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

Optimal. Leaf size=80 \[ -\frac {a^3}{x}+3 a^2 b x+\frac {3}{7} c x^7 \left (a c+b^2\right )+\frac {1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 80, 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 = {1108} \begin {gather*} 3 a^2 b x-\frac {a^3}{x}+\frac {3}{7} c x^7 \left (a c+b^2\right )+\frac {1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^2,x]

[Out]

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

Rule 1108

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 80, normalized size = 1.00 \begin {gather*} -\frac {a^3}{x}+3 a^2 b x+\frac {3}{7} c x^7 \left (a c+b^2\right )+\frac {1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{3} b c^2 x^9+\frac {c^3 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2+c x^4\right )^3}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 83, normalized size = 1.04 \begin {gather*} \frac {105 \, c^{3} x^{12} + 385 \, b c^{2} x^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 231 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 3465 \, a^{2} b x^{2} + 1155 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 1155 \, a^{3}}{1155 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(105*c^3*x^12 + 385*b*c^2*x^10 + 495*(b^2*c + a*c^2)*x^8 + 231*(b^3 + 6*a*b*c)*x^6 + 3465*a^2*b*x^2 + 1
155*(a*b^2 + a^2*c)*x^4 - 1155*a^3)/x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 83, normalized size = 1.04 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{7} \, b^{2} c x^{7} + \frac {3}{7} \, a c^{2} x^{7} + \frac {1}{5} \, b^{3} x^{5} + \frac {6}{5} \, a b c x^{5} + a b^{2} x^{3} + a^{2} c x^{3} + 3 \, a^{2} b x - \frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 84, normalized size = 1.05 \begin {gather*} \frac {c^{3} x^{11}}{11}+\frac {b \,c^{2} x^{9}}{3}+\frac {3 a \,c^{2} x^{7}}{7}+\frac {3 b^{2} c \,x^{7}}{7}+\frac {6 a b c \,x^{5}}{5}+\frac {b^{3} x^{5}}{5}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+3 a^{2} b x -\frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 78, normalized size = 0.98 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} + \frac {1}{3} \, b c^{2} x^{9} + \frac {3}{7} \, {\left (b^{2} c + a c^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{3} + 6 \, a b c\right )} x^{5} + 3 \, a^{2} b x + {\left (a b^{2} + a^{2} c\right )} x^{3} - \frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 73, normalized size = 0.91 \begin {gather*} x^5\,\left (\frac {b^3}{5}+\frac {6\,a\,c\,b}{5}\right )-\frac {a^3}{x}+\frac {c^3\,x^{11}}{11}+\frac {b\,c^2\,x^9}{3}+a\,x^3\,\left (b^2+a\,c\right )+\frac {3\,c\,x^7\,\left (b^2+a\,c\right )}{7}+3\,a^2\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 82, normalized size = 1.02 \begin {gather*} - \frac {a^{3}}{x} + 3 a^{2} b x + \frac {b c^{2} x^{9}}{3} + \frac {c^{3} x^{11}}{11} + x^{7} \left (\frac {3 a c^{2}}{7} + \frac {3 b^{2} c}{7}\right ) + x^{5} \left (\frac {6 a b c}{5} + \frac {b^{3}}{5}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________