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

Optimal. Leaf size=86 \[ -\frac {a^3}{2 x^2}+3 a^2 b \log (x)+\frac {1}{2} c x^6 \left (a c+b^2\right )+\frac {1}{4} b x^4 \left (6 a c+b^2\right )+\frac {3}{2} a x^2 \left (a c+b^2\right )+\frac {3}{8} b c^2 x^8+\frac {c^3 x^{10}}{10} \]

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Rubi [A]  time = 0.08, antiderivative size = 86, 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} \begin {gather*} 3 a^2 b \log (x)-\frac {a^3}{2 x^2}+\frac {1}{2} c x^6 \left (a c+b^2\right )+\frac {1}{4} b x^4 \left (6 a c+b^2\right )+\frac {3}{2} a x^2 \left (a c+b^2\right )+\frac {3}{8} b c^2 x^8+\frac {c^3 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^3,x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 78, normalized size = 0.91 \begin {gather*} \frac {1}{40} \left (-\frac {20 a^3}{x^2}+120 a^2 b \log (x)+20 c x^6 \left (a c+b^2\right )+10 b x^4 \left (6 a c+b^2\right )+60 a x^2 \left (a c+b^2\right )+15 b c^2 x^8+4 c^3 x^{10}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-20*a^3)/x^2 + 60*a*(b^2 + a*c)*x^2 + 10*b*(b^2 + 6*a*c)*x^4 + 20*c*(b^2 + a*c)*x^6 + 15*b*c^2*x^8 + 4*c^3*x
^10 + 120*a^2*b*Log[x])/40

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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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 85, normalized size = 0.99 \begin {gather*} \frac {4 \, c^{3} x^{12} + 15 \, b c^{2} x^{10} + 20 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 10 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 120 \, a^{2} b x^{2} \log \relax (x) + 60 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 20 \, a^{3}}{40 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(4*c^3*x^12 + 15*b*c^2*x^10 + 20*(b^2*c + a*c^2)*x^8 + 10*(b^3 + 6*a*b*c)*x^6 + 120*a^2*b*x^2*log(x) + 60
*(a*b^2 + a^2*c)*x^4 - 20*a^3)/x^2

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giac [A]  time = 0.16, size = 98, normalized size = 1.14 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} + \frac {3}{8} \, b c^{2} x^{8} + \frac {1}{2} \, b^{2} c x^{6} + \frac {1}{2} \, a c^{2} x^{6} + \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b c x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {3}{2} \, a^{2} c x^{2} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac {3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 87, normalized size = 1.01 \begin {gather*} \frac {c^{3} x^{10}}{10}+\frac {3 b \,c^{2} x^{8}}{8}+\frac {a \,c^{2} x^{6}}{2}+\frac {b^{2} c \,x^{6}}{2}+\frac {3 a b c \,x^{4}}{2}+\frac {b^{3} x^{4}}{4}+\frac {3 a^{2} c \,x^{2}}{2}+\frac {3 a \,b^{2} x^{2}}{2}+3 a^{2} b \ln \relax (x )-\frac {a^{3}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 82, normalized size = 0.95 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} + \frac {3}{8} \, b c^{2} x^{8} + \frac {1}{2} \, {\left (b^{2} c + a c^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{3} + 6 \, a b c\right )} x^{4} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) + \frac {3}{2} \, {\left (a b^{2} + a^{2} c\right )} x^{2} - \frac {a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 75, normalized size = 0.87 \begin {gather*} x^4\,\left (\frac {b^3}{4}+\frac {3\,a\,c\,b}{2}\right )-\frac {a^3}{2\,x^2}+\frac {c^3\,x^{10}}{10}+\frac {3\,b\,c^2\,x^8}{8}+3\,a^2\,b\,\ln \relax (x)+\frac {3\,a\,x^2\,\left (b^2+a\,c\right )}{2}+\frac {c\,x^6\,\left (b^2+a\,c\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^3,x)

[Out]

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

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sympy [A]  time = 0.27, size = 92, normalized size = 1.07 \begin {gather*} - \frac {a^{3}}{2 x^{2}} + 3 a^{2} b \log {\relax (x )} + \frac {3 b c^{2} x^{8}}{8} + \frac {c^{3} x^{10}}{10} + x^{6} \left (\frac {a c^{2}}{2} + \frac {b^{2} c}{2}\right ) + x^{4} \left (\frac {3 a b c}{2} + \frac {b^{3}}{4}\right ) + x^{2} \left (\frac {3 a^{2} c}{2} + \frac {3 a b^{2}}{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**3,x)

[Out]

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

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