3.25.62 \(\int \frac {(a+b x^n)^2}{x} \, dx\) [2462]

Optimal. Leaf size=32 \[ \frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n}+a^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} a^2 \log (x)+\frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2/x,x]

[Out]

(2*a*b*x^n)/n + (b^2*x^(2*n))/(2*n) + a^2*Log[x]

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+b x^n\right )^2}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {2 a b x^n}{n}+\frac {b^2 x^{2 n}}{2 n}+a^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} \frac {b x^n \left (4 a+b x^n\right )}{2 n}+\frac {a^2 \log \left (x^n\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^2/x,x]

[Out]

(b*x^n*(4*a + b*x^n))/(2*n) + (a^2*Log[x^n])/n

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Maple [A]
time = 0.20, size = 31, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {\frac {b^{2} x^{2 n}}{2}+2 a b \,x^{n}+a^{2} \ln \left (x^{n}\right )}{n}\)	\(31\)
default	\(\frac {\frac {b^{2} x^{2 n}}{2}+2 a b \,x^{n}+a^{2} \ln \left (x^{n}\right )}{n}\)	\(31\)
risch	\(\frac {2 a b \,x^{n}}{n}+\frac {b^{2} x^{2 n}}{2 n}+a^{2} \ln \left (x \right )\)	\(31\)
norman	\(a^{2} \ln \left (x \right )+\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}+\frac {2 a b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 34, normalized size = 1.06 \begin {gather*} \frac {a^{2} \log \left (x^{n}\right )}{n} + \frac {b^{2} x^{2 \, n} + 4 \, a b x^{n}}{2 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.55, size = 30, normalized size = 0.94 \begin {gather*} \frac {2 \, a^{2} n \log \left (x\right ) + b^{2} x^{2 \, n} + 4 \, a b x^{n}}{2 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.09, size = 36, normalized size = 1.12 \begin {gather*} \begin {cases} a^{2} \log {\left (x \right )} + \frac {2 a b x^{n}}{n} + \frac {b^{2} x^{2 n}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*log(x) + 2*a*b*x**n/n + b**2*x**(2*n)/(2*n), Ne(n, 0)), ((a + b)**2*log(x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^2/x, x)

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Mupad [B]
time = 1.18, size = 30, normalized size = 0.94 \begin {gather*} \frac {b^2\,x^{2\,n}+4\,a\,b\,x^n+2\,a^2\,n\,\ln \left (x\right )}{2\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^2/x,x)

[Out]

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

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