\(\int \frac {(a+b x)^7}{x^7} \, dx\) [113]

Optimal result

Integrand size = 11, antiderivative size = 85 \[ \int \frac {(a+b x)^7}{x^7} \, dx=-\frac {a^7}{6 x^6}-\frac {7 a^6 b}{5 x^5}-\frac {21 a^5 b^2}{4 x^4}-\frac {35 a^4 b^3}{3 x^3}-\frac {35 a^3 b^4}{2 x^2}-\frac {21 a^2 b^5}{x}+b^7 x+7 a b^6 \log (x) \]

[Out]

-1/6*a^7/x^6-7/5*a^6*b/x^5-21/4*a^5*b^2/x^4-35/3*a^4*b^3/x^3-35/2*a^3*b^4/x^2-21*a^2*b^5/x+b^7*x+7*a*b^6*ln(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^7}{x^7} \, dx=-\frac {a^7}{6 x^6}-\frac {7 a^6 b}{5 x^5}-\frac {21 a^5 b^2}{4 x^4}-\frac {35 a^4 b^3}{3 x^3}-\frac {35 a^3 b^4}{2 x^2}-\frac {21 a^2 b^5}{x}+7 a b^6 \log (x)+b^7 x \]

[In]

Int[(a + b*x)^7/x^7,x]

[Out]

-1/6*a^7/x^6 - (7*a^6*b)/(5*x^5) - (21*a^5*b^2)/(4*x^4) - (35*a^4*b^3)/(3*x^3) - (35*a^3*b^4)/(2*x^2) - (21*a^
2*b^5)/x + b^7*x + 7*a*b^6*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*} \text {integral}& = \int \left (b^7+\frac {a^7}{x^7}+\frac {7 a^6 b}{x^6}+\frac {21 a^5 b^2}{x^5}+\frac {35 a^4 b^3}{x^4}+\frac {35 a^3 b^4}{x^3}+\frac {21 a^2 b^5}{x^2}+\frac {7 a b^6}{x}\right ) \, dx \\ & = -\frac {a^7}{6 x^6}-\frac {7 a^6 b}{5 x^5}-\frac {21 a^5 b^2}{4 x^4}-\frac {35 a^4 b^3}{3 x^3}-\frac {35 a^3 b^4}{2 x^2}-\frac {21 a^2 b^5}{x}+b^7 x+7 a b^6 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^7} \, dx=-\frac {a^7}{6 x^6}-\frac {7 a^6 b}{5 x^5}-\frac {21 a^5 b^2}{4 x^4}-\frac {35 a^4 b^3}{3 x^3}-\frac {35 a^3 b^4}{2 x^2}-\frac {21 a^2 b^5}{x}+b^7 x+7 a b^6 \log (x) \]

[In]

Integrate[(a + b*x)^7/x^7,x]

[Out]

-1/6*a^7/x^6 - (7*a^6*b)/(5*x^5) - (21*a^5*b^2)/(4*x^4) - (35*a^4*b^3)/(3*x^3) - (35*a^3*b^4)/(2*x^2) - (21*a^
2*b^5)/x + b^7*x + 7*a*b^6*Log[x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{7}}{6 x^{6}}-\frac {7 a^{6} b}{5 x^{5}}-\frac {21 a^{5} b^{2}}{4 x^{4}}-\frac {35 a^{4} b^{3}}{3 x^{3}}-\frac {35 a^{3} b^{4}}{2 x^{2}}-\frac {21 a^{2} b^{5}}{x}+b^{7} x +7 a \,b^{6} \ln \left (x \right )\)	\(76\)
risch	\(b^{7} x +\frac {-21 a^{2} b^{5} x^{5}-\frac {35}{2} a^{3} b^{4} x^{4}-\frac {35}{3} a^{4} b^{3} x^{3}-\frac {21}{4} a^{5} b^{2} x^{2}-\frac {7}{5} a^{6} b x -\frac {1}{6} a^{7}}{x^{6}}+7 a \,b^{6} \ln \left (x \right )\)	\(76\)
norman	\(\frac {b^{7} x^{7}-\frac {1}{6} a^{7}-21 a^{2} b^{5} x^{5}-\frac {35}{2} a^{3} b^{4} x^{4}-\frac {35}{3} a^{4} b^{3} x^{3}-\frac {21}{4} a^{5} b^{2} x^{2}-\frac {7}{5} a^{6} b x}{x^{6}}+7 a \,b^{6} \ln \left (x \right )\)	\(78\)
parallelrisch	\(\frac {420 a \,b^{6} \ln \left (x \right ) x^{6}+60 b^{7} x^{7}-1260 a^{2} b^{5} x^{5}-1050 a^{3} b^{4} x^{4}-700 a^{4} b^{3} x^{3}-315 a^{5} b^{2} x^{2}-84 a^{6} b x -10 a^{7}}{60 x^{6}}\)	\(82\)

