3.5.13 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{x} \, dx\) [413]

Optimal. Leaf size=23 \[ a b x^2+\frac {b^2 x^4}{4}+a^2 \log (x) \]

[Out]

a*b*x^2+1/4*b^2*x^4+a^2*ln(x)

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \begin {gather*} a^2 \log (x)+a b x^2+\frac {b^2 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x,x]

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int \frac {a^2+2 a b x^2+b^2 x^4}{x} \, dx &=\int \left (\frac {a^2}{x}+2 a b x+b^2 x^3\right ) \, dx\\ &=a b x^2+\frac {b^2 x^4}{4}+a^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} a b x^2+\frac {b^2 x^4}{4}+a^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*b*x^2 + (b^2*x^4)/4 + a^2*Log[x]

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Maple [A]
time = 0.02, size = 22, normalized size = 0.96

method	result	size
default	\(a b \,x^{2}+\frac {b^{2} x^{4}}{4}+a^{2} \ln \left (x \right )\)	\(22\)
norman	\(a b \,x^{2}+\frac {b^{2} x^{4}}{4}+a^{2} \ln \left (x \right )\)	\(22\)
risch	\(\frac {b^{2} x^{4}}{4}+a b \,x^{2}+a^{2}+a^{2} \ln \left (x \right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b*x^2+1/4*b^2*x^4+a^2*ln(x)

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Maxima [A]
time = 0.28, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + 1/2*a^2*log(x^2)

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Fricas [A]
time = 0.37, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + a^2*log(x)

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Sympy [A]
time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} a^{2} \log {\left (x \right )} + a b x^{2} + \frac {b^{2} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*log(x) + a*b*x**2 + b**2*x**4/4

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Giac [A]
time = 12.32, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + a*b*x^2 + 1/2*a^2*log(x^2)

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Mupad [B]
time = 4.10, size = 21, normalized size = 0.91 \begin {gather*} a^2\,\ln \left (x\right )+\frac {b^2\,x^4}{4}+a\,b\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)/x,x)

[Out]

a^2*log(x) + (b^2*x^4)/4 + a*b*x^2

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