,c^2\,x^3\,\left (a^2+b\,c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.11, size = 194, normalized size = 6.26 \[ \frac {a b^{4} x^{9}}{16} + \frac {b^{5} x^{10}}{160} + x^{8} \left (\frac {a^{2} b^{3}}{4} + \frac {b^{4} c}{16}\right ) + x^{7} \left (\frac {a^{3} b^{2}}{2} + \frac {a b^{3} c}{2}\right ) + x^{6} \left (\frac {a^{4} b}{2} + \frac {3 a^{2} b^{2} c}{2} + \frac {b^{3} c^{2}}{4}\right ) + x^{5} \left (\frac {a^{5}}{5} + 2 a^{3} b c + \frac {3 a b^{2} c^{2}}{2}\right ) + x^{4} \left (a^{4} c + 3 a^{2} b c^{2} + \frac {b^{2} c^{3}}{2}\right ) + x^{3} \left (2 a^{3} c^{2} + 2 a b c^{3}\right ) + x^{2} \left (2 a^{2} c^{3} + \frac {b c^{4}}{2} + \frac {b}{2}\right ) + x \left (a c^{4} + a\right ) \]

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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