3.9.41 \(\int x^2 (a+b x^2+c x^4)^3 \, dx\) [841]

Optimal. Leaf size=89 \[ \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \]

[Out]

1/3*a^3*x^3+3/5*a^2*b*x^5+3/7*a*(a*c+b^2)*x^7+1/9*b*(6*a*c+b^2)*x^9+3/11*c*(a*c+b^2)*x^11+3/13*b*c^2*x^13+1/15
*c^3*x^15

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1122} \begin {gather*} \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{11} c x^{11} \left (a c+b^2\right )+\frac {1}{9} b x^9 \left (6 a c+b^2\right )+\frac {3}{7} a x^7 \left (a c+b^2\right )+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2 + c*x^4)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11
+ (3*b*c^2*x^13)/13 + (c^3*x^15)/15

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]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^4+3 a \left (b^2+a c\right ) x^6+b \left (b^2+6 a c\right ) x^8+3 c \left (b^2+a c\right ) x^{10}+3 b c^2 x^{12}+c^3 x^{14}\right ) \, dx\\ &=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 89, normalized size = 1.00 \begin {gather*} \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*(b^2 + a*c)*x^7)/7 + (b*(b^2 + 6*a*c)*x^9)/9 + (3*c*(b^2 + a*c)*x^11)/11
+ (3*b*c^2*x^13)/13 + (c^3*x^15)/15

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 111, normalized size = 1.25

method	result	size
norman	\(\frac {a^{3} x^{3}}{3}+\frac {3 a^{2} b \,x^{5}}{5}+\left (\frac {3}{7} a^{2} c +\frac {3}{7} a \,b^{2}\right ) x^{7}+\left (\frac {2}{3} a b c +\frac {1}{9} b^{3}\right ) x^{9}+\left (\frac {3}{11} c^{2} a +\frac {3}{11} b^{2} c \right ) x^{11}+\frac {3 b \,c^{2} x^{13}}{13}+\frac {c^{3} x^{15}}{15}\)	\(85\)
gosper	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{5} a^{2} b \,x^{5}+\frac {3}{7} x^{7} a^{2} c +\frac {3}{7} a \,b^{2} x^{7}+\frac {2}{3} x^{9} a b c +\frac {1}{9} b^{3} x^{9}+\frac {3}{11} x^{11} c^{2} a +\frac {3}{11} x^{11} b^{2} c +\frac {3}{13} b \,c^{2} x^{13}+\frac {1}{15} c^{3} x^{15}\)	\(88\)
risch	\(\frac {1}{3} a^{3} x^{3}+\frac {3}{5} a^{2} b \,x^{5}+\frac {3}{7} x^{7} a^{2} c +\frac {3}{7} a \,b^{2} x^{7}+\frac {2}{3} x^{9} a b c +\frac {1}{9} b^{3} x^{9}+\frac {3}{11} x^{11} c^{2} a +\frac {3}{11} x^{11} b^{2} c +\frac {3}{13} b \,c^{2} x^{13}+\frac {1}{15} c^{3} x^{15}\)	\(88\)
default	\(\frac {c^{3} x^{15}}{15}+\frac {3 b \,c^{2} x^{13}}{13}+\frac {\left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{11}}{11}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{9}}{9}+\frac {\left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) x^{7}}{7}+\frac {3 a^{2} b \,x^{5}}{5}+\frac {a^{3} x^{3}}{3}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15+3/13*b*c^2*x^13+1/11*(c^2*a+2*b^2*c+c*(2*a*c+b^2))*x^11+1/9*(4*a*b*c+b*(2*a*c+b^2))*x^9+1/7*(a*(
2*a*c+b^2)+2*a*b^2+a^2*c)*x^7+3/5*a^2*b*x^5+1/3*a^3*x^3

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 81, normalized size = 0.91 \begin {gather*} \frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{7} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15 + 3/13*b*c^2*x^13 + 3/11*(b^2*c + a*c^2)*x^11 + 1/9*(b^3 + 6*a*b*c)*x^9 + 3/5*a^2*b*x^5 + 3/7*(a
*b^2 + a^2*c)*x^7 + 1/3*a^3*x^3

________________________________________________________________________________________

Fricas [A]
time = 0.32, size = 81, normalized size = 0.91 \begin {gather*} \frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{7} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15 + 3/13*b*c^2*x^13 + 3/11*(b^2*c + a*c^2)*x^11 + 1/9*(b^3 + 6*a*b*c)*x^9 + 3/5*a^2*b*x^5 + 3/7*(a
*b^2 + a^2*c)*x^7 + 1/3*a^3*x^3

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 97, normalized size = 1.09 \begin {gather*} \frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{5}}{5} + \frac {3 b c^{2} x^{13}}{13} + \frac {c^{3} x^{15}}{15} + x^{11} \cdot \left (\frac {3 a c^{2}}{11} + \frac {3 b^{2} c}{11}\right ) + x^{9} \cdot \left (\frac {2 a b c}{3} + \frac {b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c}{7} + \frac {3 a b^{2}}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**5/5 + 3*b*c**2*x**13/13 + c**3*x**15/15 + x**11*(3*a*c**2/11 + 3*b**2*c/11) + x**9*(
2*a*b*c/3 + b**3/9) + x**7*(3*a**2*c/7 + 3*a*b**2/7)

________________________________________________________________________________________

Giac [A]
time = 5.11, size = 87, normalized size = 0.98 \begin {gather*} \frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, b^{2} c x^{11} + \frac {3}{11} \, a c^{2} x^{11} + \frac {1}{9} \, b^{3} x^{9} + \frac {2}{3} \, a b c x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{7} \, a^{2} c x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*c^3*x^15 + 3/13*b*c^2*x^13 + 3/11*b^2*c*x^11 + 3/11*a*c^2*x^11 + 1/9*b^3*x^9 + 2/3*a*b*c*x^9 + 3/7*a*b^2*
x^7 + 3/7*a^2*c*x^7 + 3/5*a^2*b*x^5 + 1/3*a^3*x^3

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 76, normalized size = 0.85 \begin {gather*} x^9\,\left (\frac {b^3}{9}+\frac {2\,a\,c\,b}{3}\right )+\frac {a^3\,x^3}{3}+\frac {c^3\,x^{15}}{15}+\frac {3\,a^2\,b\,x^5}{5}+\frac {3\,b\,c^2\,x^{13}}{13}+\frac {3\,a\,x^7\,\left (b^2+a\,c\right )}{7}+\frac {3\,c\,x^{11}\,\left (b^2+a\,c\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^2 + c*x^4)^3,x)

[Out]

x^9*(b^3/9 + (2*a*b*c)/3) + (a^3*x^3)/3 + (c^3*x^15)/15 + (3*a^2*b*x^5)/5 + (3*b*c^2*x^13)/13 + (3*a*x^7*(a*c
+ b^2))/7 + (3*c*x^11*(a*c + b^2))/11

________________________________________________________________________________________