[In]

int((b*x+a)^7/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^7/x^6-7/5*a^6*b/x^5-21/4*a^5*b^2/x^4-35/3*a^4*b^3/x^3-35/2*a^3*b^4/x^2-21*a^2*b^5/x+b^7*x+7*a*b^6*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^7}{x^7} \, dx=\frac {60 \, b^{7} x^{7} + 420 \, a b^{6} x^{6} \log \left (x\right ) - 1260 \, a^{2} b^{5} x^{5} - 1050 \, a^{3} b^{4} x^{4} - 700 \, a^{4} b^{3} x^{3} - 315 \, a^{5} b^{2} x^{2} - 84 \, a^{6} b x - 10 \, a^{7}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)^7/x^7,x, algorithm="fricas")

[Out]

1/60*(60*b^7*x^7 + 420*a*b^6*x^6*log(x) - 1260*a^2*b^5*x^5 - 1050*a^3*b^4*x^4 - 700*a^4*b^3*x^3 - 315*a^5*b^2*
x^2 - 84*a^6*b*x - 10*a^7)/x^6

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^7}{x^7} \, dx=7 a b^{6} \log {\left (x \right )} + b^{7} x + \frac {- 10 a^{7} - 84 a^{6} b x - 315 a^{5} b^{2} x^{2} - 700 a^{4} b^{3} x^{3} - 1050 a^{3} b^{4} x^{4} - 1260 a^{2} b^{5} x^{5}}{60 x^{6}} \]

[In]

integrate((b*x+a)**7/x**7,x)

[Out]

7*a*b**6*log(x) + b**7*x + (-10*a**7 - 84*a**6*b*x - 315*a**5*b**2*x**2 - 700*a**4*b**3*x**3 - 1050*a**3*b**4*
x**4 - 1260*a**2*b**5*x**5)/(60*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^7} \, dx=b^{7} x + 7 \, a b^{6} \log \left (x\right ) - \frac {1260 \, a^{2} b^{5} x^{5} + 1050 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 315 \, a^{5} b^{2} x^{2} + 84 \, a^{6} b x + 10 \, a^{7}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)^7/x^7,x, algorithm="maxima")

[Out]

b^7*x + 7*a*b^6*log(x) - 1/60*(1260*a^2*b^5*x^5 + 1050*a^3*b^4*x^4 + 700*a^4*b^3*x^3 + 315*a^5*b^2*x^2 + 84*a^
6*b*x + 10*a^7)/x^6

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^7}{x^7} \, dx=b^{7} x + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) - \frac {1260 \, a^{2} b^{5} x^{5} + 1050 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 315 \, a^{5} b^{2} x^{2} + 84 \, a^{6} b x + 10 \, a^{7}}{60 \, x^{6}} \]

[In]

integrate((b*x+a)^7/x^7,x, algorithm="giac")

[Out]

b^7*x + 7*a*b^6*log(abs(x)) - 1/60*(1260*a^2*b^5*x^5 + 1050*a^3*b^4*x^4 + 700*a^4*b^3*x^3 + 315*a^5*b^2*x^2 +
84*a^6*b*x + 10*a^7)/x^6

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^7}{x^7} \, dx=-\frac {10\,a^7-60\,b^7\,x^7+315\,a^5\,b^2\,x^2+700\,a^4\,b^3\,x^3+1050\,a^3\,b^4\,x^4+1260\,a^2\,b^5\,x^5+84\,a^6\,b\,x-420\,a\,b^6\,x^6\,\ln \left (x\right )}{60\,x^6} \]

[In]

int((a + b*x)^7/x^7,x)

[Out]

-(10*a^7 - 60*b^7*x^7 + 315*a^5*b^2*x^2 + 700*a^4*b^3*x^3 + 1050*a^3*b^4*x^4 + 1260*a^2*b^5*x^5 + 84*a^6*b*x -
420*a*b^6*x^6*log(x))/(60*x^6)