\(\int \frac {(a+b x)^7}{x^3} \, dx\) [109]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.03 (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} \[ \int \frac {(a+b x)^7}{x^3} \, dx=-\frac {a^7}{2 x^2}-\frac {7 a^6 b}{x}+21 a^5 b^2 \log (x)+35 a^4 b^3 x+\frac {35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+\frac {7}{4} a b^6 x^4+\frac {b^7 x^5}{5} \]

[In]

Int[(a + b*x)^7/x^3,x]

[Out]

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

Mathematica [A] (verified)

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

[In]

Integrate[(a + b*x)^7/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92

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

[In]

int((b*x+a)^7/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

integrate((b*x+a)^7/x^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^7}{x^3} \, dx=21 a^{5} b^{2} \log {\left (x \right )} + 35 a^{4} b^{3} x + \frac {35 a^{3} b^{4} x^{2}}{2} + 7 a^{2} b^{5} x^{3} + \frac {7 a b^{6} x^{4}}{4} + \frac {b^{7} x^{5}}{5} + \frac {- a^{7} - 14 a^{6} b x}{2 x^{2}} \]

[In]

integrate((b*x+a)**7/x**3,x)

[Out]

21*a**5*b**2*log(x) + 35*a**4*b**3*x + 35*a**3*b**4*x**2/2 + 7*a**2*b**5*x**3 + 7*a*b**6*x**4/4 + b**7*x**5/5
+ (-a**7 - 14*a**6*b*x)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^3} \, dx=\frac {1}{5} \, b^{7} x^{5} + \frac {7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac {35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2} \log \left (x\right ) - \frac {14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)^7/x^3,x, algorithm="maxima")

[Out]

1/5*b^7*x^5 + 7/4*a*b^6*x^4 + 7*a^2*b^5*x^3 + 35/2*a^3*b^4*x^2 + 35*a^4*b^3*x + 21*a^5*b^2*log(x) - 1/2*(14*a^
6*b*x + a^7)/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^7}{x^3} \, dx=\frac {1}{5} \, b^{7} x^{5} + \frac {7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac {35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2} \log \left ({\left | x \right |}\right ) - \frac {14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)^7/x^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^7}{x^3} \, dx=\frac {b^7\,x^5}{5}-\frac {\frac {a^7}{2}+7\,b\,x\,a^6}{x^2}+35\,a^4\,b^3\,x+\frac {7\,a\,b^6\,x^4}{4}+\frac {35\,a^3\,b^4\,x^2}{2}+7\,a^2\,b^5\,x^3+21\,a^5\,b^2\,\ln \left (x\right ) \]

[In]

int((a + b*x)^7/x^3,x)

[Out]

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