\(\int \frac {(a+b x)^7}{x^6} \, dx\) [112]

Optimal result

Integrand size = 11, antiderivative size = 84 \[ \int \frac {(a+b x)^7}{x^6} \, dx=-\frac {a^7}{5 x^5}-\frac {7 a^6 b}{4 x^4}-\frac {7 a^5 b^2}{x^3}-\frac {35 a^4 b^3}{2 x^2}-\frac {35 a^3 b^4}{x}+7 a b^6 x+\frac {b^7 x^2}{2}+21 a^2 b^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, 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} \[ \int \frac {(a+b x)^7}{x^6} \, dx=-\frac {a^7}{5 x^5}-\frac {7 a^6 b}{4 x^4}-\frac {7 a^5 b^2}{x^3}-\frac {35 a^4 b^3}{2 x^2}-\frac {35 a^3 b^4}{x}+21 a^2 b^5 \log (x)+7 a b^6 x+\frac {b^7 x^2}{2} \]

[In]

Int[(a + b*x)^7/x^6,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^6} \, dx=-\frac {a^7}{5 x^5}-\frac {7 a^6 b}{4 x^4}-\frac {7 a^5 b^2}{x^3}-\frac {35 a^4 b^3}{2 x^2}-\frac {35 a^3 b^4}{x}+7 a b^6 x+\frac {b^7 x^2}{2}+21 a^2 b^5 \log (x) \]

[In]

Integrate[(a + b*x)^7/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a^{7}}{5 x^{5}}-\frac {7 a^{6} b}{4 x^{4}}-\frac {7 a^{5} b^{2}}{x^{3}}-\frac {35 a^{4} b^{3}}{2 x^{2}}-\frac {35 a^{3} b^{4}}{x}+7 a \,b^{6} x +\frac {b^{7} x^{2}}{2}+21 a^{2} b^{5} \ln \left (x \right )\)	\(77\)
risch	\(\frac {b^{7} x^{2}}{2}+7 a \,b^{6} x +\frac {-35 a^{3} b^{4} x^{4}-\frac {35}{2} a^{4} b^{3} x^{3}-7 a^{5} b^{2} x^{2}-\frac {7}{4} a^{6} b x -\frac {1}{5} a^{7}}{x^{5}}+21 a^{2} b^{5} \ln \left (x \right )\)	\(77\)
norman	\(\frac {-\frac {1}{5} a^{7}+\frac {1}{2} b^{7} x^{7}+7 a \,b^{6} x^{6}-35 a^{3} b^{4} x^{4}-\frac {35}{2} a^{4} b^{3} x^{3}-7 a^{5} b^{2} x^{2}-\frac {7}{4} a^{6} b x}{x^{5}}+21 a^{2} b^{5} \ln \left (x \right )\)	\(79\)
parallelrisch	\(\frac {10 b^{7} x^{7}+420 a^{2} b^{5} \ln \left (x \right ) x^{5}+140 a \,b^{6} x^{6}-700 a^{3} b^{4} x^{4}-350 a^{4} b^{3} x^{3}-140 a^{5} b^{2} x^{2}-35 a^{6} b x -4 a^{7}}{20 x^{5}}\)	\(82\)

[In]

int((b*x+a)^7/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^7}{x^6} \, dx=\frac {10 \, b^{7} x^{7} + 140 \, a b^{6} x^{6} + 420 \, a^{2} b^{5} x^{5} \log \left (x\right ) - 700 \, a^{3} b^{4} x^{4} - 350 \, a^{4} b^{3} x^{3} - 140 \, a^{5} b^{2} x^{2} - 35 \, a^{6} b x - 4 \, a^{7}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^7/x^6,x, algorithm="fricas")

[Out]

1/20*(10*b^7*x^7 + 140*a*b^6*x^6 + 420*a^2*b^5*x^5*log(x) - 700*a^3*b^4*x^4 - 350*a^4*b^3*x^3 - 140*a^5*b^2*x^
2 - 35*a^6*b*x - 4*a^7)/x^5

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^7}{x^6} \, dx=21 a^{2} b^{5} \log {\left (x \right )} + 7 a b^{6} x + \frac {b^{7} x^{2}}{2} + \frac {- 4 a^{7} - 35 a^{6} b x - 140 a^{5} b^{2} x^{2} - 350 a^{4} b^{3} x^{3} - 700 a^{3} b^{4} x^{4}}{20 x^{5}} \]

[In]

integrate((b*x+a)**7/x**6,x)

[Out]

21*a**2*b**5*log(x) + 7*a*b**6*x + b**7*x**2/2 + (-4*a**7 - 35*a**6*b*x - 140*a**5*b**2*x**2 - 350*a**4*b**3*x
**3 - 700*a**3*b**4*x**4)/(20*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^7}{x^6} \, dx=\frac {1}{2} \, b^{7} x^{2} + 7 \, a b^{6} x + 21 \, a^{2} b^{5} \log \left (x\right ) - \frac {700 \, a^{3} b^{4} x^{4} + 350 \, a^{4} b^{3} x^{3} + 140 \, a^{5} b^{2} x^{2} + 35 \, a^{6} b x + 4 \, a^{7}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^7/x^6,x, algorithm="maxima")

[Out]

1/2*b^7*x^2 + 7*a*b^6*x + 21*a^2*b^5*log(x) - 1/20*(700*a^3*b^4*x^4 + 350*a^4*b^3*x^3 + 140*a^5*b^2*x^2 + 35*a
^6*b*x + 4*a^7)/x^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^7}{x^6} \, dx=\frac {1}{2} \, b^{7} x^{2} + 7 \, a b^{6} x + 21 \, a^{2} b^{5} \log \left ({\left | x \right |}\right ) - \frac {700 \, a^{3} b^{4} x^{4} + 350 \, a^{4} b^{3} x^{3} + 140 \, a^{5} b^{2} x^{2} + 35 \, a^{6} b x + 4 \, a^{7}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^7/x^6,x, algorithm="giac")

[Out]

1/2*b^7*x^2 + 7*a*b^6*x + 21*a^2*b^5*log(abs(x)) - 1/20*(700*a^3*b^4*x^4 + 350*a^4*b^3*x^3 + 140*a^5*b^2*x^2 +
35*a^6*b*x + 4*a^7)/x^5

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^7}{x^6} \, dx=\frac {b^7\,x^2}{2}-\frac {\frac {a^7}{5}+\frac {7\,a^6\,b\,x}{4}+7\,a^5\,b^2\,x^2+\frac {35\,a^4\,b^3\,x^3}{2}+35\,a^3\,b^4\,x^4}{x^5}+21\,a^2\,b^5\,\ln \left (x\right )+7\,a\,b^6\,x \]

[In]

int((a + b*x)^7/x^6,x)

[Out]

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