\(\int (a x^2+b x^3+c x^4)^2 \, dx\) [8]

Optimal result

Integrand size = 18, antiderivative size = 54 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {1}{3} a b x^6+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {1}{4} b c x^8+\frac {c^2 x^9}{9} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1608, 712} \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {1}{7} x^7 \left (2 a c+b^2\right )+\frac {1}{3} a b x^6+\frac {1}{4} b c x^8+\frac {c^2 x^9}{9} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

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

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 1608

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

Rubi steps \begin{align*} \text {integral}& = \int x^4 \left (a+b x+c x^2\right )^2 \, dx \\ & = \int \left (a^2 x^4+2 a b x^5+\left (b^2+2 a c\right ) x^6+2 b c x^7+c^2 x^8\right ) \, dx \\ & = \frac {a^2 x^5}{5}+\frac {1}{3} a b x^6+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {1}{4} b c x^8+\frac {c^2 x^9}{9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {1}{3} a b x^6+\frac {1}{7} \left (b^2+2 a c\right ) x^7+\frac {1}{4} b c x^8+\frac {c^2 x^9}{9} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {a^{2} x^{5}}{5}+\frac {a b \,x^{6}}{3}+\frac {\left (2 a c +b^{2}\right ) x^{7}}{7}+\frac {b c \,x^{8}}{4}+\frac {c^{2} x^{9}}{9}\)	\(45\)
norman	\(\frac {c^{2} x^{9}}{9}+\frac {b c \,x^{8}}{4}+\left (\frac {2 a c}{7}+\frac {b^{2}}{7}\right ) x^{7}+\frac {a b \,x^{6}}{3}+\frac {a^{2} x^{5}}{5}\)	\(46\)
gosper	\(\frac {x^{5} \left (140 c^{2} x^{4}+315 b c \,x^{3}+360 a c \,x^{2}+180 b^{2} x^{2}+420 a b x +252 a^{2}\right )}{1260}\)	\(47\)
risch	\(\frac {1}{5} a^{2} x^{5}+\frac {1}{3} a b \,x^{6}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {1}{4} b c \,x^{8}+\frac {1}{9} c^{2} x^{9}\)	\(47\)
parallelrisch	\(\frac {1}{5} a^{2} x^{5}+\frac {1}{3} a b \,x^{6}+\frac {2}{7} x^{7} a c +\frac {1}{7} b^{2} x^{7}+\frac {1}{4} b c \,x^{8}+\frac {1}{9} c^{2} x^{9}\)	\(47\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {1}{4} \, b c x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{7} \, {\left (b^{2} + 2 \, a c\right )} x^{7} + \frac {1}{5} \, a^{2} x^{5} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^{2} x^{5}}{5} + \frac {a b x^{6}}{3} + \frac {b c x^{8}}{4} + \frac {c^{2} x^{9}}{9} + x^{7} \cdot \left (\frac {2 a c}{7} + \frac {b^{2}}{7}\right ) \]

[In]

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

[Out]

a**2*x**5/5 + a*b*x**6/3 + b*c*x**8/4 + c**2*x**9/9 + x**7*(2*a*c/7 + b**2/7)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {1}{4} \, b c x^{8} + \frac {1}{7} \, b^{2} x^{7} + \frac {1}{5} \, a^{2} x^{5} + \frac {1}{21} \, {\left (6 \, c x^{7} + 7 \, b x^{6}\right )} a \]

[In]

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

[Out]

1/9*c^2*x^9 + 1/4*b*c*x^8 + 1/7*b^2*x^7 + 1/5*a^2*x^5 + 1/21*(6*c*x^7 + 7*b*x^6)*a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{9} \, c^{2} x^{9} + \frac {1}{4} \, b c x^{8} + \frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a c x^{7} + \frac {1}{3} \, a b x^{6} + \frac {1}{5} \, a^{2} x^{5} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int \left (a x^2+b x^3+c x^4\right )^2 \, dx=x^7\,\left (\frac {b^2}{7}+\frac {2\,a\,c}{7}\right )+\frac {a^2\,x^5}{5}+\frac {c^2\,x^9}{9}+\frac {a\,b\,x^6}{3}+\frac {b\,c\,x^8}{4} \]

[In]

int((a*x^2 + b*x^3 + c*x^4)^2,x)

[Out]

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