\(\int x^2 (a x+b x^3+c x^5)^2 \, dx\) [73]

Optimal result

Integrand size = 20, antiderivative size = 54 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {2}{11} b c x^{11}+\frac {c^2 x^{13}}{13} \]

[Out]

1/5*a^2*x^5+2/7*a*b*x^7+1/9*(2*a*c+b^2)*x^9+2/11*b*c*x^11+1/13*c^2*x^13

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1599, 1122} \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {1}{9} x^9 \left (2 a c+b^2\right )+\frac {2}{7} a b x^7+\frac {2}{11} b c x^{11}+\frac {c^2 x^{13}}{13} \]

[In]

Int[x^2*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^9)/9 + (2*b*c*x^11)/11 + (c^2*x^13)/13

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

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^4 \left (a+b x^2+c x^4\right )^2 \, dx \\ & = \int \left (a^2 x^4+2 a b x^6+\left (b^2+2 a c\right ) x^8+2 b c x^{10}+c^2 x^{12}\right ) \, dx \\ & = \frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {2}{11} b c x^{11}+\frac {c^2 x^{13}}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 x^5}{5}+\frac {2}{7} a b x^7+\frac {1}{9} \left (b^2+2 a c\right ) x^9+\frac {2}{11} b c x^{11}+\frac {c^2 x^{13}}{13} \]

[In]

Integrate[x^2*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^9)/9 + (2*b*c*x^11)/11 + (c^2*x^13)/13

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {a^{2} x^{5}}{5}+\frac {2 a b \,x^{7}}{7}+\frac {\left (2 a c +b^{2}\right ) x^{9}}{9}+\frac {2 b c \,x^{11}}{11}+\frac {c^{2} x^{13}}{13}\)	\(45\)
norman	\(\frac {c^{2} x^{13}}{13}+\frac {2 b c \,x^{11}}{11}+\left (\frac {2 a c}{9}+\frac {b^{2}}{9}\right ) x^{9}+\frac {2 a b \,x^{7}}{7}+\frac {a^{2} x^{5}}{5}\)	\(46\)
risch	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{7} a b \,x^{7}+\frac {2}{9} x^{9} a c +\frac {1}{9} b^{2} x^{9}+\frac {2}{11} b c \,x^{11}+\frac {1}{13} c^{2} x^{13}\)	\(47\)
parallelrisch	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{7} a b \,x^{7}+\frac {2}{9} x^{9} a c +\frac {1}{9} b^{2} x^{9}+\frac {2}{11} b c \,x^{11}+\frac {1}{13} c^{2} x^{13}\)	\(47\)
gosper	\(\frac {x^{5} \left (3465 c^{2} x^{8}+8190 b c \,x^{6}+10010 a c \,x^{4}+5005 b^{2} x^{4}+12870 a b \,x^{2}+9009 a^{2}\right )}{45045}\)	\(49\)

[In]

int(x^2*(c*x^5+b*x^3+a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/5*a^2*x^5+2/7*a*b*x^7+1/9*(2*a*c+b^2)*x^9+2/11*b*c*x^11+1/13*c^2*x^13

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{13} \, c^{2} x^{13} + \frac {2}{11} \, b c x^{11} + \frac {1}{9} \, {\left (b^{2} + 2 \, a c\right )} x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")

[Out]

1/13*c^2*x^13 + 2/11*b*c*x^11 + 1/9*(b^2 + 2*a*c)*x^9 + 2/7*a*b*x^7 + 1/5*a^2*x^5

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^{2} x^{5}}{5} + \frac {2 a b x^{7}}{7} + \frac {2 b c x^{11}}{11} + \frac {c^{2} x^{13}}{13} + x^{9} \cdot \left (\frac {2 a c}{9} + \frac {b^{2}}{9}\right ) \]

[In]

integrate(x**2*(c*x**5+b*x**3+a*x)**2,x)

[Out]

a**2*x**5/5 + 2*a*b*x**7/7 + 2*b*c*x**11/11 + c**2*x**13/13 + x**9*(2*a*c/9 + b**2/9)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{13} \, c^{2} x^{13} + \frac {2}{11} \, b c x^{11} + \frac {1}{9} \, {\left (b^{2} + 2 \, a c\right )} x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")

[Out]

1/13*c^2*x^13 + 2/11*b*c*x^11 + 1/9*(b^2 + 2*a*c)*x^9 + 2/7*a*b*x^7 + 1/5*a^2*x^5

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {1}{13} \, c^{2} x^{13} + \frac {2}{11} \, b c x^{11} + \frac {1}{9} \, b^{2} x^{9} + \frac {2}{9} \, a c x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{5} \, a^{2} x^{5} \]

[In]

integrate(x^2*(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")

[Out]

1/13*c^2*x^13 + 2/11*b*c*x^11 + 1/9*b^2*x^9 + 2/9*a*c*x^9 + 2/7*a*b*x^7 + 1/5*a^2*x^5

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx=x^9\,\left (\frac {b^2}{9}+\frac {2\,a\,c}{9}\right )+\frac {a^2\,x^5}{5}+\frac {c^2\,x^{13}}{13}+\frac {2\,a\,b\,x^7}{7}+\frac {2\,b\,c\,x^{11}}{11} \]

[In]

int(x^2*(a*x + b*x^3 + c*x^5)^2,x)

[Out]

x^9*((2*a*c)/9 + b^2/9) + (a^2*x^5)/5 + (c^2*x^13)/13 + (2*a*b*x^7)/7 + (2*b*c*x^11)/11