3.9.29 \(\int \frac {(a+b x^2+c x^4)^2}{x^2} \, dx\) [829]

Optimal. Leaf size=48 \[ -\frac {a^2}{x}+2 a b x+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 48, 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^2}{x}+\frac {1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 48, normalized size = 1.00 \begin {gather*} -\frac {a^2}{x}+2 a b x+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.03, size = 45, normalized size = 0.94

method	result	size
default	\(\frac {c^{2} x^{7}}{7}+\frac {2 b c \,x^{5}}{5}+\frac {2 a c \,x^{3}}{3}+\frac {b^{2} x^{3}}{3}+2 a b x -\frac {a^{2}}{x}\)	\(45\)
risch	\(\frac {c^{2} x^{7}}{7}+\frac {2 b c \,x^{5}}{5}+\frac {2 a c \,x^{3}}{3}+\frac {b^{2} x^{3}}{3}+2 a b x -\frac {a^{2}}{x}\)	\(45\)
norman	\(\frac {\frac {c^{2} x^{8}}{7}+\frac {2 b c \,x^{6}}{5}+\left (\frac {2 a c}{3}+\frac {b^{2}}{3}\right ) x^{4}+2 a b \,x^{2}-a^{2}}{x}\)	\(47\)
gosper	\(-\frac {-15 c^{2} x^{8}-42 b c \,x^{6}-70 a c \,x^{4}-35 b^{2} x^{4}-210 a b \,x^{2}+105 a^{2}}{105 x}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 42, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 46, normalized size = 0.96 \begin {gather*} \frac {15 \, c^{2} x^{8} + 42 \, b c x^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 210 \, a b x^{2} - 105 \, a^{2}}{105 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*c^2*x^8 + 42*b*c*x^6 + 35*(b^2 + 2*a*c)*x^4 + 210*a*b*x^2 - 105*a^2)/x

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Sympy [A]
time = 0.04, size = 44, normalized size = 0.92 \begin {gather*} - \frac {a^{2}}{x} + 2 a b x + \frac {2 b c x^{5}}{5} + \frac {c^{2} x^{7}}{7} + x^{3} \cdot \left (\frac {2 a c}{3} + \frac {b^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.06, size = 44, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a c x^{3} + 2 \, a b x - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 43, normalized size = 0.90 \begin {gather*} x^3\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )-\frac {a^2}{x}+\frac {c^2\,x^7}{7}+2\,a\,b\,x+\frac {2\,b\,c\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^2,x)

[Out]

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

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