3.5.28 \(\int \frac {(a^2+2 a b x^2+b^2 x^4)^2}{x} \, dx\) [428]

Optimal. Leaf size=50 \[ 2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8}+a^4 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45} \begin {gather*} a^4 \log (x)+2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x} \, dx}{b^4}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\text {Subst}\left (\int \left (4 a^3 b^5+\frac {a^4 b^4}{x}+6 a^2 b^6 x+4 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8}+a^4 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 50, normalized size = 1.00 \begin {gather*} 2 a^3 b x^2+\frac {3}{2} a^2 b^2 x^4+\frac {2}{3} a b^3 x^6+\frac {b^4 x^8}{8}+a^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 45, normalized size = 0.90

method	result	size
default	\(2 a^{3} b \,x^{2}+\frac {3 a^{2} b^{2} x^{4}}{2}+\frac {2 a \,b^{3} x^{6}}{3}+\frac {b^{4} x^{8}}{8}+a^{4} \ln \left (x \right )\)	\(45\)
norman	\(2 a^{3} b \,x^{2}+\frac {3 a^{2} b^{2} x^{4}}{2}+\frac {2 a \,b^{3} x^{6}}{3}+\frac {b^{4} x^{8}}{8}+a^{4} \ln \left (x \right )\)	\(45\)
risch	\(\frac {3 a^{2} b^{2} x^{4}}{2}+2 a^{3} b \,x^{2}+\frac {4 a^{4}}{3}+\frac {2 a \,b^{3} x^{6}}{3}+\frac {b^{4} x^{8}}{8}+a^{4} \ln \left (x \right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 47, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac {1}{2} \, a^{4} \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)^2/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 44, normalized size = 0.88 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4} \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)^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.03, size = 49, normalized size = 0.98 \begin {gather*} a^{4} \log {\left (x \right )} + 2 a^{3} b x^{2} + \frac {3 a^{2} b^{2} x^{4}}{2} + \frac {2 a b^{3} x^{6}}{3} + \frac {b^{4} x^{8}}{8} \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)**2/x,x)

[Out]

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

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Giac [A]
time = 6.36, size = 47, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} + \frac {2}{3} \, a b^{3} x^{6} + \frac {3}{2} \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + \frac {1}{2} \, a^{4} \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)^2/x,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 44, normalized size = 0.88 \begin {gather*} a^4\,\ln \left (x\right )+\frac {b^4\,x^8}{8}+2\,a^3\,b\,x^2+\frac {2\,a\,b^3\,x^6}{3}+\frac {3\,a^2\,b^2\,x^4}{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)^2/x,x)

[Out]

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

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