\(\int \frac {(a x^2+b x^3+c x^4)^2}{x} \, dx\) [9]

Optimal result

Integrand size = 22, antiderivative size = 54 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {1}{6} \left (b^2+2 a c\right ) x^6+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1599, 712} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {a^2 x^4}{4}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2/x,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^6)/6 + (2*b*c*x^7)/7 + (c^2*x^8)/8

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rubi steps \begin{align*} \text {integral}& = \int x^3 \left (a+b x+c x^2\right )^2 \, dx \\ & = \int \left (a^2 x^3+2 a b x^4+\left (b^2+2 a c\right ) x^5+2 b c x^6+c^2 x^7\right ) \, dx \\ & = \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {1}{6} \left (b^2+2 a c\right ) x^6+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {1}{6} \left (b^2+2 a c\right ) x^6+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \]

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^2/x,x]

[Out]

(a^2*x^4)/4 + (2*a*b*x^5)/5 + ((b^2 + 2*a*c)*x^6)/6 + (2*b*c*x^7)/7 + (c^2*x^8)/8

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {a^{2} x^{4}}{4}+\frac {2 a b \,x^{5}}{5}+\frac {\left (2 a c +b^{2}\right ) x^{6}}{6}+\frac {2 b c \,x^{7}}{7}+\frac {c^{2} x^{8}}{8}\)	\(45\)
norman	\(\frac {c^{2} x^{8}}{8}+\frac {2 b c \,x^{7}}{7}+\left (\frac {a c}{3}+\frac {b^{2}}{6}\right ) x^{6}+\frac {2 a b \,x^{5}}{5}+\frac {a^{2} x^{4}}{4}\)	\(46\)
gosper	\(\frac {x^{4} \left (105 c^{2} x^{4}+240 b c \,x^{3}+280 a c \,x^{2}+140 b^{2} x^{2}+336 a b x +210 a^{2}\right )}{840}\)	\(47\)
risch	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{5} a b \,x^{5}+\frac {1}{3} x^{6} a c +\frac {1}{6} b^{2} x^{6}+\frac {2}{7} b c \,x^{7}+\frac {1}{8} c^{2} x^{8}\)	\(47\)
parallelrisch	\(\frac {1}{4} a^{2} x^{4}+\frac {2}{5} a b \,x^{5}+\frac {1}{3} x^{6} a c +\frac {1}{6} b^{2} x^{6}+\frac {2}{7} b c \,x^{7}+\frac {1}{8} c^{2} x^{8}\)	\(47\)

[In]

int((c*x^4+b*x^3+a*x^2)^2/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{4} \, a^{2} x^{4} \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {a^{2} x^{4}}{4} + \frac {2 a b x^{5}}{5} + \frac {2 b c x^{7}}{7} + \frac {c^{2} x^{8}}{8} + x^{6} \left (\frac {a c}{3} + \frac {b^{2}}{6}\right ) \]

[In]

integrate((c*x**4+b*x**3+a*x**2)**2/x,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{4} \, a^{2} x^{4} \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=\frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {1}{6} \, b^{2} x^{6} + \frac {1}{3} \, a c x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \]

[In]

integrate((c*x^4+b*x^3+a*x^2)^2/x,x, algorithm="giac")

[Out]

1/8*c^2*x^8 + 2/7*b*c*x^7 + 1/6*b^2*x^6 + 1/3*a*c*x^6 + 2/5*a*b*x^5 + 1/4*a^2*x^4

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx=x^6\,\left (\frac {b^2}{6}+\frac {a\,c}{3}\right )+\frac {a^2\,x^4}{4}+\frac {c^2\,x^8}{8}+\frac {2\,a\,b\,x^5}{5}+\frac {2\,b\,c\,x^7}{7} \]

[In]

int((a*x^2 + b*x^3 + c*x^4)^2/x,x)

[Out]

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