3.1.69 \(\int \frac {(a+b x)^3}{x} \, dx\) [69]

Optimal. Leaf size=35 \[ 3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3}+a^3 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} a^3 \log (x)+3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x,x]

[Out]

3*a^2*b*x + (3*a*b^2*x^2)/2 + (b^3*x^3)/3 + a^3*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])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} 3 a^2 b x+\frac {3}{2} a b^2 x^2+\frac {b^3 x^3}{3}+a^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x,x]

[Out]

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

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Maple [A]
time = 0.08, size = 32, normalized size = 0.91

method	result	size
default	\(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\)	\(32\)
norman	\(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\)	\(32\)
risch	\(3 a^{2} b x +\frac {3 a \,b^{2} x^{2}}{2}+\frac {b^{3} x^{3}}{3}+a^{3} \ln \left (x \right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.14, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.03, size = 34, normalized size = 0.97 \begin {gather*} a^{3} \log {\left (x \right )} + 3 a^{2} b x + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x,x)

[Out]

a**3*log(x) + 3*a**2*b*x + 3*a*b**2*x**2/2 + b**3*x**3/3

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Giac [A]
time = 1.76, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3 + 3/2*a*b^2*x^2 + 3*a^2*b*x + a^3*log(abs(x))

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Mupad [B]
time = 0.03, size = 31, normalized size = 0.89 \begin {gather*} a^3\,\ln \left (x\right )+\frac {b^3\,x^3}{3}+\frac {3\,a\,b^2\,x^2}{2}+3\,a^2\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x,x)

[Out]

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

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