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

Optimal. Leaf size=85 \[ a^3 \log (x)+\frac {3}{2} a^2 b x^2+\frac {3}{8} c x^8 \left (a c+b^2\right )+\frac {1}{6} b x^6 \left (6 a c+b^2\right )+\frac {3}{4} a x^4 \left (a c+b^2\right )+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12} \]

[Out]

3/2*a^2*b*x^2+3/4*a*(a*c+b^2)*x^4+1/6*b*(6*a*c+b^2)*x^6+3/8*c*(a*c+b^2)*x^8+3/10*b*c^2*x^10+1/12*c^3*x^12+a^3*
ln(x)

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Rubi [A]  time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1114, 698} \[ \frac {3}{2} a^2 b x^2+a^3 \log (x)+\frac {3}{8} c x^8 \left (a c+b^2\right )+\frac {1}{6} b x^6 \left (6 a c+b^2\right )+\frac {3}{4} a x^4 \left (a c+b^2\right )+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)
/10 + (c^3*x^12)/12 + a^3*Log[x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^3}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^3}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a \left (b^2+a c\right ) x+b \left (b^2+6 a c\right ) x^2+3 c \left (b^2+a c\right ) x^3+3 b c^2 x^4+c^3 x^5\right ) \, dx,x,x^2\right )\\ &=\frac {3}{2} a^2 b x^2+\frac {3}{4} a \left (b^2+a c\right ) x^4+\frac {1}{6} b \left (b^2+6 a c\right ) x^6+\frac {3}{8} c \left (b^2+a c\right ) x^8+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12}+a^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 85, normalized size = 1.00 \[ a^3 \log (x)+\frac {3}{2} a^2 b x^2+\frac {3}{8} c x^8 \left (a c+b^2\right )+\frac {1}{6} b x^6 \left (6 a c+b^2\right )+\frac {3}{4} a x^4 \left (a c+b^2\right )+\frac {3}{10} b c^2 x^{10}+\frac {c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)
/10 + (c^3*x^12)/12 + a^3*Log[x]

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fricas [A]  time = 0.89, size = 79, normalized size = 0.93 \[ \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac {3}{2} \, a^{2} b x^{2} + \frac {3}{4} \, {\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b
^2 + a^2*c)*x^4 + a^3*log(x)

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giac [A]  time = 0.16, size = 87, normalized size = 1.02 \[ \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, b^{2} c x^{8} + \frac {3}{8} \, a c^{2} x^{8} + \frac {1}{6} \, b^{3} x^{6} + a b c x^{6} + \frac {3}{4} \, a b^{2} x^{4} + \frac {3}{4} \, a^{2} c x^{4} + \frac {3}{2} \, a^{2} b x^{2} + \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*b^2*c*x^8 + 3/8*a*c^2*x^8 + 1/6*b^3*x^6 + a*b*c*x^6 + 3/4*a*b^2*x^4 + 3/
4*a^2*c*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*log(x^2)

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maple [A]  time = 0.00, size = 85, normalized size = 1.00 \[ \frac {c^{3} x^{12}}{12}+\frac {3 b \,c^{2} x^{10}}{10}+\frac {3 a \,c^{2} x^{8}}{8}+\frac {3 b^{2} c \,x^{8}}{8}+a b c \,x^{6}+\frac {b^{3} x^{6}}{6}+\frac {3 a^{2} c \,x^{4}}{4}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {3 a^{2} b \,x^{2}}{2}+a^{3} \ln \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^3*x^12+3/10*b*c^2*x^10+3/8*x^8*a*c^2+3/8*x^8*b^2*c+x^6*a*b*c+1/6*b^3*x^6+3/4*x^4*a^2*c+3/4*a*b^2*x^4+3/
2*a^2*b*x^2+a^3*ln(x)

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maxima [A]  time = 1.39, size = 82, normalized size = 0.96 \[ \frac {1}{12} \, c^{3} x^{12} + \frac {3}{10} \, b c^{2} x^{10} + \frac {3}{8} \, {\left (b^{2} c + a c^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac {3}{2} \, a^{2} b x^{2} + \frac {3}{4} \, {\left (a b^{2} + a^{2} c\right )} x^{4} + \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b
^2 + a^2*c)*x^4 + 1/2*a^3*log(x^2)

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mupad [B]  time = 0.03, size = 73, normalized size = 0.86 \[ a^3\,\ln \relax (x)+x^6\,\left (\frac {b^3}{6}+a\,c\,b\right )+\frac {c^3\,x^{12}}{12}+\frac {3\,a^2\,b\,x^2}{2}+\frac {3\,b\,c^2\,x^{10}}{10}+\frac {3\,a\,x^4\,\left (b^2+a\,c\right )}{4}+\frac {3\,c\,x^8\,\left (b^2+a\,c\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x,x)

[Out]

a^3*log(x) + x^6*(b^3/6 + a*b*c) + (c^3*x^12)/12 + (3*a^2*b*x^2)/2 + (3*b*c^2*x^10)/10 + (3*a*x^4*(a*c + b^2))
/4 + (3*c*x^8*(a*c + b^2))/8

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sympy [A]  time = 0.22, size = 92, normalized size = 1.08 \[ a^{3} \log {\relax (x )} + \frac {3 a^{2} b x^{2}}{2} + \frac {3 b c^{2} x^{10}}{10} + \frac {c^{3} x^{12}}{12} + x^{8} \left (\frac {3 a c^{2}}{8} + \frac {3 b^{2} c}{8}\right ) + x^{6} \left (a b c + \frac {b^{3}}{6}\right ) + x^{4} \left (\frac {3 a^{2} c}{4} + \frac {3 a b^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) + 3*a**2*b*x**2/2 + 3*b*c**2*x**10/10 + c**3*x**12/12 + x**8*(3*a*c**2/8 + 3*b**2*c/8) + x**6*(a*b
*c + b**3/6) + x**4*(3*a**2*c/4 + 3*a*b**2/4)

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