3.1.86 \(\int \frac {(a+b x)^5}{x^3} \, dx\) [86]

Optimal. Leaf size=60 \[ -\frac {a^5}{2 x^2}-\frac {5 a^4 b}{x}+10 a^2 b^3 x+\frac {5}{2} a b^4 x^2+\frac {b^5 x^3}{3}+10 a^3 b^2 \log (x) \]

[Out]

-1/2*a^5/x^2-5*a^4*b/x+10*a^2*b^3*x+5/2*a*b^4*x^2+1/3*b^5*x^3+10*a^3*b^2*ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 60, 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^5}{2 x^2}-\frac {5 a^4 b}{x}+10 a^3 b^2 \log (x)+10 a^2 b^3 x+\frac {5}{2} a b^4 x^2+\frac {b^5 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^3,x]

[Out]

-1/2*a^5/x^2 - (5*a^4*b)/x + 10*a^2*b^3*x + (5*a*b^4*x^2)/2 + (b^5*x^3)/3 + 10*a^3*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)^5}{x^3} \, dx &=\int \left (10 a^2 b^3+\frac {a^5}{x^3}+\frac {5 a^4 b}{x^2}+\frac {10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx\\ &=-\frac {a^5}{2 x^2}-\frac {5 a^4 b}{x}+10 a^2 b^3 x+\frac {5}{2} a b^4 x^2+\frac {b^5 x^3}{3}+10 a^3 b^2 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 60, normalized size = 1.00 \begin {gather*} -\frac {a^5}{2 x^2}-\frac {5 a^4 b}{x}+10 a^2 b^3 x+\frac {5}{2} a b^4 x^2+\frac {b^5 x^3}{3}+10 a^3 b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^3,x]

[Out]

-1/2*a^5/x^2 - (5*a^4*b)/x + 10*a^2*b^3*x + (5*a*b^4*x^2)/2 + (b^5*x^3)/3 + 10*a^3*b^2*Log[x]

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Maple [A]
time = 0.08, size = 55, normalized size = 0.92

method	result	size
default	\(-\frac {a^{5}}{2 x^{2}}-\frac {5 a^{4} b}{x}+10 a^{2} b^{3} x +\frac {5 a \,b^{4} x^{2}}{2}+\frac {b^{5} x^{3}}{3}+10 a^{3} b^{2} \ln \left (x \right )\)	\(55\)
risch	\(\frac {b^{5} x^{3}}{3}+\frac {5 a \,b^{4} x^{2}}{2}+10 a^{2} b^{3} x +\frac {-5 a^{4} b x -\frac {1}{2} a^{5}}{x^{2}}+10 a^{3} b^{2} \ln \left (x \right )\)	\(55\)
norman	\(\frac {-\frac {1}{2} a^{5}+\frac {1}{3} b^{5} x^{5}+\frac {5}{2} a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}-5 a^{4} b x}{x^{2}}+10 a^{3} b^{2} \ln \left (x \right )\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^5/x^2-5*a^4*b/x+10*a^2*b^3*x+5/2*a*b^4*x^2+1/3*b^5*x^3+10*a^3*b^2*ln(x)

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Maxima [A]
time = 0.28, size = 53, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, b^{5} x^{3} + \frac {5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2} \log \left (x\right ) - \frac {10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="maxima")

[Out]

1/3*b^5*x^3 + 5/2*a*b^4*x^2 + 10*a^2*b^3*x + 10*a^3*b^2*log(x) - 1/2*(10*a^4*b*x + a^5)/x^2

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Fricas [A]
time = 0.95, size = 59, normalized size = 0.98 \begin {gather*} \frac {2 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} \log \left (x\right ) - 30 \, a^{4} b x - 3 \, a^{5}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*x^5 + 15*a*b^4*x^4 + 60*a^2*b^3*x^3 + 60*a^3*b^2*x^2*log(x) - 30*a^4*b*x - 3*a^5)/x^2

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Sympy [A]
time = 0.07, size = 60, normalized size = 1.00 \begin {gather*} 10 a^{3} b^{2} \log {\left (x \right )} + 10 a^{2} b^{3} x + \frac {5 a b^{4} x^{2}}{2} + \frac {b^{5} x^{3}}{3} + \frac {- a^{5} - 10 a^{4} b x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**3,x)

[Out]

10*a**3*b**2*log(x) + 10*a**2*b**3*x + 5*a*b**4*x**2/2 + b**5*x**3/3 + (-a**5 - 10*a**4*b*x)/(2*x**2)

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Giac [A]
time = 2.02, size = 54, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, b^{5} x^{3} + \frac {5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) - \frac {10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^3,x, algorithm="giac")

[Out]

1/3*b^5*x^3 + 5/2*a*b^4*x^2 + 10*a^2*b^3*x + 10*a^3*b^2*log(abs(x)) - 1/2*(10*a^4*b*x + a^5)/x^2

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.92 \begin {gather*} \frac {b^5\,x^3}{3}-\frac {\frac {a^5}{2}+5\,b\,x\,a^4}{x^2}+10\,a^2\,b^3\,x+\frac {5\,a\,b^4\,x^2}{2}+10\,a^3\,b^2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^3,x)

[Out]

(b^5*x^3)/3 - (a^5/2 + 5*a^4*b*x)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^2)/2 + 10*a^3*b^2*log(x)

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