\(\int \frac {(a+b x)^7}{x^5} \, dx\) [111]

Optimal result

Integrand size = 11, antiderivative size = 86 \[ \int \frac {(a+b x)^7}{x^5} \, dx=-\frac {a^7}{4 x^4}-\frac {7 a^6 b}{3 x^3}-\frac {21 a^5 b^2}{2 x^2}-\frac {35 a^4 b^3}{x}+21 a^2 b^5 x+\frac {7}{2} a b^6 x^2+\frac {b^7 x^3}{3}+35 a^3 b^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 86, 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} \[ \int \frac {(a+b x)^7}{x^5} \, dx=-\frac {a^7}{4 x^4}-\frac {7 a^6 b}{3 x^3}-\frac {21 a^5 b^2}{2 x^2}-\frac {35 a^4 b^3}{x}+35 a^3 b^4 \log (x)+21 a^2 b^5 x+\frac {7}{2} a b^6 x^2+\frac {b^7 x^3}{3} \]

[In]

Int[(a + b*x)^7/x^5,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^5} \, dx=-\frac {a^7}{4 x^4}-\frac {7 a^6 b}{3 x^3}-\frac {21 a^5 b^2}{2 x^2}-\frac {35 a^4 b^3}{x}+21 a^2 b^5 x+\frac {7}{2} a b^6 x^2+\frac {b^7 x^3}{3}+35 a^3 b^4 \log (x) \]

[In]

Integrate[(a + b*x)^7/x^5,x]

[Out]

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {a^{7}}{4 x^{4}}-\frac {7 a^{6} b}{3 x^{3}}-\frac {21 a^{5} b^{2}}{2 x^{2}}-\frac {35 a^{4} b^{3}}{x}+21 a^{2} b^{5} x +\frac {7 a \,b^{6} x^{2}}{2}+\frac {b^{7} x^{3}}{3}+35 a^{3} b^{4} \ln \left (x \right )\)	\(77\)
risch	\(\frac {b^{7} x^{3}}{3}+\frac {7 a \,b^{6} x^{2}}{2}+21 a^{2} b^{5} x +\frac {-35 a^{4} b^{3} x^{3}-\frac {21}{2} a^{5} b^{2} x^{2}-\frac {7}{3} a^{6} b x -\frac {1}{4} a^{7}}{x^{4}}+35 a^{3} b^{4} \ln \left (x \right )\)	\(77\)
norman	\(\frac {-\frac {1}{4} a^{7}+\frac {1}{3} b^{7} x^{7}+\frac {7}{2} a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}-35 a^{4} b^{3} x^{3}-\frac {21}{2} a^{5} b^{2} x^{2}-\frac {7}{3} a^{6} b x}{x^{4}}+35 a^{3} b^{4} \ln \left (x \right )\)	\(79\)
parallelrisch	\(\frac {4 b^{7} x^{7}+42 a \,b^{6} x^{6}+420 a^{3} b^{4} \ln \left (x \right ) x^{4}+252 a^{2} b^{5} x^{5}-420 a^{4} b^{3} x^{3}-126 a^{5} b^{2} x^{2}-28 a^{6} b x -3 a^{7}}{12 x^{4}}\)	\(82\)

[In]

int((b*x+a)^7/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^7}{x^5} \, dx=\frac {4 \, b^{7} x^{7} + 42 \, a b^{6} x^{6} + 252 \, a^{2} b^{5} x^{5} + 420 \, a^{3} b^{4} x^{4} \log \left (x\right ) - 420 \, a^{4} b^{3} x^{3} - 126 \, a^{5} b^{2} x^{2} - 28 \, a^{6} b x - 3 \, a^{7}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^7/x^5,x, algorithm="fricas")

[Out]

1/12*(4*b^7*x^7 + 42*a*b^6*x^6 + 252*a^2*b^5*x^5 + 420*a^3*b^4*x^4*log(x) - 420*a^4*b^3*x^3 - 126*a^5*b^2*x^2
- 28*a^6*b*x - 3*a^7)/x^4

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^7}{x^5} \, dx=35 a^{3} b^{4} \log {\left (x \right )} + 21 a^{2} b^{5} x + \frac {7 a b^{6} x^{2}}{2} + \frac {b^{7} x^{3}}{3} + \frac {- 3 a^{7} - 28 a^{6} b x - 126 a^{5} b^{2} x^{2} - 420 a^{4} b^{3} x^{3}}{12 x^{4}} \]

[In]

integrate((b*x+a)**7/x**5,x)

[Out]

35*a**3*b**4*log(x) + 21*a**2*b**5*x + 7*a*b**6*x**2/2 + b**7*x**3/3 + (-3*a**7 - 28*a**6*b*x - 126*a**5*b**2*
x**2 - 420*a**4*b**3*x**3)/(12*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^5} \, dx=\frac {1}{3} \, b^{7} x^{3} + \frac {7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left (x\right ) - \frac {420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^7/x^5,x, algorithm="maxima")

[Out]

1/3*b^7*x^3 + 7/2*a*b^6*x^2 + 21*a^2*b^5*x + 35*a^3*b^4*log(x) - 1/12*(420*a^4*b^3*x^3 + 126*a^5*b^2*x^2 + 28*
a^6*b*x + 3*a^7)/x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^7}{x^5} \, dx=\frac {1}{3} \, b^{7} x^{3} + \frac {7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left ({\left | x \right |}\right ) - \frac {420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \]

[In]

integrate((b*x+a)^7/x^5,x, algorithm="giac")

[Out]

1/3*b^7*x^3 + 7/2*a*b^6*x^2 + 21*a^2*b^5*x + 35*a^3*b^4*log(abs(x)) - 1/12*(420*a^4*b^3*x^3 + 126*a^5*b^2*x^2
+ 28*a^6*b*x + 3*a^7)/x^4

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^5} \, dx=\frac {b^7\,x^3}{3}-\frac {\frac {a^7}{4}+\frac {7\,a^6\,b\,x}{3}+\frac {21\,a^5\,b^2\,x^2}{2}+35\,a^4\,b^3\,x^3}{x^4}+21\,a^2\,b^5\,x+\frac {7\,a\,b^6\,x^2}{2}+35\,a^3\,b^4\,\ln \left (x\right ) \]

[In]

int((a + b*x)^7/x^5,x)

[Out]

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