\(\int \frac {(a+b x)^7}{x^{15}} \, dx\) [121]

Optimal result

Integrand size = 11, antiderivative size = 95 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {a^7}{14 x^{14}}-\frac {7 a^6 b}{13 x^{13}}-\frac {7 a^5 b^2}{4 x^{12}}-\frac {35 a^4 b^3}{11 x^{11}}-\frac {7 a^3 b^4}{2 x^{10}}-\frac {7 a^2 b^5}{3 x^9}-\frac {7 a b^6}{8 x^8}-\frac {b^7}{7 x^7} \]

[Out]

-1/14*a^7/x^14-7/13*a^6*b/x^13-7/4*a^5*b^2/x^12-35/11*a^4*b^3/x^11-7/2*a^3*b^4/x^10-7/3*a^2*b^5/x^9-7/8*a*b^6/
x^8-1/7*b^7/x^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 95, 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} \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {a^7}{14 x^{14}}-\frac {7 a^6 b}{13 x^{13}}-\frac {7 a^5 b^2}{4 x^{12}}-\frac {35 a^4 b^3}{11 x^{11}}-\frac {7 a^3 b^4}{2 x^{10}}-\frac {7 a^2 b^5}{3 x^9}-\frac {7 a b^6}{8 x^8}-\frac {b^7}{7 x^7} \]

[In]

Int[(a + b*x)^7/x^15,x]

[Out]

-1/14*a^7/x^14 - (7*a^6*b)/(13*x^13) - (7*a^5*b^2)/(4*x^12) - (35*a^4*b^3)/(11*x^11) - (7*a^3*b^4)/(2*x^10) -
(7*a^2*b^5)/(3*x^9) - (7*a*b^6)/(8*x^8) - b^7/(7*x^7)

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^{15}}+\frac {7 a^6 b}{x^{14}}+\frac {21 a^5 b^2}{x^{13}}+\frac {35 a^4 b^3}{x^{12}}+\frac {35 a^3 b^4}{x^{11}}+\frac {21 a^2 b^5}{x^{10}}+\frac {7 a b^6}{x^9}+\frac {b^7}{x^8}\right ) \, dx \\ & = -\frac {a^7}{14 x^{14}}-\frac {7 a^6 b}{13 x^{13}}-\frac {7 a^5 b^2}{4 x^{12}}-\frac {35 a^4 b^3}{11 x^{11}}-\frac {7 a^3 b^4}{2 x^{10}}-\frac {7 a^2 b^5}{3 x^9}-\frac {7 a b^6}{8 x^8}-\frac {b^7}{7 x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {a^7}{14 x^{14}}-\frac {7 a^6 b}{13 x^{13}}-\frac {7 a^5 b^2}{4 x^{12}}-\frac {35 a^4 b^3}{11 x^{11}}-\frac {7 a^3 b^4}{2 x^{10}}-\frac {7 a^2 b^5}{3 x^9}-\frac {7 a b^6}{8 x^8}-\frac {b^7}{7 x^7} \]

[In]

Integrate[(a + b*x)^7/x^15,x]

[Out]

-1/14*a^7/x^14 - (7*a^6*b)/(13*x^13) - (7*a^5*b^2)/(4*x^12) - (35*a^4*b^3)/(11*x^11) - (7*a^3*b^4)/(2*x^10) -
(7*a^2*b^5)/(3*x^9) - (7*a*b^6)/(8*x^8) - b^7/(7*x^7)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83

