\(\int (a+b x^2+c x^4)^3 \, dx\) [843]

Optimal result

Integrand size = 14, antiderivative size = 81 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=a^3 x+a^2 b x^3+\frac {3}{5} a \left (b^2+a c\right ) x^5+\frac {1}{7} b \left (b^2+6 a c\right ) x^7+\frac {1}{3} c \left (b^2+a c\right ) x^9+\frac {3}{11} b c^2 x^{11}+\frac {c^3 x^{13}}{13} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1104} \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=a^3 x+a^2 b x^3+\frac {1}{3} c x^9 \left (a c+b^2\right )+\frac {1}{7} b x^7 \left (6 a c+b^2\right )+\frac {3}{5} a x^5 \left (a c+b^2\right )+\frac {3}{11} b c^2 x^{11}+\frac {c^3 x^{13}}{13} \]

[In]

Int[(a + b*x^2 + c*x^4)^3,x]

[Out]

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

Rule 1104

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]
/; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=a^3 x+a^2 b x^3+\frac {3}{5} a \left (b^2+a c\right ) x^5+\frac {1}{7} b \left (b^2+6 a c\right ) x^7+\frac {1}{3} c \left (b^2+a c\right ) x^9+\frac {3}{11} b c^2 x^{11}+\frac {c^3 x^{13}}{13} \]

[In]

Integrate[(a + b*x^2 + c*x^4)^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {c^{3} x^{13}}{13}+\frac {3 b \,c^{2} x^{11}}{11}+\left (\frac {1}{3} a \,c^{2}+\frac {1}{3} b^{2} c \right ) x^{9}+\left (\frac {6}{7} a b c +\frac {1}{7} b^{3}\right ) x^{7}+\left (\frac {3}{5} c \,a^{2}+\frac {3}{5} b^{2} a \right ) x^{5}+a^{2} b \,x^{3}+a^{3} x\)	\(81\)
gosper	\(\frac {1}{13} c^{3} x^{13}+\frac {3}{11} b \,c^{2} x^{11}+\frac {1}{3} x^{9} a \,c^{2}+\frac {1}{3} x^{9} b^{2} c +\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {3}{5} x^{5} c \,a^{2}+\frac {3}{5} b^{2} x^{5} a +a^{2} b \,x^{3}+a^{3} x\)	\(84\)
risch	\(\frac {1}{13} c^{3} x^{13}+\frac {3}{11} b \,c^{2} x^{11}+\frac {1}{3} x^{9} a \,c^{2}+\frac {1}{3} x^{9} b^{2} c +\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {3}{5} x^{5} c \,a^{2}+\frac {3}{5} b^{2} x^{5} a +a^{2} b \,x^{3}+a^{3} x\)	\(84\)
parallelrisch	\(\frac {1}{13} c^{3} x^{13}+\frac {3}{11} b \,c^{2} x^{11}+\frac {1}{3} x^{9} a \,c^{2}+\frac {1}{3} x^{9} b^{2} c +\frac {6}{7} x^{7} a b c +\frac {1}{7} b^{3} x^{7}+\frac {3}{5} x^{5} c \,a^{2}+\frac {3}{5} b^{2} x^{5} a +a^{2} b \,x^{3}+a^{3} x\)	\(84\)
default	\(\frac {c^{3} x^{13}}{13}+\frac {3 b \,c^{2} x^{11}}{11}+\frac {\left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{9}}{9}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right ) x^{5}}{5}+a^{2} b \,x^{3}+a^{3} x\)	\(107\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{13} \, c^{3} x^{13} + \frac {3}{11} \, b c^{2} x^{11} + \frac {1}{3} \, {\left (b^{2} c + a c^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{7} + a^{2} b x^{3} + \frac {3}{5} \, {\left (a b^{2} + a^{2} c\right )} x^{5} + a^{3} x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=a^{3} x + a^{2} b x^{3} + \frac {3 b c^{2} x^{11}}{11} + \frac {c^{3} x^{13}}{13} + x^{9} \left (\frac {a c^{2}}{3} + \frac {b^{2} c}{3}\right ) + x^{7} \cdot \left (\frac {6 a b c}{7} + \frac {b^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c}{5} + \frac {3 a b^{2}}{5}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{13} \, c^{3} x^{13} + \frac {3}{11} \, b c^{2} x^{11} + \frac {1}{3} \, b^{2} c x^{9} + \frac {1}{7} \, b^{3} x^{7} + a^{3} x + \frac {1}{5} \, {\left (3 \, c x^{5} + 5 \, b x^{3}\right )} a^{2} + \frac {1}{105} \, {\left (35 \, c^{2} x^{9} + 90 \, b c x^{7} + 63 \, b^{2} x^{5}\right )} a \]

[In]

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

[Out]

1/13*c^3*x^13 + 3/11*b*c^2*x^11 + 1/3*b^2*c*x^9 + 1/7*b^3*x^7 + a^3*x + 1/5*(3*c*x^5 + 5*b*x^3)*a^2 + 1/105*(3
5*c^2*x^9 + 90*b*c*x^7 + 63*b^2*x^5)*a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=\frac {1}{13} \, c^{3} x^{13} + \frac {3}{11} \, b c^{2} x^{11} + \frac {1}{3} \, b^{2} c x^{9} + \frac {1}{3} \, a c^{2} x^{9} + \frac {1}{7} \, b^{3} x^{7} + \frac {6}{7} \, a b c x^{7} + \frac {3}{5} \, a b^{2} x^{5} + \frac {3}{5} \, a^{2} c x^{5} + a^{2} b x^{3} + a^{3} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^2+c x^4\right )^3 \, dx=x^7\,\left (\frac {b^3}{7}+\frac {6\,a\,c\,b}{7}\right )+a^3\,x+\frac {c^3\,x^{13}}{13}+a^2\,b\,x^3+\frac {3\,b\,c^2\,x^{11}}{11}+\frac {3\,a\,x^5\,\left (b^2+a\,c\right )}{5}+\frac {c\,x^9\,\left (b^2+a\,c\right )}{3} \]

[In]

int((a + b*x^2 + c*x^4)^3,x)

[Out]

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