\(\int \frac {(a+b x)^6}{x^{10}} \, dx\) [245]

Optimal result

Integrand size = 11, antiderivative size = 56 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {(a+b x)^7}{9 a x^9}+\frac {b (a+b x)^7}{36 a^2 x^8}-\frac {b^2 (a+b x)^7}{252 a^3 x^7} \]

[Out]

-1/9*(b*x+a)^7/a/x^9+1/36*b*(b*x+a)^7/a^2/x^8-1/252*b^2*(b*x+a)^7/a^3/x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {b^2 (a+b x)^7}{252 a^3 x^7}+\frac {b (a+b x)^7}{36 a^2 x^8}-\frac {(a+b x)^7}{9 a x^9} \]

[In]

Int[(a + b*x)^6/x^10,x]

[Out]

-1/9*(a + b*x)^7/(a*x^9) + (b*(a + b*x)^7)/(36*a^2*x^8) - (b^2*(a + b*x)^7)/(252*a^3*x^7)

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)^7}{9 a x^9}-\frac {(2 b) \int \frac {(a+b x)^6}{x^9} \, dx}{9 a} \\ & = -\frac {(a+b x)^7}{9 a x^9}+\frac {b (a+b x)^7}{36 a^2 x^8}+\frac {b^2 \int \frac {(a+b x)^6}{x^8} \, dx}{36 a^2} \\ & = -\frac {(a+b x)^7}{9 a x^9}+\frac {b (a+b x)^7}{36 a^2 x^8}-\frac {b^2 (a+b x)^7}{252 a^3 x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {a^6}{9 x^9}-\frac {3 a^5 b}{4 x^8}-\frac {15 a^4 b^2}{7 x^7}-\frac {10 a^3 b^3}{3 x^6}-\frac {3 a^2 b^4}{x^5}-\frac {3 a b^5}{2 x^4}-\frac {b^6}{3 x^3} \]

[In]

Integrate[(a + b*x)^6/x^10,x]

[Out]

-1/9*a^6/x^9 - (3*a^5*b)/(4*x^8) - (15*a^4*b^2)/(7*x^7) - (10*a^3*b^3)/(3*x^6) - (3*a^2*b^4)/x^5 - (3*a*b^5)/(
2*x^4) - b^6/(3*x^3)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21

method	result	size
norman	\(\frac {-\frac {1}{3} b^{6} x^{6}-\frac {3}{2} a \,x^{5} b^{5}-3 a^{2} x^{4} b^{4}-\frac {10}{3} a^{3} x^{3} b^{3}-\frac {15}{7} a^{4} x^{2} b^{2}-\frac {3}{4} a^{5} x b -\frac {1}{9} a^{6}}{x^{9}}\)	\(68\)
risch	\(\frac {-\frac {1}{3} b^{6} x^{6}-\frac {3}{2} a \,x^{5} b^{5}-3 a^{2} x^{4} b^{4}-\frac {10}{3} a^{3} x^{3} b^{3}-\frac {15}{7} a^{4} x^{2} b^{2}-\frac {3}{4} a^{5} x b -\frac {1}{9} a^{6}}{x^{9}}\)	\(68\)
gosper	\(-\frac {84 b^{6} x^{6}+378 a \,x^{5} b^{5}+756 a^{2} x^{4} b^{4}+840 a^{3} x^{3} b^{3}+540 a^{4} x^{2} b^{2}+189 a^{5} x b +28 a^{6}}{252 x^{9}}\)	\(69\)
default	\(-\frac {10 a^{3} b^{3}}{3 x^{6}}-\frac {15 a^{4} b^{2}}{7 x^{7}}-\frac {a^{6}}{9 x^{9}}-\frac {b^{6}}{3 x^{3}}-\frac {3 a \,b^{5}}{2 x^{4}}-\frac {3 a^{2} b^{4}}{x^{5}}-\frac {3 a^{5} b}{4 x^{8}}\)	\(69\)
parallelrisch	\(\frac {-84 b^{6} x^{6}-378 a \,x^{5} b^{5}-756 a^{2} x^{4} b^{4}-840 a^{3} x^{3} b^{3}-540 a^{4} x^{2} b^{2}-189 a^{5} x b -28 a^{6}}{252 x^{9}}\)	\(69\)

[In]

int((b*x+a)^6/x^10,x,method=_RETURNVERBOSE)

[Out]

1/x^9*(-1/3*b^6*x^6-3/2*a*x^5*b^5-3*a^2*x^4*b^4-10/3*a^3*x^3*b^3-15/7*a^4*x^2*b^2-3/4*a^5*x*b-1/9*a^6)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

[In]

integrate((b*x+a)^6/x^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=\frac {- 28 a^{6} - 189 a^{5} b x - 540 a^{4} b^{2} x^{2} - 840 a^{3} b^{3} x^{3} - 756 a^{2} b^{4} x^{4} - 378 a b^{5} x^{5} - 84 b^{6} x^{6}}{252 x^{9}} \]

[In]

integrate((b*x+a)**6/x**10,x)

[Out]

(-28*a**6 - 189*a**5*b*x - 540*a**4*b**2*x**2 - 840*a**3*b**3*x**3 - 756*a**2*b**4*x**4 - 378*a*b**5*x**5 - 84
*b**6*x**6)/(252*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

[In]

integrate((b*x+a)^6/x^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

[In]

integrate((b*x+a)^6/x^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4*b^2*x^2 + 189*a^5*b*x + 28*a^
6)/x^9

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^6}{x^{10}} \, dx=-\frac {\frac {a^6}{9}+\frac {3\,a^5\,b\,x}{4}+\frac {15\,a^4\,b^2\,x^2}{7}+\frac {10\,a^3\,b^3\,x^3}{3}+3\,a^2\,b^4\,x^4+\frac {3\,a\,b^5\,x^5}{2}+\frac {b^6\,x^6}{3}}{x^9} \]

[In]

int((a + b*x)^6/x^10,x)

[Out]

-(a^6/9 + (b^6*x^6)/3 + (3*a*b^5*x^5)/2 + (15*a^4*b^2*x^2)/7 + (10*a^3*b^3*x^3)/3 + 3*a^2*b^4*x^4 + (3*a^5*b*x
)/4)/x^9