\(\int \frac {(a+b x)^{10}}{x^{13}} \, dx\) [147]

Optimal result

Integrand size = 11, antiderivative size = 36 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {(a+b x)^{11}}{12 a x^{12}}+\frac {b (a+b x)^{11}}{132 a^2 x^{11}} \]

[Out]

-1/12*(b*x+a)^11/a/x^12+1/132*b*(b*x+a)^11/a^2/x^11

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} \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}} \]

[In]

Int[(a + b*x)^10/x^13,x]

[Out]

-1/12*(a + b*x)^11/(a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^11)

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)^{11}}{12 a x^{12}}-\frac {b \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{12 a} \\ & = -\frac {(a+b x)^{11}}{12 a x^{12}}+\frac {b (a+b x)^{11}}{132 a^2 x^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(36)=72\).

Time = 0.00 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.56 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {a^{10}}{12 x^{12}}-\frac {10 a^9 b}{11 x^{11}}-\frac {9 a^8 b^2}{2 x^{10}}-\frac {40 a^7 b^3}{3 x^9}-\frac {105 a^6 b^4}{4 x^8}-\frac {36 a^5 b^5}{x^7}-\frac {35 a^4 b^6}{x^6}-\frac {24 a^3 b^7}{x^5}-\frac {45 a^2 b^8}{4 x^4}-\frac {10 a b^9}{3 x^3}-\frac {b^{10}}{2 x^2} \]

[In]

Integrate[(a + b*x)^10/x^13,x]

[Out]

-1/12*a^10/x^12 - (10*a^9*b)/(11*x^11) - (9*a^8*b^2)/(2*x^10) - (40*a^7*b^3)/(3*x^9) - (105*a^6*b^4)/(4*x^8) -
(36*a^5*b^5)/x^7 - (35*a^4*b^6)/x^6 - (24*a^3*b^7)/x^5 - (45*a^2*b^8)/(4*x^4) - (10*a*b^9)/(3*x^3) - b^10/(2*
x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(32)=64\).

Time = 0.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11

method	result	size
norman	\(\frac {-\frac {1}{2} b^{10} x^{10}-\frac {10}{3} a \,b^{9} x^{9}-\frac {45}{4} a^{2} b^{8} x^{8}-24 a^{3} b^{7} x^{7}-35 a^{4} b^{6} x^{6}-36 a^{5} b^{5} x^{5}-\frac {105}{4} a^{6} b^{4} x^{4}-\frac {40}{3} a^{7} b^{3} x^{3}-\frac {9}{2} a^{8} b^{2} x^{2}-\frac {10}{11} a^{9} b x -\frac {1}{12} a^{10}}{x^{12}}\)	\(112\)
risch	\(\frac {-\frac {1}{2} b^{10} x^{10}-\frac {10}{3} a \,b^{9} x^{9}-\frac {45}{4} a^{2} b^{8} x^{8}-24 a^{3} b^{7} x^{7}-35 a^{4} b^{6} x^{6}-36 a^{5} b^{5} x^{5}-\frac {105}{4} a^{6} b^{4} x^{4}-\frac {40}{3} a^{7} b^{3} x^{3}-\frac {9}{2} a^{8} b^{2} x^{2}-\frac {10}{11} a^{9} b x -\frac {1}{12} a^{10}}{x^{12}}\)	\(112\)
gosper	\(-\frac {66 b^{10} x^{10}+440 a \,b^{9} x^{9}+1485 a^{2} b^{8} x^{8}+3168 a^{3} b^{7} x^{7}+4620 a^{4} b^{6} x^{6}+4752 a^{5} b^{5} x^{5}+3465 a^{6} b^{4} x^{4}+1760 a^{7} b^{3} x^{3}+594 a^{8} b^{2} x^{2}+120 a^{9} b x +11 a^{10}}{132 x^{12}}\)	\(113\)
default	\(-\frac {9 a^{8} b^{2}}{2 x^{10}}-\frac {35 a^{4} b^{6}}{x^{6}}-\frac {36 a^{5} b^{5}}{x^{7}}-\frac {40 a^{7} b^{3}}{3 x^{9}}-\frac {a^{10}}{12 x^{12}}-\frac {10 a \,b^{9}}{3 x^{3}}-\frac {10 a^{9} b}{11 x^{11}}-\frac {b^{10}}{2 x^{2}}-\frac {45 a^{2} b^{8}}{4 x^{4}}-\frac {24 a^{3} b^{7}}{x^{5}}-\frac {105 a^{6} b^{4}}{4 x^{8}}\)	\(113\)
parallelrisch	\(\frac {-66 b^{10} x^{10}-440 a \,b^{9} x^{9}-1485 a^{2} b^{8} x^{8}-3168 a^{3} b^{7} x^{7}-4620 a^{4} b^{6} x^{6}-4752 a^{5} b^{5} x^{5}-3465 a^{6} b^{4} x^{4}-1760 a^{7} b^{3} x^{3}-594 a^{8} b^{2} x^{2}-120 a^{9} b x -11 a^{10}}{132 x^{12}}\)	\(113\)

[In]

int((b*x+a)^10/x^13,x,method=_RETURNVERBOSE)

[Out]

1/x^12*(-1/2*b^10*x^10-10/3*a*b^9*x^9-45/4*a^2*b^8*x^8-24*a^3*b^7*x^7-35*a^4*b^6*x^6-36*a^5*b^5*x^5-105/4*a^6*
b^4*x^4-40/3*a^7*b^3*x^3-9/2*a^8*b^2*x^2-10/11*a^9*b*x-1/12*a^10)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (32) = 64\).

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

[In]

integrate((b*x+a)^10/x^13,x, algorithm="fricas")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (29) = 58\).

Time = 0.53 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.36 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=\frac {- 11 a^{10} - 120 a^{9} b x - 594 a^{8} b^{2} x^{2} - 1760 a^{7} b^{3} x^{3} - 3465 a^{6} b^{4} x^{4} - 4752 a^{5} b^{5} x^{5} - 4620 a^{4} b^{6} x^{6} - 3168 a^{3} b^{7} x^{7} - 1485 a^{2} b^{8} x^{8} - 440 a b^{9} x^{9} - 66 b^{10} x^{10}}{132 x^{12}} \]

[In]

integrate((b*x+a)**10/x**13,x)

[Out]

(-11*a**10 - 120*a**9*b*x - 594*a**8*b**2*x**2 - 1760*a**7*b**3*x**3 - 3465*a**6*b**4*x**4 - 4752*a**5*b**5*x*
*5 - 4620*a**4*b**6*x**6 - 3168*a**3*b**7*x**7 - 1485*a**2*b**8*x**8 - 440*a*b**9*x**9 - 66*b**10*x**10)/(132*
x**12)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (32) = 64\).

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

[In]

integrate((b*x+a)^10/x^13,x, algorithm="maxima")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (32) = 64\).

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

[In]

integrate((b*x+a)^10/x^13,x, algorithm="giac")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 4620*a^4*b^6*x^6 + 4752*a^5*b^5*x
^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b x)^{10}}{x^{13}} \, dx=-\frac {\left (11\,a-b\,x\right )\,{\left (a+b\,x\right )}^{11}}{132\,a^2\,x^{12}} \]

[In]

int((a + b*x)^10/x^13,x)

[Out]

-((11*a - b*x)*(a + b*x)^11)/(132*a^2*x^12)