method	result	size
norman	\(\frac {-\frac {1}{14} a^{7}-\frac {7}{13} a^{6} b x -\frac {7}{4} a^{5} b^{2} x^{2}-\frac {35}{11} a^{4} b^{3} x^{3}-\frac {7}{2} a^{3} b^{4} x^{4}-\frac {7}{3} a^{2} b^{5} x^{5}-\frac {7}{8} a \,b^{6} x^{6}-\frac {1}{7} b^{7} x^{7}}{x^{14}}\)	\(79\)
risch	\(\frac {-\frac {1}{14} a^{7}-\frac {7}{13} a^{6} b x -\frac {7}{4} a^{5} b^{2} x^{2}-\frac {35}{11} a^{4} b^{3} x^{3}-\frac {7}{2} a^{3} b^{4} x^{4}-\frac {7}{3} a^{2} b^{5} x^{5}-\frac {7}{8} a \,b^{6} x^{6}-\frac {1}{7} b^{7} x^{7}}{x^{14}}\)	\(79\)
gosper	\(-\frac {3432 b^{7} x^{7}+21021 a \,b^{6} x^{6}+56056 a^{2} b^{5} x^{5}+84084 a^{3} b^{4} x^{4}+76440 a^{4} b^{3} x^{3}+42042 a^{5} b^{2} x^{2}+12936 a^{6} b x +1716 a^{7}}{24024 x^{14}}\)	\(80\)
default	\(-\frac {a^{7}}{14 x^{14}}-\frac {7 a^{6} b}{13 x^{13}}-\frac {7 a^{5} b^{2}}{4 x^{12}}-\frac {35 a^{4} b^{3}}{11 x^{11}}-\frac {7 a^{3} b^{4}}{2 x^{10}}-\frac {7 a^{2} b^{5}}{3 x^{9}}-\frac {7 a \,b^{6}}{8 x^{8}}-\frac {b^{7}}{7 x^{7}}\)	\(80\)
parallelrisch	\(\frac {-3432 b^{7} x^{7}-21021 a \,b^{6} x^{6}-56056 a^{2} b^{5} x^{5}-84084 a^{3} b^{4} x^{4}-76440 a^{4} b^{3} x^{3}-42042 a^{5} b^{2} x^{2}-12936 a^{6} b x -1716 a^{7}}{24024 x^{14}}\)	\(80\)

[In]

int((b*x+a)^7/x^15,x,method=_RETURNVERBOSE)

[Out]

1/x^14*(-1/14*a^7-7/13*a^6*b*x-7/4*a^5*b^2*x^2-35/11*a^4*b^3*x^3-7/2*a^3*b^4*x^4-7/3*a^2*b^5*x^5-7/8*a*b^6*x^6
-1/7*b^7*x^7)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

[In]

integrate((b*x+a)^7/x^15,x, algorithm="fricas")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4 + 76440*a^4*b^3*x^3 + 42042*a
^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=\frac {- 1716 a^{7} - 12936 a^{6} b x - 42042 a^{5} b^{2} x^{2} - 76440 a^{4} b^{3} x^{3} - 84084 a^{3} b^{4} x^{4} - 56056 a^{2} b^{5} x^{5} - 21021 a b^{6} x^{6} - 3432 b^{7} x^{7}}{24024 x^{14}} \]

[In]

integrate((b*x+a)**7/x**15,x)

[Out]

(-1716*a**7 - 12936*a**6*b*x - 42042*a**5*b**2*x**2 - 76440*a**4*b**3*x**3 - 84084*a**3*b**4*x**4 - 56056*a**2
*b**5*x**5 - 21021*a*b**6*x**6 - 3432*b**7*x**7)/(24024*x**14)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

[In]

integrate((b*x+a)^7/x^15,x, algorithm="maxima")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4 + 76440*a^4*b^3*x^3 + 42042*a
^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

[In]

integrate((b*x+a)^7/x^15,x, algorithm="giac")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4 + 76440*a^4*b^3*x^3 + 42042*a
^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^7}{x^{15}} \, dx=-\frac {\frac {a^7}{14}+\frac {7\,a^6\,b\,x}{13}+\frac {7\,a^5\,b^2\,x^2}{4}+\frac {35\,a^4\,b^3\,x^3}{11}+\frac {7\,a^3\,b^4\,x^4}{2}+\frac {7\,a^2\,b^5\,x^5}{3}+\frac {7\,a\,b^6\,x^6}{8}+\frac {b^7\,x^7}{7}}{x^{14}} \]

[In]

int((a + b*x)^7/x^15,x)

[Out]

-(a^7/14 + (b^7*x^7)/7 + (7*a*b^6*x^6)/8 + (7*a^5*b^2*x^2)/4 + (35*a^4*b^3*x^3)/11 + (7*a^3*b^4*x^4)/2 + (7*a^
2*b^5*x^5)/3 + (7*a^6*b*x)/13)/x^14