3.1.71 \(\int \frac {(a+b x)^3}{x^3} \, dx\) [71]

Optimal. Leaf size=33 \[ -\frac {a^3}{2 x^2}-\frac {3 a^2 b}{x}+b^3 x+3 a b^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, 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*} -\frac {a^3}{2 x^2}-\frac {3 a^2 b}{x}+3 a b^2 \log (x)+b^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} -\frac {a^3}{2 x^2}-\frac {3 a^2 b}{x}+b^3 x+3 a b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^3,x]

[Out]

-1/2*a^3/x^2 - (3*a^2*b)/x + b^3*x + 3*a*b^2*Log[x]

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Mathics [A]
time = 1.83, size = 31, normalized size = 0.94 \begin {gather*} -\frac {a^3}{2 x^2}-\frac {3 a^2 b}{x}+3 a b^2 \text {Log}\left [x\right ]+b^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^3/x^3,x]')

[Out]

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

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Maple [A]
time = 0.09, size = 32, normalized size = 0.97

method	result	size
default	\(-\frac {a^{3}}{2 x^{2}}-\frac {3 a^{2} b}{x}+b^{3} x +3 a \,b^{2} \ln \left (x \right )\)	\(32\)
risch	\(b^{3} x +\frac {-3 a^{2} b x -\frac {1}{2} a^{3}}{x^{2}}+3 a \,b^{2} \ln \left (x \right )\)	\(32\)
norman	\(\frac {b^{3} x^{3}-\frac {1}{2} a^{3}-3 a^{2} b x}{x^{2}}+3 a \,b^{2} \ln \left (x \right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.24, size = 30, normalized size = 0.91 \begin {gather*} b^{3} x + 3 \, a b^{2} \log \left (x\right ) - \frac {6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.31, size = 37, normalized size = 1.12 \begin {gather*} \frac {2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.09, size = 32, normalized size = 0.97 \begin {gather*} 3 a b^{2} \log {\left (x \right )} + b^{3} x + \frac {- a^{3} - 6 a^{2} b x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 36, normalized size = 1.09 \begin {gather*} x b^{3}+\frac {\frac {1}{2} \left (-6 b a^{2} x-a^{3}\right )}{x^{2}}+3 b^{2} a \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x + 3*a*b^2*log(abs(x)) - 1/2*(6*a^2*b*x + a^3)/x^2

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Mupad [B]
time = 0.03, size = 32, normalized size = 0.97 \begin {gather*} b^3\,x-\frac {\frac {a^3}{2}+3\,b\,x\,a^2}{x^2}+3\,a\,b^2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^3,x)

[Out]

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

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