\(\int \frac {(a+b x)^7}{x^{14}} \, dx\) [120]

Optimal result

Integrand size = 11, antiderivative size = 93 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \]

[Out]

-1/13*a^7/x^13-7/12*a^6*b/x^12-21/11*a^5*b^2/x^11-7/2*a^4*b^3/x^10-35/9*a^3*b^4/x^9-21/8*a^2*b^5/x^8-a*b^6/x^7
-1/6*b^7/x^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 93, 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} \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \]

[In]

Int[(a + b*x)^7/x^14,x]

[Out]

-1/13*a^7/x^13 - (7*a^6*b)/(12*x^12) - (21*a^5*b^2)/(11*x^11) - (7*a^4*b^3)/(2*x^10) - (35*a^3*b^4)/(9*x^9) -
(21*a^2*b^5)/(8*x^8) - (a*b^6)/x^7 - b^7/(6*x^6)

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^{14}}+\frac {7 a^6 b}{x^{13}}+\frac {21 a^5 b^2}{x^{12}}+\frac {35 a^4 b^3}{x^{11}}+\frac {35 a^3 b^4}{x^{10}}+\frac {21 a^2 b^5}{x^9}+\frac {7 a b^6}{x^8}+\frac {b^7}{x^7}\right ) \, dx \\ & = -\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {a^7}{13 x^{13}}-\frac {7 a^6 b}{12 x^{12}}-\frac {21 a^5 b^2}{11 x^{11}}-\frac {7 a^4 b^3}{2 x^{10}}-\frac {35 a^3 b^4}{9 x^9}-\frac {21 a^2 b^5}{8 x^8}-\frac {a b^6}{x^7}-\frac {b^7}{6 x^6} \]

[In]

Integrate[(a + b*x)^7/x^14,x]

[Out]

-1/13*a^7/x^13 - (7*a^6*b)/(12*x^12) - (21*a^5*b^2)/(11*x^11) - (7*a^4*b^3)/(2*x^10) - (35*a^3*b^4)/(9*x^9) -
(21*a^2*b^5)/(8*x^8) - (a*b^6)/x^7 - b^7/(6*x^6)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85

method	result	size
norman	\(\frac {-\frac {1}{6} b^{7} x^{7}-a \,b^{6} x^{6}-\frac {21}{8} a^{2} b^{5} x^{5}-\frac {35}{9} a^{3} b^{4} x^{4}-\frac {7}{2} a^{4} b^{3} x^{3}-\frac {21}{11} a^{5} b^{2} x^{2}-\frac {7}{12} a^{6} b x -\frac {1}{13} a^{7}}{x^{13}}\)	\(79\)
risch	\(\frac {-\frac {1}{6} b^{7} x^{7}-a \,b^{6} x^{6}-\frac {21}{8} a^{2} b^{5} x^{5}-\frac {35}{9} a^{3} b^{4} x^{4}-\frac {7}{2} a^{4} b^{3} x^{3}-\frac {21}{11} a^{5} b^{2} x^{2}-\frac {7}{12} a^{6} b x -\frac {1}{13} a^{7}}{x^{13}}\)	\(79\)
gosper	\(-\frac {1716 b^{7} x^{7}+10296 a \,b^{6} x^{6}+27027 a^{2} b^{5} x^{5}+40040 a^{3} b^{4} x^{4}+36036 a^{4} b^{3} x^{3}+19656 a^{5} b^{2} x^{2}+6006 a^{6} b x +792 a^{7}}{10296 x^{13}}\)	\(80\)
default	\(-\frac {a^{7}}{13 x^{13}}-\frac {7 a^{6} b}{12 x^{12}}-\frac {21 a^{5} b^{2}}{11 x^{11}}-\frac {7 a^{4} b^{3}}{2 x^{10}}-\frac {35 a^{3} b^{4}}{9 x^{9}}-\frac {21 a^{2} b^{5}}{8 x^{8}}-\frac {a \,b^{6}}{x^{7}}-\frac {b^{7}}{6 x^{6}}\)	\(80\)
parallelrisch	\(\frac {-1716 b^{7} x^{7}-10296 a \,b^{6} x^{6}-27027 a^{2} b^{5} x^{5}-40040 a^{3} b^{4} x^{4}-36036 a^{4} b^{3} x^{3}-19656 a^{5} b^{2} x^{2}-6006 a^{6} b x -792 a^{7}}{10296 x^{13}}\)	\(80\)

[In]

int((b*x+a)^7/x^14,x,method=_RETURNVERBOSE)

[Out]

1/x^13*(-1/6*b^7*x^7-a*b^6*x^6-21/8*a^2*b^5*x^5-35/9*a^3*b^4*x^4-7/2*a^4*b^3*x^3-21/11*a^5*b^2*x^2-7/12*a^6*b*
x-1/13*a^7)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^7/x^14,x, algorithm="fricas")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=\frac {- 792 a^{7} - 6006 a^{6} b x - 19656 a^{5} b^{2} x^{2} - 36036 a^{4} b^{3} x^{3} - 40040 a^{3} b^{4} x^{4} - 27027 a^{2} b^{5} x^{5} - 10296 a b^{6} x^{6} - 1716 b^{7} x^{7}}{10296 x^{13}} \]

[In]

integrate((b*x+a)**7/x**14,x)

[Out]

(-792*a**7 - 6006*a**6*b*x - 19656*a**5*b**2*x**2 - 36036*a**4*b**3*x**3 - 40040*a**3*b**4*x**4 - 27027*a**2*b
**5*x**5 - 10296*a*b**6*x**6 - 1716*b**7*x**7)/(10296*x**13)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^7/x^14,x, algorithm="maxima")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^7/x^14,x, algorithm="giac")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4 + 36036*a^4*b^3*x^3 + 19656*a
^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^7}{x^{14}} \, dx=-\frac {\frac {a^7}{13}+\frac {7\,a^6\,b\,x}{12}+\frac {21\,a^5\,b^2\,x^2}{11}+\frac {7\,a^4\,b^3\,x^3}{2}+\frac {35\,a^3\,b^4\,x^4}{9}+\frac {21\,a^2\,b^5\,x^5}{8}+a\,b^6\,x^6+\frac {b^7\,x^7}{6}}{x^{13}} \]

[In]

int((a + b*x)^7/x^14,x)

[Out]

-(a^7/13 + (b^7*x^7)/6 + a*b^6*x^6 + (21*a^5*b^2*x^2)/11 + (7*a^4*b^3*x^3)/2 + (35*a^3*b^4*x^4)/9 + (21*a^2*b^
5*x^5)/8 + (7*a^6*b*x)/12)/x^13