\(\int \frac {(a+b x)^7}{x^{10}} \, dx\) [116]

Optimal result

Integrand size = 11, antiderivative size = 36 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {(a+b x)^8}{9 a x^9}+\frac {b (a+b x)^8}{72 a^2 x^8} \]

[Out]

-1/9*(b*x+a)^8/a/x^9+1/72*b*(b*x+a)^8/a^2/x^8

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=\frac {b (a+b x)^8}{72 a^2 x^8}-\frac {(a+b x)^8}{9 a x^9} \]

[In]

Int[(a + b*x)^7/x^10,x]

[Out]

-1/9*(a + b*x)^8/(a*x^9) + (b*(a + b*x)^8)/(72*a^2*x^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{9 a x^9}-\frac {b \int \frac {(a+b x)^7}{x^9} \, dx}{9 a} \\ & = -\frac {(a+b x)^8}{9 a x^9}+\frac {b (a+b x)^8}{72 a^2 x^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(36)=72\).

Time = 0.00 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {a^7}{9 x^9}-\frac {7 a^6 b}{8 x^8}-\frac {3 a^5 b^2}{x^7}-\frac {35 a^4 b^3}{6 x^6}-\frac {7 a^3 b^4}{x^5}-\frac {21 a^2 b^5}{4 x^4}-\frac {7 a b^6}{3 x^3}-\frac {b^7}{2 x^2} \]

[In]

Integrate[(a + b*x)^7/x^10,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).

Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19

method	result	size
norman	\(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\)	\(79\)
risch	\(\frac {-\frac {1}{2} b^{7} x^{7}-\frac {7}{3} a \,b^{6} x^{6}-\frac {21}{4} a^{2} b^{5} x^{5}-7 a^{3} b^{4} x^{4}-\frac {35}{6} a^{4} b^{3} x^{3}-3 a^{5} b^{2} x^{2}-\frac {7}{8} a^{6} b x -\frac {1}{9} a^{7}}{x^{9}}\)	\(79\)
gosper	\(-\frac {36 b^{7} x^{7}+168 a \,b^{6} x^{6}+378 a^{2} b^{5} x^{5}+504 a^{3} b^{4} x^{4}+420 a^{4} b^{3} x^{3}+216 a^{5} b^{2} x^{2}+63 a^{6} b x +8 a^{7}}{72 x^{9}}\)	\(80\)
default	\(-\frac {35 a^{4} b^{3}}{6 x^{6}}-\frac {3 a^{5} b^{2}}{x^{7}}-\frac {a^{7}}{9 x^{9}}-\frac {7 a \,b^{6}}{3 x^{3}}-\frac {b^{7}}{2 x^{2}}-\frac {21 a^{2} b^{5}}{4 x^{4}}-\frac {7 a^{3} b^{4}}{x^{5}}-\frac {7 a^{6} b}{8 x^{8}}\)	\(80\)
parallelrisch	\(\frac {-36 b^{7} x^{7}-168 a \,b^{6} x^{6}-378 a^{2} b^{5} x^{5}-504 a^{3} b^{4} x^{4}-420 a^{4} b^{3} x^{3}-216 a^{5} b^{2} x^{2}-63 a^{6} b x -8 a^{7}}{72 x^{9}}\)	\(80\)

[In]

int((b*x+a)^7/x^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]

[In]

integrate((b*x+a)^7/x^10,x, algorithm="fricas")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=\frac {- 8 a^{7} - 63 a^{6} b x - 216 a^{5} b^{2} x^{2} - 420 a^{4} b^{3} x^{3} - 504 a^{3} b^{4} x^{4} - 378 a^{2} b^{5} x^{5} - 168 a b^{6} x^{6} - 36 b^{7} x^{7}}{72 x^{9}} \]

[In]

integrate((b*x+a)**7/x**10,x)

[Out]

(-8*a**7 - 63*a**6*b*x - 216*a**5*b**2*x**2 - 420*a**4*b**3*x**3 - 504*a**3*b**4*x**4 - 378*a**2*b**5*x**5 - 1
68*a*b**6*x**6 - 36*b**7*x**7)/(72*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).

Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]

[In]

integrate((b*x+a)^7/x^10,x, algorithm="maxima")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (32) = 64\).

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 168 \, a b^{6} x^{6} + 378 \, a^{2} b^{5} x^{5} + 504 \, a^{3} b^{4} x^{4} + 420 \, a^{4} b^{3} x^{3} + 216 \, a^{5} b^{2} x^{2} + 63 \, a^{6} b x + 8 \, a^{7}}{72 \, x^{9}} \]

[In]

integrate((b*x+a)^7/x^10,x, algorithm="giac")

[Out]

-1/72*(36*b^7*x^7 + 168*a*b^6*x^6 + 378*a^2*b^5*x^5 + 504*a^3*b^4*x^4 + 420*a^4*b^3*x^3 + 216*a^5*b^2*x^2 + 63
*a^6*b*x + 8*a^7)/x^9

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x)^7}{x^{10}} \, dx=-\frac {\left (8\,a-b\,x\right )\,{\left (a+b\,x\right )}^8}{72\,a^2\,x^9} \]

[In]

int((a + b*x)^7/x^10,x)

[Out]

-((8*a - b*x)*(a + b*x)^8)/(72*a^2*x^9)