\(\int \frac {(a+b x)^7}{x^8} \, dx\) [114]

Optimal result

Integrand size = 11, antiderivative size = 89 \[ \int \frac {(a+b x)^7}{x^8} \, dx=-\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 89, 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} \[ \int \frac {(a+b x)^7}{x^8} \, dx=-\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \]

[In]

Int[(a + b*x)^7/x^8,x]

[Out]

-1/7*a^7/x^7 - (7*a^6*b)/(6*x^6) - (21*a^5*b^2)/(5*x^5) - (35*a^4*b^3)/(4*x^4) - (35*a^3*b^4)/(3*x^3) - (21*a^
2*b^5)/(2*x^2) - (7*a*b^6)/x + b^7*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 (\frac {a^7}{x^8}+\frac {7 a^6 b}{x^7}+\frac {21 a^5 b^2}{x^6}+\frac {35 a^4 b^3}{x^5}+\frac {35 a^3 b^4}{x^4}+\frac {21 a^2 b^5}{x^3}+\frac {7 a b^6}{x^2}+\frac {b^7}{x}\right ) \, dx \\ & = -\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^8} \, dx=-\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \]

[In]

Integrate[(a + b*x)^7/x^8,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {a^{7}}{7 x^{7}}-\frac {7 a^{6} b}{6 x^{6}}-\frac {21 a^{5} b^{2}}{5 x^{5}}-\frac {35 a^{4} b^{3}}{4 x^{4}}-\frac {35 a^{3} b^{4}}{3 x^{3}}-\frac {21 a^{2} b^{5}}{2 x^{2}}-\frac {7 a \,b^{6}}{x}+b^{7} \ln \left (x \right )\)	\(78\)
norman	\(\frac {-\frac {1}{7} a^{7}-7 a \,b^{6} x^{6}-\frac {21}{2} a^{2} b^{5} x^{5}-\frac {35}{3} a^{3} b^{4} x^{4}-\frac {35}{4} a^{4} b^{3} x^{3}-\frac {21}{5} a^{5} b^{2} x^{2}-\frac {7}{6} a^{6} b x}{x^{7}}+b^{7} \ln \left (x \right )\)	\(78\)
risch	\(\frac {-\frac {1}{7} a^{7}-7 a \,b^{6} x^{6}-\frac {21}{2} a^{2} b^{5} x^{5}-\frac {35}{3} a^{3} b^{4} x^{4}-\frac {35}{4} a^{4} b^{3} x^{3}-\frac {21}{5} a^{5} b^{2} x^{2}-\frac {7}{6} a^{6} b x}{x^{7}}+b^{7} \ln \left (x \right )\)	\(78\)
parallelrisch	\(\frac {420 b^{7} \ln \left (x \right ) x^{7}-2940 a \,b^{6} x^{6}-4410 a^{2} b^{5} x^{5}-4900 a^{3} b^{4} x^{4}-3675 a^{4} b^{3} x^{3}-1764 a^{5} b^{2} x^{2}-490 a^{6} b x -60 a^{7}}{420 x^{7}}\)	\(82\)

[In]

int((b*x+a)^7/x^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^7}{x^8} \, dx=\frac {420 \, b^{7} x^{7} \log \left (x\right ) - 2940 \, a b^{6} x^{6} - 4410 \, a^{2} b^{5} x^{5} - 4900 \, a^{3} b^{4} x^{4} - 3675 \, a^{4} b^{3} x^{3} - 1764 \, a^{5} b^{2} x^{2} - 490 \, a^{6} b x - 60 \, a^{7}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^7/x^8,x, algorithm="fricas")

[Out]

1/420*(420*b^7*x^7*log(x) - 2940*a*b^6*x^6 - 4410*a^2*b^5*x^5 - 4900*a^3*b^4*x^4 - 3675*a^4*b^3*x^3 - 1764*a^5
*b^2*x^2 - 490*a^6*b*x - 60*a^7)/x^7

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^7}{x^8} \, dx=b^{7} \log {\left (x \right )} + \frac {- 60 a^{7} - 490 a^{6} b x - 1764 a^{5} b^{2} x^{2} - 3675 a^{4} b^{3} x^{3} - 4900 a^{3} b^{4} x^{4} - 4410 a^{2} b^{5} x^{5} - 2940 a b^{6} x^{6}}{420 x^{7}} \]

[In]

integrate((b*x+a)**7/x**8,x)

[Out]

b**7*log(x) + (-60*a**7 - 490*a**6*b*x - 1764*a**5*b**2*x**2 - 3675*a**4*b**3*x**3 - 4900*a**3*b**4*x**4 - 441
0*a**2*b**5*x**5 - 2940*a*b**6*x**6)/(420*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^7}{x^8} \, dx=b^{7} \log \left (x\right ) - \frac {2940 \, a b^{6} x^{6} + 4410 \, a^{2} b^{5} x^{5} + 4900 \, a^{3} b^{4} x^{4} + 3675 \, a^{4} b^{3} x^{3} + 1764 \, a^{5} b^{2} x^{2} + 490 \, a^{6} b x + 60 \, a^{7}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^7/x^8,x, algorithm="maxima")

[Out]

b^7*log(x) - 1/420*(2940*a*b^6*x^6 + 4410*a^2*b^5*x^5 + 4900*a^3*b^4*x^4 + 3675*a^4*b^3*x^3 + 1764*a^5*b^2*x^2
+ 490*a^6*b*x + 60*a^7)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^8} \, dx=b^{7} \log \left ({\left | x \right |}\right ) - \frac {2940 \, a b^{6} x^{6} + 4410 \, a^{2} b^{5} x^{5} + 4900 \, a^{3} b^{4} x^{4} + 3675 \, a^{4} b^{3} x^{3} + 1764 \, a^{5} b^{2} x^{2} + 490 \, a^{6} b x + 60 \, a^{7}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^7/x^8,x, algorithm="giac")

[Out]

b^7*log(abs(x)) - 1/420*(2940*a*b^6*x^6 + 4410*a^2*b^5*x^5 + 4900*a^3*b^4*x^4 + 3675*a^4*b^3*x^3 + 1764*a^5*b^
2*x^2 + 490*a^6*b*x + 60*a^7)/x^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^7}{x^8} \, dx=b^7\,\ln \left (x\right )-\frac {\frac {a^7}{7}+\frac {7\,a^6\,b\,x}{6}+\frac {21\,a^5\,b^2\,x^2}{5}+\frac {35\,a^4\,b^3\,x^3}{4}+\frac {35\,a^3\,b^4\,x^4}{3}+\frac {21\,a^2\,b^5\,x^5}{2}+7\,a\,b^6\,x^6}{x^7} \]

[In]

int((a + b*x)^7/x^8,x)

[Out]

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